Free Response of a Rotational Nonlinear Energy Sink Coupled to a Linear Oscillator: Fractality, Riddling, and Initial-Condition Sensitivity at Large Initial Displacements

We consider free response of a linearly sprung vibrating primary mass inertially coupled to a rotational nonlinear energy sink (NES). Coupling of rectilinear oscillation of the primary mass to viscously damped rotation of the NES mass about a vertical axis leads to targeted energy transfer (TET), in which vibrational energy of the primary mass is transferred to rotational energy of the NES mass, which is then dissipated. The nonlinear dynamics has been investigated numerically [1–5] and experimentally [1]. This approach to TET has been explored in simulations of structural vibration suppression [6] and flow-induced vibration suppression [7–10].

If B_∞ > 0, then θ_∞ = nπ, where n is an integer [1–5], and we have a “semi-trivial” solution [11] of the type reported in early work [1–3]. There are also motionless (“zero-energy”) asymptotic solutions with B_∞ = 0 (and θ_∞ typically not an integer multiple of π) [4]. Floquet analysis [4] shows that the semi-trivial solutions are orbitally stable if B_∞ lies below a lowest critical value A₀(α) or in a “tongue” A_2k−1(α) < B_∞ < A_2k(α), k = 1, 2, 3, …, whose extent decreases rapidly as k increases or as α decreases. For all other values of B_∞, semi-trivial solutions are unstable.

For two numerical combinations of α and σ (including that focused on by Gendelman et al. [1] and Sigalov et al. [2,3]) with no rectilinear damping, we reported [4,5] a number of results not found in the previous work [1–3]. For 0 < η_i ≤ A₀(α) and both (α, σ, λ) = (1, 0.01, 0) and (0.1, 0.1, 0), there is a wide and connected part of the motionless initial-condition (MIC) space (wherein ν_i = Ω_i = 0) in which damped rotation of the NES mass dissipates all initial energy of the primary mass, giving a motionless final state [4]. For larger initial rectilinear displacements, in the range A₀(α) ≤ η_i ≤ A₁(α), results for (α, σ, λ) = (1, 0.01, 0) and (0.1, 0.1, 0) are quite different [5]. For (α, σ, λ) = (1, 0.01, 0), a wide and connected set of MICs lead to orbitally stable semi-trivial solutions with η_i-dependent asymptotic amplitudes just below A₀(1), or for larger η_i in the range A₀(1) ≤ η_i ≤ A₁(1), to asymptotic solutions with η_i-invariant amplitude B_∞,s,1 = 1.1712734… (with θ_∞ = nπ for a range of integer n). We refer to the latter as “ground-state time-harmonic special solutions,” with a single value of B_∞ and different θ_∞. For A₀(1) ≤ η_i ≤ A₁(1), no MICs were found for which all initial energy is dissipated. On the other hand, for (α, σ, λ) = (0.1, 0.1, 0) and A₀(0.1) ≤ η_i ≤ A₁(0.1), virtually all MICs lead to B_∞ < A₀(0.1), a large fraction of MICs lead to complete dissipation, and no MICs leading to special solutions are found. Fractality and riddling of asymptotic solutions for both combinations in MIC space were characterized [5].

Here, we consider initial rectilinear displacements exceeding A₁(α), the upper limit of oscillation amplitude of the first range of orbital instability, and show that Eqs. 1(a) and 1(b) support rich dynamical behavior not found earlier [1–5]. For (α, σ, λ) = (1, 0.01, 0), a wide range of MICs lead to a second set of time-harmonic “special” solutions, each with a different value of θ_∞ = nπ, but all having the same amplitude within T₁(1), giving rise to fractal and riddled basins of attraction. (In the second tongue of orbital stability, T₂(1), there is also a set of time-harmonic special solutions with a single amplitude, but the range of MICs leading to them is very limited.) For (α, σ, λ) = (0.1, 0.1, 0), the rotational damping parameter is sufficiently small that the two lowest-lying tongues (see Fig. 1 of Ref. [4]) are very narrow. As a result, the only MICs approaching a semi-trivial solution in T₁(0.1) lie in small, connected sets very close to θ_i = 0 or π, or are apparently isolated. No semi-trivial solutions are found with asymptotic amplitudes in T₂(0.1). For all other initial angular positions, asymptotic behavior is qualitatively similar to that observed for smaller initial rectilinear displacements [5].

The remainder of the paper is organized as follows. For (α, σ, λ) = (1, 0.01, 0), we briefly discuss representative trajectories in Sec. 2, followed in Sec. 3 by a discussion of two additional sets of time-harmonic special solutions. In Sec. 4, we divide the MIC space into regions according to asymptotic behavior of trajectories originating within those regions, and in Sec. 5 characterize fine-scale structure in the MIC space. In Sec. 6, we present results for (α, σ, λ) = (0.1, 0.1, 0), and in Sec. 7 consider direct damping of rectilinear oscillation of the primary mass for the two combinations of α and σ. A discussion and conclusions follow in Secs. 8 and 9.

For (α, σ, λ) = (1, 0.01, 0), we will show in Sec. 5 that initial-condition sensitivity in broad portions of the MIC space leads to fractal or riddled basins for certain asymptotic solutions. Because initial-condition sensitivity gives rise to sensitivity with respect to numerical integration details [5,12–16], we do not attempt to track the “exact” solution over an extended time interval using a higher-order accurate integrator, a small time-step size, and multiple precision arithmetic. Rather, we recognize that initial conditions cannot be precisely specified in experiments, and instead focus on understanding sensitivity and associating it with the underlying dynamics. All trajectories here are integrated using matlab subroutine ode45 with, as before [4,5], the absolute and relative error tolerances in this explicit (4,5) Runge–Kutta integrator taken to be equal, with their common value denoted by δ, equal to 10⁻⁸ unless otherwise stated. We use 64-bit arithmetic. The asymptotic B_∞ (sometimes zero) is extracted by closely following the method previously described [4] using the Levenberg–Marquardt algorithm [17] with 50 iterations. Wavelet transforms were performed as previously described [4,5].

For (α, σ, λ) = (1, 0.01, 0) with A₁(1) ≤ η_i ≤ 15, a Floquet analysis [4] identified the three lowest ranges of oscillation amplitude in which semi-trivial solutions are orbitally stable. These are 0 ≤ B_∞ < A₀(1), A₁(1) < B_∞ < A₂(1), and A₃(1) < B_∞ < A₄(1), with A₀(1) = 1.18341…, A₁(1) = 4.20291…, A₂(1) = 4.40642…, A₃(1) = 11.18428…, and A₄(1) = 11.20204…. For η_i considered here, there exist MICs whose trajectories asymptotically approach solutions in each orbitally stable range. In addition to the countable set of ground-state time-harmonic special solutions with asymptotic amplitude B_∞,s,1 = 1.1712734… (with members of the set having θ_∞ = nπ for different integer n) [5], we find two additional sets of time-harmonic special solutions lying within T₁(1) and T₂(1), with asymptotic rectilinear amplitudes B_∞,s,2 = 4.4056717… and B_∞,s,3 = 11.2020156…, respectively, discussed in Sec. 3. Here, we discuss representative trajectories for (α, σ, λ) = (1, 0.01, 0), some of which show sensitivity of asymptotic amplitude to initial conditions, not found for 0 < η_i < A₁(1) [4,5].

We first consider MICs for which the baseline (BL) trajectory computed with the default δ is graphically indistinguishable over 0 ≤ τ ≤ 2 × 10⁴ from those computed when δ is reduced from 10⁻⁸ to 10⁻¹², or η_i is increased by 10⁻⁸, referred to as the reduced integration tolerance (RIT) and perturbed initial-condition (PIC) trajectories, respectively.

Case 1. Capture inT₁(1) withB_∞ < B_∞,s,2: For (η_i, θ_i) = (4.43, 0.15π), Figs. 1(a)–1(d) show a short irregular rotation between τ = 0 and 20, with significant variation of θ and B. For τ > 20, the angular velocity undergoes a decaying oscillation, and the trajectory asymptotes to harmonic rectilinear oscillation with (B_∞, θ_∞) = (4.386…, − 4π).

Case 2. Capture inT₁(1) withB_∞ = B_∞,s,2: Figures 1(e)–1(h) show the trajectory for (η_i, θ_i) = (4.5, 0.725π). The rotation rate undergoes an irregular transient for 0 ≤ τ ≤ 25 (shown in detail in Figs. S1(a)–S1(f) available in the Supplemental Materials), followed by abrupt decay, by slow growth of its envelope to a local maximum near τ = 140, and by slow decay to zero (Fig. S1(g)), leading to (B_∞, θ_∞) = (4.4056717…, − 5π) with the amplitude of a time-harmonic special solution.

Case 3. Capture inT₂(1) withB_∞ < B_∞,s,3: For (η_i, θ_i) = (11.2, 5 × 10⁻⁴π), with θ_i ≈ 0, Figs. 1(i)–1(l) show that energy dissipation is limited, with B_∞ lying within T₂(1).

