Singular Continuous Spectra in Dissipative Dynamics

Strange nonchaotic attractors (SNAs) can be regarded as a special class of attractors between quasiperiodic attractors and chaotic attractors. The word strange means that the dependence of the dynamical variables to the phase is not given by the smooth relations but constitutes some fractal ones. The word nonchaotic means that the maximum Lyapunov exponent is nonpositive. However, there are few studies on SNAs in nonsmooth dynamical systems. In this work, we study a quasiperiodically forced piecewise linear oscillator. The complicated and interesting strange nonchaotic dynamics phenomena are revealed via numerical methods.

I. INTRODUCTION

Section:

SNAs have a geometric fractal structure, but the maximum Lyapunov exponent is nonpositive, which exhibits nonchaotic properties in a dynamic sense. Since SNAs were uncovered by Grebogi et al., ¹ 1. C. Grebogi, E. Ott, S. Pelikan, and J. A. Yorke, "Strange attractors that are not chaotic," Physica D 13, 261–268 (1984). https://doi.org/10.1016/0167-2789(84)90282-3 it was realized that strangeness is not equivalent to chaos. SNAs have been widely studied experimentally and numerically. Ding et al. ² 2. M. Ding, C. Grebogi, and E. Ott, "Evolution of attractors in quasiperiodically forced systems: From quasiperiodic to strange nonchaotic to chaotic," Phys. Rev. A 39, 2593–2598 (1989). https://doi.org/10.1103/PhysRevA.39.2593 confirmed the existence of SNAs from numerical and analytical methods and illustrated various dynamical behaviors in a typical quasiperiodically forced system. Pikovsky and Feudel ³ 3. A. S. Pikovsky and U. Feudel, "Characterizing strange nonchaotic attractors," Chaos 5, 253–260 (1995). https://doi.org/10.1063/1.166074 described strange properties of such attractors by calculating the phase sensitivity exponents. With the aid of a function equation for the invariant curve, Nishikawa and Kaneko ⁴ 4. T. Nishikawa and K. Kaneko, "Fractalization of a torus as a strange nonchaotic attractor," Phys. Rev. E 54, 6114–6124 (1996). https://doi.org/10.1103/PhysRevE.54.6114 studied the fractal structure and chaotic evolution of a class of SNAs in a quasiperiodically forced logistic map. Ditto et al. ⁵ 5. W. L. Ditto, M. L. Spano, H. T. Savage, S. N. Rauseo, J. Heagy, and E. Ott, "Experimental observation of a strange nonchaotic attractor," Phys. Rev. Lett. 65, 533 (1990). https://doi.org/10.1103/PhysRevLett.65.533 observed SNAs in a two-frequency quasiperiodically driven, buckled, magnetoelastic ribbon experiment. According to an experiment and numerical simulation, Thamilmaran et al. ⁶ 6. K. Thamilmaran, D. V. Senthilkumar, A. Venkatesan, and M. Lakshmanan, "Experimental realization of strange nonchaotic attractors in a quasiperiodically forced electronic circuit," Phys. Rev. E 74, 036205 (2006). https://doi.org/10.1103/PhysRevE.74.036205 identified chaotic attractors and SNAs in a circuit system and distinguished these two kinds of attractors by the Poincaré map, maximum Lyapunov exponents, and their variance. In addition, Linder et al. ⁷ 7. J. F. Lindner, V. Kohar, J. G. Learned, and W. L. Ditto, "Strange nonchaotic stars," Phys. Rev. Lett. 114, 054–101 (2015). https://doi.org/10.1103/PhysRevLett.114.054101 established the existence of strange nonchaotic stars through the Kepler space telescope, which further indicates the presence of strange nonchaotic phenomena in nature.