Case 4. Capture inT₂(1) withB_∞ = B_∞,s,3: For (η_i, θ_i) = (11.21, 10⁻⁴π), with η_i > A₄(1) = 11.20204, Figs. 1(m)–1(p) show that after initial energy transfer to the rotational mode, during which dθ/dτ increases to its maximum, 0.237, at τ = 68.53, rotational energy decays asymptotically to zero, but much more slowly than in Case 3. The difference in energy dissipation rate between this and Case 3 can be understood as follows. Figures S2(a)–S2(f) available in the Supplemental Materials show, in addition to trajectories for Cases 3 and 4, one for an “intermediate” MIC. Figure S2(c) available in the Supplemental Materials shows that for (η_i, θ_i) = (11.2, 10⁻⁴π), dθ/dτ decays quite rapidly, and with much greater similarity to Case 3 than to Case 4, showing that for those MICs giving rise to B_∞ within T₂(1), decay of dθ/dτ is closely related to whether η_i lies within the orbitally stable range of rectilinear displacements, rather than to the closeness of θ_i to an integer multiple of π. (We used δ = 10⁻¹² for Case 4, because over the very long time scale demanded by the weak decay, numerical error for δ = 10⁻⁸ leads to long-time linear decrease of B(τ) rather than asymptotic decay to B_∞,s,3).

Here we discuss two MICs for which transient chaos leads to initial-condition sensitivity of the asymptotic solutions. For these nominal MICs, we apply a wavelet transform to dθ/dτ and characterize the transient chaos by computing the correlation dimension, closely following procedures used earlier [4,5].

Case 5. Capture inT₁(1) withB_∞ < B_∞,s,2: For (η_i, θ_i) = (8.1, 0.7π), Figs. 2(a)–2(c) show time series of the dynamical variables, while Fig. 2(d) shows a wavelet transform of dθ/dτ. In Figs. 2(b) and 2(d), we see that the NES mass undergoes fairly regular rotation over 0 ≤ τ ≤ 145, with significant energy at two frequencies (second and sixth harmonics f = 0.16 and 0.48, respectively, having a fundamental frequency f = 0.08; Fig. 2(d)). For 145 ≤ τ ≤ 655, the trajectory is more irregular (Fig. 2(b)), and the wavelet transform is broadband (Fig. 2(d)), suggesting chaotic behavior. Beyond τ = 655, the time series of dθ/dτ becomes more regular, with frequencies f = 0.08 and f = 0.40 (fifth harmonic) being the most prominent over the range 655 ≤ τ ≤ 824. Rotation is predominantly counterclockwise for 838 ≤ τ ≤ 879 and predominantly clockwise for 892 ≤ τ ≤ 918. During those intervals, the most energetic frequency is about 0.33 (fourth harmonic). Beyond τ = 920, dθ/dτ rapidly decays to zero (Fig. 2(b)), with (B_∞, θ_∞) = (4.384…, − 2π).

For the same nominal MIC, Figs. S3(a)–S3(c) available in the Supplemental Materials show the sensitivity of the trajectories and asymptotic solutions if we reduce the integration tolerance δ from 10⁻⁸ to 10⁻¹² (RIT case) or perturb η_i to 8.1 + 10⁻⁹ (PIC case). Time series of η(τ) for the RIT and PIC cases are graphically indistinguishable, with both leading to B_∞ = B_∞,s,1, whereas B_∞ for the BL trajectory lies inside T₁(1), just below B_∞,s,2, showing that the asymptotic amplitude depends sensitively on initial conditions and integration details. Figures S3(b) and S3(c) available in the Supplemental Materials show that time series for θ/π are distinct for the BL, RIT, and PIC cases, as are those for dθ/dτ. The angular velocity (Fig. S3(c) available in the Supplemental Materials) is essentially identical for all three cases until about τ = 190 (including the beginning of transient chaos), beyond which trajectories begin to diverge. Wavelet transformation (Figs. S4(b), S4(d), and S4(f) available in the Supplemental Materials) shows that the frequency content in each case is very similar through the end of the second interval of transient chaos (ending near τ = 920). Beyond that time, the angular velocity in the BL case rapidly decays to zero, while in the RIT and PIC cases, the oppositely signed envelopes of dθ/dτ (Figs. S3(c), S4(c), and S4(e) available in the Supplemental Materials) correspond to nearly identical wavelet transformations, with the most energetic frequencies being about 0.17, 0.33, and 0.49. For the RIT and PIC cases, after the irregularity during 1470 ≤ τ ≤ 1680, the frequency content during oscillatory decay (1705 ≤ τ ≤ 2500) is dominated by f = 0.08 and 0.24. For each trajectory during each chaotic transient, we calculated the correlation dimension d_corr [5], taken as the limiting value for embedding dimensions up to and including 56. For 200 ≤ τ ≤ 600, the BL, RIT, and PIC trajectories give d_corr = 2.28, 2.27, and 2.48, respectively. Later intervals of irregular behavior coincide for the RIT and PIC trajectories, giving d_corr = 2.98 and 3.02, respectively, over 835 ≤ τ ≤ 960, and 4.22 and 3.77, respectively, over 1500 ≤ τ ≤ 1680. That these values all exceed 2.0 and are in each interval in good agreement, even over the latter two intervals whose shortness (about 20 and 35 oscillations, respectively) complicates extraction of an accurate d_corr, indicates that the irregularity is due to transient chaos.

Case 6. Capture inT₁(1) withB_∞ = B_∞,s,2: For (η_i, θ_i) = (7.5, 0.4π), Figs. 3(a)–3(d) show that the motion undergoes an irregular transient for 0 ≤ τ ≤ 820, beyond which B and θ rapidly approach their asymptotes. The envelope of dθ/dτ then grows slowly until τ = 870, after which it slowly decays to zero. The asymptotic solution is (B_∞, θ_∞) = (4.4056717…, − 13π), with amplitude B_∞,s,2 lying just below A₂(1), the upper bound of the first tongue of stability. Analogous to the ground-state time-harmonic special solutions [5] giving rise to B_∞ = B_∞,s,1, the special amplitude B_∞,s,2 is the single limiting asymptotic amplitude of a large fraction of MICs (Sec. 3). Trajectories for the BL, RIT, and PIC cases integrated with (η_i, θ_i, δ) = (7.5, 0.4π, 10⁻⁸), (7.5, 0.4π, 10⁻¹²), and (7.5 + 10⁻⁸, 0.4π, 10⁻⁸), respectively, show that the rectilinear displacements of the BL and RIT trajectories are graphically indistinguishable (Fig. S5(a) available in the Supplemental Materials), with asymptotic amplitude B_∞,s,2, whereas the PIC case has asymptotic amplitude B_∞,s,1. On the other hand, θ and dθ/dτ for the three trajectories diverge near τ = 76 (Figs. S5(b) and S5(c) available in the Supplemental Materials), and η diverges near τ = 740 (Fig. S5(a) available in the Supplemental Materials), leading to three different combinations (B_∞, θ_∞), again indicating sensitivity with respect to integration tolerance and initial conditions.

As discussed in Ref. [5], there is a large, connected range of MICs lying below A₁(1) = 4.20291… for which the asymptotic rectilinear amplitude is B_∞,s,1 = 1.1712734…, just below A₀(1) = 1.18341…. Here, we discuss two other sets of time-harmonic special solutions with amplitudes B_∞,s,2 = 4.4056717… and B_∞,s,3 = 11.2020156…, lying just below A₂(1) = 4.40642… and A₄(1) = 11.20204…, respectively.

For A₁(1) ≤ η_i ≤ 4.6 and θ_i near 0 or π, B_∞ depends continuously on MICs in two connected parts of MIC space, qualitatively similar to Regions IIA and IIB [4,5]. There is a range of MICs (Fig. 4(a)) with η_i lying just above A₂(1) and with θ_i near zero, for which the limiting behavior is B_∞ → B_∞,s,2 as η_i increases, similar to the limit B_∞ → B_∞,s,1 as η_i increases just above A₀(1) [5]. To further verify that B_∞,s,2 is a single value rather than a narrow range, Table 1 provides statistics (mean, variance, and difference between maximum and minimum) for amplitudes of the T₁(1) time-harmonic special solution, B_∞,s,2, for 243 MICs shown in Table S1(a) available in the Supplemental Materials lying within 4.42 ≤ η_i ≤ 4.5 with θ_i = 10⁻³π, 5 × 10⁻⁴π, and 10⁻⁴π, each integrated with δ = 10^−m, for 8 ≤ m ≤ 12, and extrapolated to large m using Shanks' transformation [18], as in Ref. [5]. As the integration tolerance is tightened, the values of B_∞ lie in an increasingly narrow range, corresponding to a well-defined limiting value of B_∞,s,2 = 4.4056717… (Table 1), analogous to the value (1.1712734…) determined earlier for B_∞,s,1 [5].