The mathematical research of SNAs mainly focused on skew product systems (see, e.g., Refs. 8–10 8. P. Glendinning, "Global attractors of pinched skew products," Dyn. Syst. 17, 287–294 (2002). https://doi.org/10.1080/14689360210160878 9. J. Stark, "Transitive sets for quasi-periodically forced monotone maps," Dyn. Syst. 18, 351–364 (2003). https://doi.org/10.1080/14689360310001610155 10. P. Glendinning, T. Jäeger, and G. Keller, "How chaotic are strange nonchaotic attractors," Nonlinearity 19, 2005–2022 (2006). https://doi.org/10.1088/0951-7715/19/9/001 ). Keller ¹¹ 11. G. Keller, "A note on strange nonchaotic attractors," Fund. Math. 151, 139–148 (1996). investigated a class of skew product maps that have monotonically increasing and strictly concave fiber maps. It was proved that the system has an attracting invariant curve, which is almost everywhere discontinuous and, thus, an SNA. Ding et al. ¹² 12. M. Ding, C. Grebogi, and E. Ott, "Dimensions of strange nonchaotic attractors," Phys. Lett. A 137, 167–172 (1989). https://doi.org/10.1016/0375-9601(89)90204-1 and Fuhrmann et al. ¹³ 13. G. Fuhrmann, M. Gröger, and T. Jäger, "Non-smooth saddle-node bifurcations II: Dimensions of strange attractors," Ergod. Theory Dyn. Syst. 30, 2989–3011 (2018). https://doi.org/10.1017/etds.2017.4 studied the fractal dimension of SNAs in the quasiperiodically forced monotone interval maps, which are created through nonsmooth saddle-node bifurcation, and determined that the Hausdorff dimension and box-counting dimension have different values. Other theoretical results of SNAs are about discrete Schrödinger equation with quasiperiodic potential in which the Harper map is a typical example. Haro and Puig ¹⁴ 14. Ã. Haro and J. Puig, "Strange nonchaotic attractors in Harper maps," Chaos 16, 033127 (2006). https://doi.org/10.1063/1.2259821 proved that the Harper map has SNAs in a set of parameter values of positive Lebesgue measure. These proofs involved spectrum analysis of the Schrödinger operator and the estimate of the Lyapunov exponent.

Although the existence of SNAs has been studied widely in quasiperiodically forced systems (cf. Refs. 15–18 15. F. J. Romeiras and E. Ott, "Strange nonchaotic attractors of the damped pendulum with quasiperiodic forcing," Phys. Rev. A 35, 4404–4413 (1987). https://doi.org/10.1103/PhysRevA.35.4404 16. A. Prasad, S. S. Negi, and R. Ramaswamy, "Strange nonchaotic attractors," Int. J. Bifurcation Chaos 11, 291–309 (2001). https://doi.org/10.1142/S0218127401002195 17. U. Feudel, S. Kuznetsov, and A. S. Pikovsky, Strange Nonchaotic Attractors: Dynamics between Order and Chaos in Quasiperiodically Forced Systems (World Scientific Publishing Co. Pte. Ltd., Singapore, 2006), Chap. 6, pp. 75–127. 18. G. Li, Y. Yue, J. Xie, and C. Grebogi, "Strange nonchaotic attractors in a nonsmooth dynamical system," Commun. Nonlinear Sci. Numer. Simul. 78, 104858 (2019). https://doi.org/10.1016/j.cnsns.2019.104858 ), there are still many interesting questions, for instance: how SNAs are generated and eventually evolve into chaotic attractors? Precisely, what exactly causes SNAs is somewhat vague since the bifurcation mechanisms of quasiperiodic driving systems have not been understood in detail. However, several routes to SNAs have been described in the literature, such as Heagy–Hammel route, ^19,20 19. J. F. Heagy and S. M. Hammel, "The birth of strange nonchaotic attractors," Physica D 70, 140–153 (1994). https://doi.org/10.1016/0167-2789(94)90061-2 20. Y. Zhang and G. Luo, "Torus-doubling bifurcations and strange nonchaotic attractors in a vibro-impact system," J. Sound Vibr. 332, 5462–5475 (2013). https://doi.org/10.1016/j.jsv.2013.04.046 fractal route, ²¹ 21. S. Datta, R. Ramaswamy, and A. Prasad, "Fractalization route to strange nonchaotic dynamics," Phys. Rev. E 70, 046203 (2004). https://doi.org/10.1103/PhysRevE.70.046203 intermittency route, ^22,23 22. A. Prasad, V. Mehra, and R. Ramaswamy, "Intermittency route to strange nonchaotic attractors," Phys. Rev. Lett. 79, 4127–4130 (1997). https://doi.org/10.1103/PhysRevLett.79.4127 23. S. Y. Kim, W. Lim, and E. Ott, "Mechanism for the intermittent route to strange nonchaotic attractors," Phys. Rev. E 67, 056203 (2013). https://doi.org/10.1103/PhysRevE.67.056203 Blowout bifurcation route, ²⁴ 24. T. Yalçınkaya and Y. C. Lai, "Blowout bifurcation route to strange nonchaotic attractors," Phys. Rev. Lett. 77, 5039–5042 (1996). https://doi.org/10.1103/PhysRevLett.77.5039 and so on. The literature offers overviews and further references for other routes (see, e.g., Refs. 25–29 25. D. V. Senthilkumar, V. Srinivasan, K. Thamilmaran, and M. Lakshmanan, "Birth of strange nonchaotic attractors through formation and merging of bubbles in a quasiperiodically forced Chua's oscillator," Phys. Rev. E 78, 066211 (2008). https://doi.org/10.1103/PhysRevE.78.066211 26. K. Suresh, A. Prasad, and K. Thamilmaran, "Bubbling route to strange nonchaotic attractor in a nonlinear series LCR circuit with a nonsinusoidal force," Phys. Lett. A 377, 612–621 (2013). https://doi.org/10.1016/j.physleta.2012.12.026 27. G. Li, Y. Yue, D. Li, J. Xie, and C. Grebogi, "The existence of strange nonchaotic attractors in the quasiperiodically forced Ricker family," Chaos 30, 053124 (2020). https://doi.org/10.1063/1.5144145 28. Y. Yue, P. Miao, and J. Xie, "Coexistence of strange nonchaotic attractors and a special mixed attractor caused by a new intermittency in a periodically driven vibro-impact system," Nonlinear Dyn. 87, 1–21 (2017). https://doi.org/10.1007/s11071-016-3109-2 29. T. Cheng, Y. Zhang, and Y. Shen, "Infinite number of parameter regions with fractal nonchaotic attractors in a piecewise map," Fractals 29, 2150087 (2021). https://doi.org/10.1142/S0218348X21500870 ).

The studies of SNAs are mainly limited to smooth systems. The mechanism of SNA generation is more complicated in nonsmooth systems since there exist many nontypical bifurcations, such as grazing bifurcation, sliding bifurcation, period-adding bifurcation, and so on. Li et al. ³⁰ 30. G. Li, Y. Yue, J. Xie, and C. Grebogi, "Multistability in a quasiperiodically forced piecewise smooth dynamical system," Commun. Nonlinear Sci. Numer. Simul. 84, 105165 (2020). https://doi.org/10.1016/j.cnsns.2019.105165 studied a piecewise smooth map quasiperiodically forced and found three routes (Heagy–Hammel route, fractal route, and type-I intermittency route) to SNAs in the parameter space. Shen and Zhang ³¹ 31. Y. Shen and Y. Zhang, "Mechanisms of strange nonchaotic attractors in a nonsmooth system with border-collision bifurcations," Nonlinear Dyn. 96, 1405–1428 (2019). https://doi.org/10.1007/s11071-019-04862-5 revealed several types of routes to SNAs in two critical tongue-type regions in a quasiperiodically forced piecewise logistic map. Zhang et al. ³² 32. Y. Zhang and Y. Shen, "A new route to strange nonchaotic attractors in an interval map," Int. J. Bifurcation Chaos 30, 2050063 (2020). https://doi.org/10.1142/S0218127420500637 studied a class of quasiperiodically forced interval maps and showed that smooth quasiperiodic torus attractors lose their smoothness by grazing bifurcation and eventually become SNAs. Li et al. ³³ 33. G. Li, Y. Yue, C. Grebogi, D. Li, and J. Xie, "Strange nonchaotic attractors and multistability in a two-degree-of-freedom quasiperiodically forced vibro-impact system," Fractals 29, 2150103 (2021). https://doi.org/10.1142/S0218348X21501036 discovered that SNAs exist between two parameter regions corresponding to chaotic motions and further discussed the coexistence of SNAs, quasiperiodic attractors, and chaotic attractors in a nonsmooth mechanical system.