Figures 1(i)–1(p) also show that trajectories emanating within or slightly above T₂(1) with θ_i close to an integer multiple of π lead to values of B_∞ lying in T₂(1), with almost no energy dissipation. As with η_i values just above either A₀(1) or A₂(1) giving rise to time-harmonic special solutions, having B_∞ = B_∞,s,1 or B_∞,s,2, respectively, here there exist connected MIC regions, above A₄(1), corresponding to a new set of time-harmonic special solutions, with B_∞ = B_∞,s,3 = 11.2020156… (Table 2, and Table S1(b) available in the Supplemental Materials), lying within T₂(1). Figure 4(b) shows that B_∞ approaches B_∞,s,3 for θ_i near zero. For 23 η_i values within 11.208 < η_i < 11.219 with θ_i = 10⁻⁴π, 5 × 10⁻⁵π, and 10⁻⁵π, statistics for 69 trajectories shown in Table 2 suggest that B_∞,s,3 is a single limiting value, and does not correspond to a narrow range of nonzero extent.

As mentioned in Ref. [5], the mechanism by which each of a large number of MICs leads to an orbitally (but not asymptotically) stable time-harmonic solution with a single rectilinear amplitude involves a countable set of attracting manifolds, each leading to an orbitally stable solution with precisely one amplitude, but different θ_∞. For the special solutions with B_∞ = B_∞,s,2 and B_∞,s,3, these attracting manifolds lie in parts of the phase space where the energy E (defined in Eq. (5) of Ref. [4]) exceeds [A₂(1)]²/2 and [A₄(1)]²/2, respectively. The details of these manifolds are beyond the scope of this work.

For both combinations of α and σ and smaller initial energies [4,5], there are three regions in the MIC space (Regions I, IIA, and IIB) in which B_∞ and θ_∞ vary continuously, with the behavior becoming fractal or riddled for larger initial energies (in Region III). For (α, σ, λ) = (1, 0.01, 0), results in Region III show that θ_∞ (but not B_∞) is sensitive to initial conditions, and there is a large connected portion of MIC space corresponding to the single limiting rectilinear amplitude B_∞,s,1 = 1.1712734…., with no zero-energy solutions being found.

For (α, σ, λ) = (1, 0.01, 0), we show in Secs. 4.1 and 4.2 that at larger initial energies, the asymptotic amplitude B_∞ also becomes sensitive to initial conditions, and identify two additional regions (Region IV; separated by η_i = A₁(1) from Region III; and Region V, separated by η_i = A₂(1) from Region IV; in both cases as shown in Fig. 5(a)) in the MIC space. In Region IV, most MICs lead to the ground-state time-harmonic special solution with asymptotic amplitude B_∞ = B_∞,s,1, while the remainder lead to solutions having A₁(1) < B_∞ < A₂(1). Region V includes MICs leading to special solutions having two other amplitudes, namely, B_∞ = B_∞,s,2 and B_∞,s,3, with the latter approached only for a narrow range of MICs, unlike the former, which is reached by a wide range of MICs. Like solutions with asymptotic amplitude B_∞,s,1, these additional special solutions have amplitudes in ranges of orbital stability established by the linear stability analysis [4]. We also show (see details in Sec. 4.2) that additional ranges of orbitally stable semi-trivial solutions at still higher amplitudes are so narrow that very few MICs are led to them.

For 736 MICs over 1.64 ≤ η_i ≤ 15 and 0 < θ_i < π, Figs. 6 and 7 show values of B_∞ and θ_∞, respectively. For these and 4619 additional MICs in this range of η_i, distributions of B_∞ and θ_∞ (Tables S2(a) and S2(b) available in the Supplemental Materials, respectively) reveal no trajectory for which all initial energy is dissipated and show that B_∞ ≥ B_∞,s,1 = 1.1712734…. For 4.20291… = A₁(1) ≤ η_i ≤ A₂(1) = 4.40642…, Figs. 6 and 7 show that MICs for which B_∞ ≠ B_∞,s,1 are concentrated near θ_i = 0 and π and lead to solutions having A₁(1) ≤ B_∞ ≤ A₂(1), with θ_∞ being the integer multiple of π nearest to θ_i. In this range of η_i, the remaining MICs, for which B_∞ ≠ B_∞,s,1, are apparently scattered throughout A₁(1) ≤ η_i ≤ A₂(1) (e.g., (η_i, θ_i) = (4.34, 0.425π)), with −9π ≤ θ_∞ ≤ 3π (Tables S2(a) and S2(b) available in the Supplemental Materials).

Trajectories emanating from MICs close to θ_i = 0 or π undergo initially rapid growth in |dθ/dτ|. This is followed by an interval of relatively slow decay, before approaching a non − B_∞,s,1 solution, with θ_∞ being the integer multiple of π nearest to θ_i, as illustrated for (η_i, θ_i) = (4.4, 0.001π) in Figs. 8(a) and 8(b). For these MICs, the NES mass rotates about θ_i and then settles to θ_∞ = 0 or π. No transient chaos is observed in dθ/dτ, whose relatively small magnitude (Fig. 8(b)) limits energy dissipation. This differs markedly from the situation for 1.64 ≤ η_i ≤ A₁(1), where MICs near θ_i = 0 or π cannot “settle” into a nonrotating asymptotic state without rotationally dissipating enough energy (during transient chaos) to fall below B = A₀(1). On the other hand, trajectories emanating from apparently isolated MICs giving rise to non − B_∞,s,1 solutions undergo only a few irregular rotations before approaching their asymptotes, as shown in Figs. 8(c) and 8(d) for (η_i, θ_i) = (4.34, 0.425π). The short duration of rotational motion means that very little initial energy is dissipated, and the magnitude of θ_∞ is small. Compared with Fig. 8(b), Fig. 8(d) shows a much shorter duration of rotation, with greater variation in rotational rate. In both cases, the magnitude of θ_∞ is small, with limited dissipation of initial energy.

For A₂(1) ≤ η_i ≤ 15, Fig. 6 shows that the distribution of B_∞ is more complicated, with most MICs leading to (B_∞,s,1, nπ) or (B_∞,s,2, nπ), with the latter becoming dominant as η_i increases. There are also small ranges of (η_i, θ_i) for which the asymptotic solution is (B_∞, nπ) with A₁(1) < B_∞ < B_∞,s,2, B_∞,s,2 < B_∞ < A₂(1), or A₃(1) < B_∞ < A₄(1). The significant dependence of B_∞ on η_i shown in Fig. 6 leads us to identify regions of MIC space exhibiting distinct distributions of asymptotic amplitudes, corresponding to A₁(1) ≤ η_i ≤ A₂(1) (Region IV), and A₂(1) < η_i ≤ 15 (Region V), as shown in Figs. 9(a)–9(c). (As discussed below, the upper limit shown for Region V is not a critical value beyond which the trajectories exhibit qualitatively different behavior.)

Figure 10(a) shows how θ_∞ depends on η_i for 5355 MICs in Table S2(b) available in the Supplemental Materials, with Figs. 10(b) and 10(c) providing more detail for 4.2 ≤ η_i ≤ 4.6 and 11.17 ≤ η_i ≤ 11.25, respectively. Points are colored according to B_∞ values (Table S2(a) available in the Supplemental Materials), with the overwhelming majority being B_∞ = B_∞,s,1 (orange) or B_∞,s,2 (gray), and the remainder lying in the ranges shown in the figure caption. Variation of θ_∞ for 1.6 ≤ η_i ≤ 4.2 was discussed earlier [5]. For larger initial displacements lying within T₁(1) (Region IV; see Figs. 9(a)–9(c)), 888 of 1035 MICs lead to B_∞ = B_∞,s,1 with 146π ≤ θ_∞ ≤ 174π or −178π ≤ θ_∞ ≤ −145π, and 147 MICs lead to A₁(1) < B_∞ < A₂(1) with θ_∞ = nπ for much smaller values of |n| (Fig. 10(b)). For A₂(1) < η_i ≤ 4.6, 231 of 360 MICs lead to B_∞ = B_∞,s,1 and the others to A₁(1) < B_∞ < A₂(1). In the latter case, |θ_∞| is relatively small, with the range widening as η_i increases. For 4.6 < η_i ≤ 15, 1424 of 2160 trajectories lead to B_∞ = B_∞,s,2 with −45π ≤ θ_∞ ≤ 50π (Fig. 10(a)), and 674 to B_∞ = B_∞,s,1 with 94π ≤ θ_∞ ≤ 217π or −213π ≤ θ_∞ ≤ −111π. Fifty-one other MICs have A₁(1) < B_∞ < B_∞,s,2 or B_∞,s,2 < B_∞ < A₂(1), and 11 (all with η_i lying in or slightly above T₂(1), with θ_i close to 0 or π) have B_∞ within T₂(1), with θ_∞ = 0 or π. (Note that B_∞ = B_∞,s,3 for none of the MICs considered.)

For (α, σ, λ) = (1, 0.01, 0), Tables S3(a) and S3(b) available in the Supplemental Materials show, analogously to Tables S2(a) and S2(b) available in the Supplemental Materials, the dependence of B_∞ and θ_∞ on MICs for δ = 10⁻¹⁰. For η_i ≤ 4.45, the values of B_∞ are essentially identical for δ = 10⁻⁸ and 10⁻¹⁰, even in parts of Region IV where B_∞ values lie in T₁(1), sometimes at apparently isolated MICs. For all η_i ≤ 1.7, θ_∞ is the same for both values of δ, as discussed previously for η_i ≤ 1.64 [5]. In regions of MIC space where B_∞ or θ_∞ is sensitive to δ, general trends of asymptotic states are qualitatively similar for both δ values (e.g., |θ_∞| increases with increasing η_i in Regions III and IV, with the MIC dependence of B_∞ and θ_∞ becoming more complex in Region V). Consistency of values and trends of asymptotic states for two values of δ strongly suggests that irregularity in the results is a property of the dynamical system, rather than an artifact of numerical integration.

For (α, σ, λ) = (1, 0.01, 0), we found that (B_∞, θ_∞) is a continuous function of η_i and θ_i in some regions of the MIC space (e.g., in Region I, where B_∞ = 0 and θ_∞ depends on η_i and θ_i, and in Regions IIA and IIB, where θ_∞ = 0 or π and B_∞ depends on η_i and θ_i) [4]. In those cases, there is, at best, a curve (rather than a basin) in the MIC space from which trajectories lead to a particular asymptotic solution, the number of final states is not countable, and the concept of a basin of attraction is not meaningful.

As discussed in Sec. 3 and shown in Fig. 9(a), a large fraction of MICs lead to time-harmonic special solutions with a small number of asymptotic rectilinear amplitudes, namely, B_∞,s,1, B_∞,s,2, or B_∞,s,3, in each case with θ_∞ equal to an integer multiple of π, and with dynamical characteristics (e.g., energy, stability) depending only on the rectilinear amplitude. These solutions thus constitute a set for which the number of asymptotic solutions is countable, and for which the basins of attraction can be characterized.

In applications, B_∞ is likely to be of greater interest than θ_∞. Thus, if nearby MICs have different asymptotic solutions, we will want to know if all have the same B_∞, and differ only in the multiple of π in θ_∞. To address this, we introduced the concepts of (a) “true basins of attraction” [5] for the set of MICs giving rise to a solution (B_∞, nπ) for a particular combination of nπ and B_∞, and (b) “amplitude-only basins of attraction” for the union of MICs giving rise to one value of B_∞ regardless of θ_∞.

The amplitude-only basin for solutions with B_∞,s,3 = 11.2020156… is confined to small portions of MIC space slightly above the second tongue near (η_i, θ_i) = (B_∞,s,3, 0) and (B_∞,s,3, π), unlike at smaller η_i where a broad range of MICs lying above T₁(1) lead to B_∞ = B_∞,s,2. The range of MICs leading to solutions with B_∞ in the third tongue, T₃(1), 21.507875… = A₅(1) ≤ B_∞ ≤ A₆(1) = 21.509052… (whose limits we determined by a Floquet analysis identical to that performed earlier [4]), is even smaller. The few MICs leading to solutions with B_∞ ≠ B_∞,s,1 and B_∞ not lying within T₁(1) are confined to an exceedingly narrow range of η_i in or slightly above the third tongue and to values of θ_i exceedingly close to an integer multiple of π (Table S4 available in the Supplemental Materials). (No MICs lying within or above T₃(1) are found to lead to B_∞ within T₂(1).)

Defining the tongue width w(T_k(α)) = A_2k(α) − A_2k−1(α), we see that w(T₃(1)) = 1.18 × 10⁻³ is much less than w(T₂(1)) = 1.78 × 10⁻² and w(T₁(1)) = 0.204. Progressive narrowing of these tongues of orbital stability as amplitude increases coincides with narrowing of the two ranges of θ_i (near 0 and π) for which an MIC can lead to B_∞ lying within that tongue. This shows that, except for MICs in progressively minute regions near an integer multiple of π within or above progressively narrow tongues of asymptotic amplitude, no MIC leads to an asymptotic state with B_∞ > A₂(1). These results lead us to conclude that semi-trivial solutions within the second and third tongues of orbital stability identified by the Floquet analysis are unstable (by a nonlinear mechanism) when subjected to very small but noninfinitesimal angular disturbances.

For (α, σ, λ) = (1, 0.01, 0), these results show that, except in exceedingly narrow parts of the MIC space, MICs with even higher energy will have asymptotic rectilinear amplitudes in the same ranges as the MICs considered here.

As discussed in Sec. 4.1, (a) B_∞ ≠ B_∞,s,1 for several scattered MICs in Region IV and (b) the dependence of B_∞ on η_i and θ_i in Region V is complicated. For ensembles of trajectories, we show here that these scattered MICs are associated with absence of transient chaos in narrow parts of MIC space, where trajectories are invariant to MIC perturbation and integration tolerance. In contrast, complicated dependence of B_∞ on η_i and θ_i in Region V is associated with “riddled” behavior of amplitude-only basins of attraction, corresponding to initial-condition sensitivity mediated by transient chaos. While these behaviors seem to be qualitatively similar, the underlying mechanisms differ in Regions IV and V, as discussed in Secs. 5.1 and 5.2.

As shown in Fig. 6 and Table S2(a) available in the Supplemental Materials, apparently isolated MICs in Region IV lead to non − B_∞,s,1 values of B_∞ lying in T₁(1), surrounded by MICs leading to B_∞ = B_∞,s,1. More detailed examination shows that the non − B_∞,s,1 MICs are not isolated and instead exist within thin bands in the MIC space wherein all asymptotic solutions have non − B_∞,s,1 amplitudes lying in T₁(1). For the MIC rectangle (4.32 ≤ η_i ≤ 4.50, 0.35π ≤ θ_i ≤ 0.45π), Fig. 11(a) shows that (η_i, θ_i) = (4.34, 0.425π), which in Fig. 6 appears to be an isolated MIC leading to a non − B_∞,s,1 solution, instead lies on a “primary band” extending from η_i just below 4.34, up into Region V. Within this band of MICs, all values of B_∞ lie in T₁(1) and B_∞ appears to be a continuous function of η_i and θ_i, approaching B_∞,s,2 = 4.4056717… in both the left and right “branches” as η_i ≈ 4.48 is reached in Region V. All MICs lying within this primary band have the asymptotic orientation θ_∞ = 6π. As discussed in Sec. 4.1, the only MICs leading to asymptotic trajectories with B_∞ = B_∞,s,2 are either (a) close to θ_i = 0 or π with A₁(1) < η_i < A₂(1), for which B_∞ varies continuously between A₁(1) and A₂(1), and B_∞ = B_∞,s,2 only on a curve in this part of MIC space, or (b) above Region IV, and hence with η_i above T₁(1).

Figure 11(b) expands the view of part of the MIC space (4.478 ≤ η_i ≤ 4.490, 0.3584π ≤ θ_i ≤ 0.3614π) shown in Fig. 11(a). This includes the part of the left branch within which B_∞ generally increases with increasing η_i and decreasing θ_i, and then assumes a constant value B_∞,s,2, visible in gray in Figs. 11(b) and 11(c). This transition involves internal structure within the band, and considerable dependence of B_∞ on a direction “transverse” to the band. Recalling that MICs lying outside the band approach asymptotic solutions with B_∞ = B_∞,s,1, it is clear that as one traverses the band (e.g., by increasing η_i and θ_i), MICs lead to asymptotic solutions with B_∞ = B_∞,s,1 followed by A₁(1) < B_∞ < A₂(1), and then B_∞ = B_∞,s,1.

Figure 11(a) also shows secondary bands with finer scales, depicted by a series of points, lying on curves locally parallel to the primary band. Details of one secondary band are shown in Fig. S6 available in the Supplemental Materials. For MICs lying in this secondary band (shown as green points), θ_∞ = −7π. Near this secondary band but not within it, we find six apparently isolated points for which the asymptotic solution lies within T₁(1) but below B_∞,s,2, suggesting the possibility that there are additional unresolved bands with even finer structure. Other than these six points, trajectories emanating from all MICs lying near, but outside of, the secondary band have B_∞ = B_∞,s,1, with −174π ≤ θ_∞ ≤ −154π or 141π ≤ θ_∞ ≤ 163π, ranges similar to those found in Sec. 4.1 with larger increments of η_i and θ_i over a larger part of the MIC space.

Comparing results for δ = 10⁻⁸ (Tables S2(a) and S2(b) available in the Supplemental Materials) and δ = 10⁻¹⁰ (Tables S3(a) and S3(b) available in the Supplemental Materials), we see that asymptotic states corresponding to “isolated points” in Region IV are invariant to integration tolerance. To verify that the band structure described above is insensitive to integration tolerance, we computed B_∞ and θ_∞ for 201 MICs, for 4.372 ≤ η_i ≤ 4.392 and θ_i = 0.4π integrated with δ = 10⁻⁸ and 10⁻¹⁰ (Table S5 available in the Supplemental Materials). The locations of band boundaries are essentially identical, as are values of B_∞ and θ_∞ within the bands. Outside of the bands, where transient chaos leads to asymptotic states with B_∞ = B_∞,s,1, the values of θ_∞ vary slightly with integration tolerance.

In Region V, for MICs with η_i > 4.5 lying on or near the extension of the left branch of the primary band shown in Fig. 11(a), Figs. 13(a)–13(c) show B_∞ distributions in rectangular neighborhoods centered at (η_i, θ_i) = (4.5530, 0.341π), (4.5960, 0.331π), and (4.6582, 0.318π). As one moves upward and leftward in Region IV along the primary band (from Figs. 13(a) to 13(b) to 13(c)) toward Region V, the relatively wide, well-defined internal structure of the primary band becomes increasingly complex, suggesting transition from fractality to riddling.

Analogous to the asymptotic amplitudes B_∞ in T₁(1) for isolated MICs in Region IV, MICs in Region V with A₃(1) ≤ η_i ≤ A₄(1) and θ_i departing from 0 and π can lead to trajectories with A₃(1) ≤ B_∞ ≤ A₄(1) (see Table S2(a) available in the Supplemental Materials). A key difference for these Region V MICs, in T₂(1), is that sampling the MIC space with the same increment in θ_i and an increment in η_i comparable with that used in T₁(1) reveals no additional MICs (i.e., not close to θ_i = 0 or π) for which A₃(1) ≤ B_∞ ≤ A₄(1). If such MICs exist, in bands even thinner than their T₁(1) analogues, none were found.

To quantitatively understand the structure of basins leading to special-amplitude solutions with B_∞ = B_∞,s,1 or B_∞,s,2, we follow the procedure described earlier [5] to approximate scales of the true basins of attraction (denoted by $s(B∞,nπ)$⁠) and amplitude-only basins (denoted by $sB∞$⁠), in which (B_∞, nπ) and B_∞, respectively, are locally insensitive to initial-condition perturbations. Note, however, that the values of $s(B∞,nπ)$ and $sB∞$ do not directly establish fractality or riddling in the MIC space, and that small values can only suggest riddling. The distinction between fractality and riddling is more clearly established by an uncertainty exponent [19], computed following our earlier procedure [5] presented in Sec. 5.3.

In Region V, the scales $s(B∞,nπ)$ and $sB∞$ for which (B_∞, θ_∞) or just B_∞, respectively, are insensitive to initial-condition perturbations, are calculated by an approach similar to that used earlier [5]. For 4 ≤ K ≤ 11, we compute (B_∞, θ_∞) at a nominal MIC (η_i,nom, θ_i,nom) and four nearby MICs (η_i,nom, θ_i,nom ± 10^−K) and (η_i,nom ± 10^−K, θ_i,nom). We begin with K = 12, for which we compute (B_∞, θ_∞) at these five and 12 other MICs, (η_i,nom ± 10⁻¹², θ_i,nom + 10⁻¹²), (η_i,nom ± 10⁻¹², θ_i,nom − 10⁻¹²), (η_i,nom + 10⁻¹², θ_i,nom ± 5 × 10⁻¹³), (η_i,nom − 10⁻¹², θ_i,nom ± 5 × 10⁻¹³), (η_i,nom ± 5 × 10⁻¹³, θ_i,nom + 10⁻¹²), and (η_i,nom ± 5 × 10⁻¹³, θ_i,nom − 10⁻¹²). We then decrease K by unity, each time computing (B_∞, θ_∞) at four new points until not all solutions are identical. The scale $s(B∞,nπ)$ corresponds to the largest among nine values of ɛ_p = 10^−K for which (η_i,nom, θ_i,nom) and all 16 + 4(12 − K) perturbed MICs lead to the same (B_∞, θ_∞). Similarly, $sB∞$ is the largest ɛ_p for which B_∞ is identical at the same MICs.

Figure 14(a) shows distributions of B_∞ and $sB∞$ (the scale of a neighborhood in MIC space in which B_∞ is insensitive), with values of the latter generally decreasing with increasing η_i, suggesting that the amplitude-only basins of attraction become more fractal, consistent with Figs. 13(a)–13(c). In Fig. 14(a), all MICs for which $sB∞≥10−12$ and η_i ≥ 5.48 are seen to have B_∞ = B_∞,s,2. This suggests that for η_i ≥ 5.48, all MICs lying in connected portions of an amplitude-only basin lead to a T₁(1) time-harmonic special solution. For η_i ≥ 5.72, no point with $sB∞≥10−12$ is found, suggesting that this portion of the MIC space is riddled.

Figure 14(b) shows the distributions of B_∞ and $s(B∞,nπ)$ for 4.4 ≤ η_i ≤ 6.0. In this range, all MICs leading to a ground-state time-harmonic special solution have $s(B∞,nπ)<10−12$⁠, suggesting that the true basins of attraction for MICs leading to solutions with this special amplitude are riddled. This is consistent with the riddling associated with θ_∞ sensitivity identified in Region III [5], in which all trajectories have asymptotic rectilinear amplitude B_∞,s,1. As η_i increases, $s(B∞,nπ)$ generally decreases for points leading to B_∞,s,2, indicating that the true basins either remain fractal but become smaller, or transition from fractal to riddled. Again, no MIC gives $s(B∞,nπ)≥10−12$ for η_i ≥ 5.72, suggesting that this portion of the MIC space is riddled.

The general trends of $sB∞$ and $s(B∞,nπ)$ are the same when trajectories are integrated with δ = 10⁻¹⁰ (compare Figs. 14(a) and S7(a), and Figs. 14(b) and S7(b) available in the Supplemental Materials), indicating that the observed behavior is a property of the differential equation system and is not due to numerical artifacts.

In Region III, B_∞ assumes a single nonzero value (B_∞,s,1) over a range of MICs, while θ_∞ is sensitive to initial conditions [5]. This is also true in parts of Regions IV and V. Elsewhere in Region V, both B_∞ and θ_∞ can be sensitive to initial conditions. The scales of the true and amplitude-only basins of attraction generally decrease and become exceedingly small as η_i increases, suggesting that the behavior for large η_i is either fractal or riddled. However, as mentioned in Sec. 5.2, basin structure cannot be established directly by values of $s(B∞,nπ)$ (or $sB∞$⁠), but can be characterized by the related uncertainty exponent [19], which we compute for the true basins (denoted by a_true) and amplitude-only basins (⁠$aB∞$⁠), closely following our previous procedure [5]. In each square neighborhood of MICs centered about (η_i,cen, θ_i,cen), we determine a_true and $aB∞$ from 400 randomly selected nominal MICs, each with four neighboring perturbed MICs for each of several values of the perturbation magnitude ɛ_p. For each ɛ_p and each B_∞ (or combination of B_∞ and θ_∞), we compute $gB∞$ (or g_true) as the fraction of the points at which there is “amplitude sensitivity” (or “solution sensitivity”). The slope of a plot of the logarithm of $gB∞$ (or g_true) versus the logarithm of ɛ_p provides $aB∞$ (or a_true), whose value identifies fractal behavior (⁠$aB∞$ or a_true not close to zero) or riddling behavior (⁠$aB∞$ or a_true close to zero) [19].

Neighborhood 1. Fractal amplitude-only basins of attraction and riddled true basins of attraction. For a square neighborhood of edge length 2 × 10⁻⁵ centered at (η_i,cen, θ_i,cen) = (4.7, 0.5π), we compute (B_∞, nπ) for 400 randomly selected MICs. For 13 values of 10⁻¹² ≤ ɛ_p ≤ 10⁻⁶, the distribution of B_∞ for all 21,200 (=400 × (13 × 4 + 1)) MICs is shown in Fig. 15(a), where a series of parallel bands giving rise to identical B_∞ is evident. From the 400 nominal MICs, we excluded 15 for which B_∞ is neither B_∞,s,1 nor B_∞,s,2. We calculated $gB∞$ and g_true by using the 385 nominal MICs leading to either B_∞,s,1 (of which there are 180) or B_∞,s,2 (of which there are 205) to generate 20,405 (=385 × (13 × 4 + 1)) perturbed MICs. The results give $aB∞=0.248$⁠, showing that the amplitude-only basins for B_∞,s,1 and B_∞,s,2 are fractal (see Fig. 15(b)).

In this neighborhood of fractal amplitude-only basins, we characterized the true basins. Figure 15(c) shows the dependence of g_true on ɛ_p for 10⁻¹² ≤ ɛ_p ≤ 10⁻⁶. For 10⁻⁹ ≤ ɛ_p ≤ 10⁻⁶, log g_true increases linearly with increasing log ɛ_p, having slope a_true = 0.105. For 10⁻¹² ≤ ɛ_p ≤ 10^−10.5, log g_true is nearly constant (g_true ≈ 0.4675), indicating riddling for sufficiently small ɛ_p. This disjointedness of g_true indicates coexistence of fractality and riddling of the true basins in this neighborhood.

To establish this, we consider separately the true basins of attraction for B_∞,s,1 and those for B_∞,s,2. Among the 400 randomly selected MICs, examination of the 180 for which B_∞ = B_∞,s,1 shows that each is sensitive to initial conditions (i.e., g_true = 1 for ɛ_p ≤ 10⁻⁶). Thus, in the neighborhood of the central MIC (η_i, θ_i) = (4.7, 0.5π), all randomly selected MICs leading to B_∞,s,1 lie in riddled true basins, consistent with the result in Fig. 15(c) that the limiting value g_true ≈ 0.4675 is the fraction of MICs leading to a ground-state time-harmonic special solution (180/385 = 0.4675…). Figure 15(d) shows log g_true calculated using the 205 random MICs for which B_∞ = B_∞,s,2. Linear dependence of log g_true on log ɛ_p establishes the fractal nature of all true basins giving B_∞,s,2 and further verifies that fractal and riddled true basins are intertwined within the neighborhood, as reported previously in another context [20].

Neighborhood 2. Portions of MIC space with riddled amplitude-only and true basins of attraction. For a square neighborhood of edge length 2 × 10⁻⁹ centered at (η_i,cen, θ_i,cen) = (5.7, 0.85π), we extracted a_true = 0 and $aB∞=0.0048$ for 10⁻¹³ ≤ ɛ_p ≤ 10⁻¹⁰ (Figs. S8(a) and S8(b) available in the Supplemental Materials), showing that the true and amplitude-only basins of attraction are riddled [19] in this neighborhood. From the distribution of B_∞ (Fig. S8(c) available in the Supplemental Materials), it is apparent that there is no systemic variation of B_∞ in this MIC neighborhood.

For (α, σ, λ) = (0.1, 0.1, 0), Fig. 16(a) shows the B_∞ distribution for 2451 MICs (43 θ_i and 57 η_i) in Region III which, as shown in Fig. 5(b), begins near η_i = 0.4638 and extends to at least η_i = 15, well beyond A₄(0.1). The distribution is qualitatively similar to that for smaller η_i in Region III (0.4638… = A₀(0.1) ≤ η_i ≤ A₁(0.1) = 3.7617…) [5]. Tables S6(a) and S6(b) available in the Supplemental Materials show B_∞ and θ_∞, respectively, for 3804 additional MICs (A₁(0.1) ≤ η_i ≤ 15 and 0 < θ_i < π). Unlike the (α, σ, λ) = (1, 0.01, 0) case, there is no large set of MICs leading to a single nonzero asymptotic amplitude. In Fig. 16(a), only three MICs, (η_i, θ_i) = (3.8, 0.99π), (3.8, 0.999π), and (5.4, 0.475π), identified by pink dots, lead to B_∞ values lying within T₁(0.1). The other 2448 MICs in Fig. 16(a) have 0 ≤ B_∞ < 0.161 (including 849 with B_∞ = 0), and thus dissipate at least 99.82% of the initial energy. These results clearly show that for (α, σ, λ) = (0.1, 0.1, 0), larger initial energies will have no effect on the asymptotic rectilinear amplitude, except in exceedingly narrow parts of the MIC space.

The observed sensitivity in asymptotic amplitude is related to the transient chaos in trajectories, resulting in a riddled zero-energy amplitude-only basin. (For (α, σ, λ) = (0.1, 0.1, 0), the true and amplitude-only basins of attraction are not well defined for B_∞ ≠ 0, as B_∞ can assume any one of an uncountable number of values and no nonzero B_∞ is the asymptotic amplitude for a large number of MICs.)

For each of 57 uniformly incremented η_i in the range 3.8 ≤ η_i ≤ 15, Fig. 16(b) gives P₀, the fraction of the 43 θ_i values shown in Fig. 16(a) leading to a zero-energy solution. This fraction does not systematically depend on η_i, continuing the behavior found for η_i ranging from about 1.7 to A₁(0.1) = 3.7617… (see Fig. 14(c) of Ref. [5]). In the neighborhood (6.6 − 10⁻⁵ ≤ η_i ≤ 6.6 + 10⁻⁵, 0.5π − 10⁻⁵ ≤ θ_i ≤ 0.5π + 10⁻⁵), Fig. 16(c) shows the B_∞ distribution for 10,201 MICs (101 uniformly incremented η_i and θ_i). In this neighborhood, there exist several apparently isolated points (pink) leading to B_∞ values lying within T₁(0.1), showing that even though the initial coupling strength is strong (with θ_i departing strongly from an integer multiple of π), there are MICs for which only a small fraction of the initial energy is dissipated. In addition to the MICs giving rise to nonzero B_∞, the complicated distribution of MICs leading to zero-energy solutions (black) in Fig. 16(c) is reminiscent of the riddled behavior for (α, σ, λ) = (1, 0.01, 0) (Sec. 5.3), due to sensitivity of B_∞ attributable to passage through transient chaos. (It is possible that these apparently isolated MICs lie on a fractal band, as shown in Figs. 16(a) and 16(b) of Ref. [5], and in Sec. 5.1 for (α, σ, λ) = (1, 0.01, 0).) To characterize the zero-energy amplitude-only basins, we examine an additional 400 randomly selected nominal MICs leading to zero-energy solutions (none from among the 10,201 points considered above). Calculation of $aB∞=0$ closely follows our previous procedure [5], with each random MIC that leads to a zero-energy solution being perturbed in each of four directions, (η_i ± ɛ_p, θ_i) and (η_i, θ_i ± ɛ_p), for nine perturbation magnitudes 10⁻⁸ ≤ ɛ_p ≤ 10⁻⁶. The uncertainty exponent of that basin is $aB∞=0=1.025×10−3$⁠, indicating riddling [19].

Near θ_i = 0 and π, there exist η_i (3.7617… = A₁(0.1) ≤ η_i ≤ 3.812) for which B_∞ depends continuously on the MIC (Table S6(a) available in the Supplemental Materials) and the asymptotic state is rapidly approached with no chaotic transient, with θ_∞ being the integer multiple of π nearest to θ_i. This continuous variation of B_∞ is similar to that found in Regions IIA and IIB for (α, σ, λ) = (0.1, 0.1, 0) and (1, 0.01, 0) [4], but unlike the (α, σ, λ) = (1, 0.01, 0) approach of B_∞ to a single value as η_i increases, for (α, σ, λ) = (0.1, 0.1, 0) there are no special oscillation amplitudes to which wide ranges of MICs are led.

Figures 17(a) and 17(b) show the η_i-dependence of θ_∞ for a wide range of MICs (162 values of η_i for 0.48 ≤ η_i ≤ 24 and 43 values of θ_i for 10⁻³π ≤ θ_i ≤ 0.999π, with η_i increments of 0.04 for 0.48 ≤ η_i ≤ 4, 0.2 for 4 ≤ η_i ≤ 15 and 0.5 for 15 ≤ η_i ≤ 24). Each point is colored according to its B_∞ value. For 2.2 ≤ η_i ≤ 12.2, the θ_∞ values for semi-trivial and zero-energy solutions lie in two nearly symmetric bands of roughly equal extent, with 0 ≤ |θ_∞| ≤ 83π. As η_i increases beyond 13.4, the ranges of θ_∞ increasingly and symmetrically depart from θ_∞ = 0. Beyond η_i = 16.5, the two ranges no longer change, with 64π ≤ |θ_∞| ≤ 411π. Comparing Figs. 17(a) and 17(b), we see that departure of the θ_∞ bands from θ_∞ = 0 at small η_i and then again near η_i = 13.4, in both cases followed by an η_i range in which band width is independent of η_i, is similar. Whether a third departure occurs for larger η_i is unclear.

With direct rectilinear damping (λ > 0), all asymptotic states have zero energy (i.e., B_∞ = 0). For η_i ≥ A₁(α), we focus on characterizing energy dissipation using the “settling time” required to dissipate 99% of the initial energy, denoted by τ_0.01 [4,5]. This is a useful measure because dissipation can occur (simultaneously or during different stages of the decay process) by means of both rotational and direct rectilinear damping, in either case with significant nonlinear effects, thus limiting use of a single-exponential decay model. Here, we discuss the dependence of τ_0.01 on MICs and the relationship to the λ = 0 results discussed above. We also show results for the fraction of the energy dissipation attributable to rotational damping, β.

For (α, σ, λ) = (0.1, 0.1, λ) with 3.5 ≤ η_i ≤ 15 and θ_i = 0.5π (for which initial “leverage” is greatest), Fig. 18(a) shows the dependence of τ_0.01 on η_i, for λ = 10⁻⁵, 10⁻⁴, 10⁻³, 3 × 10⁻³, and 10⁻², the same values considered previously [5]. For each λ, τ_0.01 generally increases as η_i increases, over the entire range. For 10⁻⁵ < λ ≤ 10⁻², the mean value of dτ_0.01/dη_i (averaged over the η_i range) decreases as λ increases. Comparison with Fig. 18(b) of Ref. [5] shows that the relatively weak and generally monotonic dependence of τ_0.01 on η_i and λ in this range of η_i continues the behavior found in Region III for 1.21 ≤ η_i ≤ 3.76, in contrast to more complex behavior found for smaller η_i in that region. “Dips” in τ_0.01 as a function of η_i are associated with initial-condition sensitivity of trajectories, as discussed for smaller η_i [5]. This sensitivity means that there are values of η_i for which increased direct rectilinear damping counterintuitively increases τ_0.01, e.g., τ_0.01(η_i = 7.25, λ = 10⁻⁴) > τ_0.01(η_i = 7.25, λ = 10⁻⁵). Similar sensitivity can be expected with respect to θ_i.

For (α, σ, λ) = (1, 0.01, λ) with θ_i = 0.5π, the dependence of τ_0.01 on η_i and λ is quite different. Figure 18(b) shows that τ_0.01 depends weakly on η_i for λ = 10⁻³, 3 × 10⁻³, and 10⁻², making clear that the stronger η_i variation for all five λ values at smaller η_i (see Fig. 17(b) of Ref. [5]) does not extend to Regions IV and V. (The results shown in Fig. 18(b) are displayed as a plot of λτ_0.01 versus η_i in Fig. S9(b) available in the Supplemental Materials, for more direct comparison to Fig. 17(b) of Ref. [5].) The dips in τ_0.01 (Fig. 18(b)), over 90% for λ = 10⁻⁴ and almost 99% for λ = 10⁻⁵, are attributable to initial-condition sensitivity that carries over from the λ = 0 case. This is illustrated for λ = 10⁻⁴ in Figs. 19(a)–19(d), where apparent randomness of these dips persists down to increments between consecutive values of η_i far smaller than the value (0.25) used in Fig. 18(b). This is directly traceable to initial-condition sensitivity exhibited by trajectories for nearby MICs (Figs. S10(a)–S10(d) available in the Supplemental Materials). Implications of this sensitivity for design are discussed in Sec. 8. The dual-value nature of the fine-scale sensitivity apparent in Figs. 19(a)–19(d), with all values of τ_0.01 being near 4.1 × 10⁴ or 1.5 × 10⁴, strongly suggests association with the initial-condition sensitivity of B_∞ in the λ = 0 case, which leads to two dominant values of B_∞, namely, B_∞,s,1 and B_∞,s,2.

For (α, σ, λ) = (0.1, 0.1, λ) with θ_i = 0.5π, the fraction of dissipation attributable to rotational damping, β, decreases nearly monotonically as η_i and λ increase (Fig. S11(a) available in the Supplemental Materials). Local dips in β as a function of η_i again result from initial-condition sensitivity. As for τ_0.01, there are combinations of η_i and λ for which β depends nonmonotonically on λ. For (α, σ, λ) = (1, 0.01, λ), the more complex dependence of β on η_i and λ (Fig. S11(b) available in the Supplemental Materials) reflects more complex dependence of B_∞ on η_i for (α, σ, λ) = (1, 0.01, 0) compared with (α, σ, λ) = (0.1, 0.1, 0). (For (α, σ, λ) = (0.1, 0.1, 0) and A₂(0.1) < η_i ≤ 15, recall from Sec. 6 that 2448 of 2451 MICs lead to trajectories dissipating at least 99.82% of the initial energy, whereas for (α, σ, 0) = (1, 0.01, 0) essentially all trajectories lead to asymptotic solutions with amplitudes lying in one of three ranges of orbital stability, with almost all in two of those ranges.)

To further explore initial-condition sensitivity of the settling time τ_0.01, and the relationship to initial-condition sensitivity for λ = 0, we examine small neighborhoods in MIC space, as we did for λ = 0 in Secs. 5.2 and 5.3.

As shown in Sec. 2 for λ = 0, the duration of transient chaos for trajectories with similar η_i is longer for those with asymptotic amplitude B_∞,s,1 than for those with B_∞ = B_∞,s,2. For λ > 0, the need to dissipate all initial energy leads to significant variation in settling time. To relate the λ = 0 MIC dependence of asymptotic amplitude to the λ > 0 MIC dependence of settling time, we consider (α, σ, λ) = (1, 0.01, 0) and extract B_∞ for 651 uniformly incremented MICs (at 31 η_i and 21 θ_i) in the rectangular neighborhood 4.43 − 5 × 10⁻³ ≤ η_i ≤ 4.43 + 10⁻², 0.475π − 5 × 10⁻³ ≤ θ_i ≤ 0.475π + 5 × 10⁻³). We then consider six nonzero λ ranging from 10⁻⁵ to 7 × 10⁻⁴, and for each λ compute the corresponding distributions of τ_0.01 (Figs. 20(a)–20(f)) and β (Figs. 21(a)–21(f)), as well as the maximum variation of λτ_0.01 in the neighborhood, denoted by Δ(λτ_0.01) = λ(τ_0.01,max − τ_0.01,min), and the corresponding Δβ = β_max − β_min.

Figures 20(a)–20(f) and 21(a)–21(f) show distributions of λτ_0.01 and β, respectively, for the same 651 MICs. Except for Fig. 20(f), each subfigure displays a band structure similar to that shown for B_∞ in Fig. 20(g) when λ = 0, where a primary band, with A₁(1) < B_∞ < A₂(1), is apparently accompanied by a series of unresolved secondary bands, indicating a fractal distribution of B_∞. For the same set of MICs, we compute the distributions of λτ_0.01 and β for λ = 10⁻⁵, whose fractal structures (Figs. 20(a) and 21(a)) are in both cases qualitatively similar to that for B_∞, with Δ(λτ_0.01) = 2.63 and Δβ = 0.908 over the range of MICs. As λ increases to 6 × 10⁻⁴, the MIC band narrows and shifts slightly upward for both λτ_0.01 and β (Figs. 20(b)–20(e) and 21(b)–21(e)), and the variations diminish (Table 3). For λ = 7 × 10⁻⁴, there is no obvious variation of λτ_0.01 (Fig. 20(f)), while the banding for β is still evident (Fig. 21(f)). This is consistent with the much more rapid decrease of Δ(λτ_0.01) between λ = 6 × 10⁻⁴ and 7 × 10⁻⁴ compared with the decrease in Δβ (Table 3), and indicates that as λ approaches 7 × 10⁻⁴, direct rectilinear damping is sufficient to suppress initial-condition sensitivity of λτ_0.01 and β caused by transient chaos in this neighborhood.

The relationship between MIC sensitivity of B_∞ for λ = 0 and MIC sensitivity of τ_0.01 and β for λ > 0 is also observed in the riddled MIC neighborhood (6 − 10⁻⁸ ≤ η_i ≤ 6 + 10⁻⁸, 0.5π − 10⁻⁸ ≤ θ_i ≤ 0.5π + 10⁻⁸). For 10⁻⁵ ≤ λ ≤ 10⁻³, Figs. 22(a)–22(f) and 23(a)–23(f) show the distributions of τ_0.01 and β, respectively. In this neighborhood, for λ ≥ 9 × 10⁻⁴, the initial-condition sensitivity of λτ_0.01 and β is greatly diminished (Table 3). For (α, σ, λ) = (1, 0.01, 0), Fig. 22(g) shows a riddled distribution of B_∞ within the neighborhood containing 441 MICs (21 uniformly incremented values each of η_i and θ_i). These results clearly show that initial-condition sensitivity of the λ = 0 response provides good guidance to the sensitivity of energy dissipation under weak direct rectilinear damping. As λ increases beyond a critical value, riddled behavior of τ_0.01 and β is suppressed, which explains why in Fig. 18(b), there is no significant variation in τ_0.01 for λ ≥ 10⁻³. As shown in Figs. 18(a) and 18(b), direct rectilinear damping reduces initial-condition sensitivity of τ_0.01 and β but does not eliminate it (see Fig. S12 available in the Supplemental Materials for λ = 10⁻³).

For (α, σ, 0) = (0.1, 0.1, 0), the initial-condition sensitivity of B_∞ to η_i is much weaker and so no strong sensitivity of τ_0.01 and β is found. The general dependence of τ_0.01 and β on η_i is qualitatively similar to that for A₀(0.1) ≤ η_i ≤ A₁(0.1) [5].

For one combination of damping coefficient and coupling coefficient (α = 1, σ = 0.01) and no direct rectilinear damping, a wide range of motionless initial conditions gives rise to three sets of asymptotically nonrotating “time-harmonic special solutions,” each with amplitude in one of the three lowest-lying ranges of orbitally stable solutions. Initial-condition sensitivity associated with transient chaos leads to fractality and riddling in the initial-condition space with respect to both asymptotic amplitude of the rectilinear oscillation and final angular orientation. For α = 1 and σ = 0.01, no motionless initial condition with initial rectilinear amplitude above the lowest value at which instability is possible leads to a zero-energy solution, with the primary and NES masses at rest.

For α = 1 and σ = 0.01, the results presented in Sec. 4 strongly suggest that the amplitude-only basin of attraction of asymptotic solutions with the special amplitude B_∞,s,3 = 11.2020156… in tongue T₂(1) does not extend beyond the small regions shown in Fig. 6(a). On the other hand, trajectories emanating from a broad class of initial conditions (including initial rectilinear displacements only slightly larger than the special amplitudes B_∞,s,1 = 1.1712734… or B_∞,s,2 = 4.4056717…) lead to solutions with B_∞,s,1 and B_∞,s,2. That is simply not the case for B_∞,s,3.

For the other combination (α = σ = 0.1), zero-energy solutions are found for a wide range of motionless initial conditions. The amplitude-only basin of attraction (i.e., the set of motionless initial conditions for which the asymptotic rectilinear amplitude is zero regardless of final angular orientation) is riddled. This behavior is qualitatively similar to that found at smaller initial rectilinear displacements [5].

For α = 1 and σ = 0.01, the results in Sec. 4.2 clearly establish that except in exceedingly small parts of the MIC space, asymptotic solutions with rectilinear amplitude greater than A₂(1) = 4.40642… will not be found regardless of how large the initial energy is. Similarly, results in Sec. 6 show that for α = σ = 0.1, asymptotic solutions with amplitude greater than A₂(0.1) = 3.7961… will be found only in very limited parts of the MIC space, regardless of initial energy. Temptation to generalize to other combinations of α and σ is tempered by the fact that the distributions of asymptotic amplitudes below A₂(α) are so different for the two combinations investigated.

Finally, initial-condition sensitivity for the case where there is no direct damping of rectilinear motion (λ = 0) provides good guidance to understand the response for λ > 0 cases where such damping is accounted for, including cases where fractality or riddling of the amplitude-only basins of attraction in the λ = 0 case correspond to fractality or riddling of the settling time for λ > 0.

Sensitivity to initial conditions can pose a significant challenge to use of a rotational NES in applications, as discussed previously [5]. One can of course attempt to avoid combinations of the parameters (α, σ, and λ) for which performance (e.g., the settling time τ_0.01) is significantly sensitive over the expected range of initial conditions. Alternatively, there are various probabilistic design approaches to dealing with the sensitivity [5], including developing bounds for best and worst cases, using the methods of uncertainty quantification [21], and using information about the probability distribution of initial energies to compute probability distributions for response measures such as settling time. For both (α, σ) combinations, our results point to probabilistic interpretation of the final state [5], and prediction of the probabilities that the asymptotic harmonic motion has an amplitude B_∞ equal to (a) zero; (b) one of the time-harmonic special solutions; or (c) a value in one of the first three ranges of orbital stability, but not equal to that of the time-harmonic special solution lying in that range. More generally, for any initial condition, one can specify the size of a neighborhood, and then compute the probabilities that the asymptotic rectilinear amplitude or angular orientation lie in a given range (or their joint probability). When trajectories depend sensitively on initial conditions, this approach can serve as the basis for probabilistic design.

From the standpoint of settling time, Fig. 18(a) clearly shows that the degree of initial-condition sensitivity of τ_0.01 is quite large for (α, σ) = (0.1, 0.1) at small values of the rectilinear damping parameter λ. For (α, σ) = (0.1, 0.1) and the two smallest values of λ considered (10⁻⁴ and 10⁻⁵), Fig. 18(a) shows that although τ_0.01 varies by approximately a factor of five over the indicated range of η_i, the variation is nearly monotonic (and approximately linear with η_i). Moreover, the local variation in τ_0.01 attributable to initial-condition sensitivity is quite small (less than 20%), especially in comparison with the variations for (α, σ) = (1, 0.01) of well over 90% for small λ shown in Fig. 18(b). These marked differences point to the importance of understanding the degree of sensitivity for any set of parameters used in design.

For (α, σ, λ) = (0.1, 0.1, 0) and no direct rectilinear damping, Fig. 16(a) and the accompanying discussion show that for 2451 initial conditions over a wide range, all of the initial energy was dissipated for 849 of them, and that for all but three of the others, at least 99.82% almost all (at least 99.82%) was dissipated. From an application standpoint, this is a much more favorable result than might be inferred for (α, σ, λ) = (0.1, 0.1, 0) results at lower initial energy reported in Ref. [1], where two of the six trajectories dissipated more than 98% of the initial energy, and the others dissipated approximately 42%, 24%, 15%, and 0%. (No trajectory shown in Ref. [1] has an initial condition for which all initial energy is dissipated [4]. Note also that the results reported in Ref. [2] for nonzero rotational damping at higher initial energy were computed at a single initial rectilinear displacement, and include no values of asymptotic rectilinear amplitude.) On the other hand, three of the 2451 initial conditions lead to asymptotic solutions having significant rectilinear amplitudes, illustrating the importance of thoroughly exploring the initial-condition space in order to identify “outliers.”

Finally, in applications, the initial conditions cannot always be expected to be motionless. For (α, σ, λ) = (0.1, 0.1, 0) with the same (η_i, θ_i) considered in Fig. 16(a) and v_i = Ω_i = 0.1, Fig. 24 shows the distribution of B_∞, with 820 of the 2451 initial conditions leading to zero-energy solutions, and the remainder having B_∞ < 0.1473. Thus, for each of the initial conditions considered, over 99.95% of the initial energy (see Eq. (5) of Ref. [4]) is dissipated. Other than the fact that none of the initial conditions gives rise to a solution with B_∞ lying in T₁(0.1) (as opposed to three such solutions for the 2451 MICs considered in Fig. 16(a)) the number of zero-energy solutions and the fraction of energy dissipation are qualitatively similar to the results for motionless initial conditions shown in Fig. 16(a) (849 zero-energy solutions, and the remainder dissipating at least 99.82% of the initial energy). This, along with similar results reported previously [4,5], supports the hypothesis that our sensitivity results are not limited to the motionless subspace of initial conditions.

For a rotational nonlinear energy sink inertially coupled to a linear oscillator, we have examined the part of the motionless projection of the initial-condition space (i.e., with the initial rectilinear and angular velocities zero) in which the initial rectilinear displacement (η_i) exceeds the second threshold amplitude above which purely rectilinear time-harmonic motion is unstable. Absent direct rectilinear damping (λ = 0), for one combination of the dimensionless coupling parameter (α = 1) and dimensionless rotational damping parameter (σ = 0.01), no motionless initial condition leads to a zero-energy solution (in which all initial energy is dissipated). For this range of η_i, almost all initial conditions lead to asymptotically nonrotating semi-trivial solutions for which the rectilinear amplitude is either the smallest “special” amplitude (B_∞,s,1) or lies in the first tongue in which nonrotating rectilinearly harmonic solutions are orbitally stable. The exceptions are in very small parts of the motionless initial-condition space where η_i lies in or just above the extremely narrow second or third tongues in which semi-trivial solutions are orbitally stable, and the initial angular position of the NES mass departs only minutely from an integer multiple of π. Most initial conditions leading to solutions with asymptotic amplitudes in the first tongue have the second-smallest special amplitude (B_∞,s,2). The basins of attraction of solutions with special asymptotic amplitudes B_∞,s,1 or B_∞,s,2 are fractal or riddled in the motionless initial-condition space. For initial rectilinear displacements beyond those considered here, our results strongly suggest that asymptotic solutions with other amplitudes (i.e., lying in the second or third tongues of orbital stability) are possible only for exceedingly small sets of the motionless initial conditions.

In stark contrast are results for the other combination of parameters (α = σ = 0.1), where the dimensionless rotational damping parameter is 90% smaller and the dimensionless coupling parameter is higher by tenfold. For this case, many initial conditions lead to zero-energy solutions, for which the amplitude-only basin of attraction (i.e., the set of motionless initial conditions for which the asymptotic amplitude is zero, irrespective of asymptotic angular orientation) is riddled. Almost all motionless initial conditions are attracted to either zero-energy solutions or to solutions for which at least 99.82% of the initial energy is dissipated. The exceptions are some with initial rectilinear displacements in or just above the first tongue of orbital stability with departure of the initial angular orientation from an integer multiple of π being exceedingly small, and some apparently isolated initial conditions for which the initial angular orientation is not close to an integer multiple of π. No special-amplitude solutions (i.e., solutions with a common nonzero amplitude, to which many initial conditions are led) are found.

For both combinations of the parameters, the results clearly establish that asymptotic solutions with rectilinear amplitude greater than the upper bound of the second tongue occur only in exceedingly limited parts of the motionless condition space, regardless of how large the initial energy is.

Finally, when one accounts for direct rectilinear damping (λ > 0), the asymptotic rectilinear amplitude is necessarily zero, and the time required to dissipate 99% of the initial energy (τ_0.01) can depend complexly and nonmonotonically on the initial conditions and rectilinear damping parameter, with the distribution of asymptotic rectilinear amplitudes for λ = 0 providing good guidance as to the distribution in the motionless initial-condition space of both τ_0.01 and the fraction of energy dissipation attributable to rotational damping.