The piecewise linear systems are a class of classical nonsmooth dynamical systems. ^34–36 34. Y. Liu, Q. Wang, and H. Xu, "Bifurcations of periodic motion in a three-degree-of-freedom vibro-impact system with clearance," Commun. Nonlinear Sci. Numer. Simul. 48, 1–17 (2017). https://doi.org/10.1016/j.cnsns.2016.12.018 35. S. Kundu, S. Banerjee, J. Ing, E. Pavlovskaia, and M. Wiercigroch, "Singularities in soft-impacting systems," Physica D 241, 553–565 (2012). https://doi.org/10.1016/j.physd.2011.11.014 36. R. R. Aguiar and H. I. Weber, "Mathematical modeling and experimental investigation of an embedded vibro-impact system," Nonlinear Dyn. 65, 317–334 (2011). https://doi.org/10.1007/s11071-010-9894-0 There are still only a few results about the complex dynamics in such systems with quasiperiodic excitations. In this work, we identify and analyze the existence of SNAs in a piecewise linear oscillator with quasiperiodic force, and the dynamics transition of the system is further discussed. The remaining of this paper is organized as follows. In Sec. II, we briefly describe the mathematical model of the piecewise linear oscillator and its Poincaré map. Then, the doubling and period-adding phenomena of the dynamics are investigated in Sec. III. The generation and existence mechanisms of SNAs are discussed in Sec. IV. The main results are summarized in Sec. V.

II. THE PIECEWISE LINEAR OSCILLATOR WITH QUASIPERIODIC EXCITATION AND ITS POINCARÉ MAP

Section:

Consider a piecewise linear oscillator shown in Fig. 1 . The mass $m$ is attached to a linear spring of stiffness $k_1$ and a damper of damping coefficient $c_1$ . It is being acted upon by a quasiperiodic force $p\sin (\omega t)+\epsilon \sin (\xi t)$ . When the displacement $x$ exceeds a certain value $B$ , the second linear spring with stiffness $k_2$ contacts the mass $m$ .

The differential equation of motion of the system can be expressed as

$$m\ddot{x}+c_1\dot{x}+K(x)=p\sin (\omega t)+\epsilon \sin (\xi t),$$

(1)

where

$$K(x)=\{\begin{array}{ll}{k}_{1}x& \text{ if}x\le B,\\ k_{1}x+k_2(x-B)& \text{ if}x>B.\end{array}$$

(2)

Let $\theta =\omega t,\varphi =\xi t$ . Then, Eq. (1) can be written in the following form:

$$\{\begin{array}{l}\dot{x}=v,\\ \dot{v}=(-c_{1}v-K(x)+p\sin (\theta )+\epsilon \sin (\varphi ))/m,\\ \dot{\varphi }=\xi ,\\ \dot{\theta }=\omega .\end{array}$$

(3)

A stroboscopic section is taken in each period $2\pi /\xi$ . We obtain the following Poincaré map of the system (3), which has the form

$$\begin{array}{c}{x}_{n+1}=f_1(x_n,v_n,{\theta }_n),\\ v_{n+1}=f_2(x_n,v_n,{\theta }_n),\\ {\theta }_{n+1}={\theta }_n+\frac{2\pi }{\xi }mod2\pi ,\end{array}$$

(4)

where $f_1$ and $f_2$ are determined by (3).

III. THE DOUBLING AND PERIOD-ADDING PHENOMENA IN THE DYNAMICS

Section:

A. The bifurcations in the unperturbed system

In order to understand the dynamical behavior of system (3), we first consider the unperturbed system, i.e., the case $\epsilon =0$ . The bifurcation diagram of the unperturbed system by varying the external excitation frequency $\omega$ is shown in Figs. 2(a) and 2(b) , where the other parameter values are $p=10,m=0.5,k_1=1.0,k_2=30,c_1=0.2,B=0.0001$ , and the initial values are $(x_0,v_0,{\theta }_0,{\varphi }_0)=(0,0,0,0)$ .

FIG. 2. The bifurcation diagram of the unperturbed system with respect to the parameter $\omega$ . (a) Period-doubling bifurcations. (b) Period-adding bifurcations.

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When the bifurcation parameter $\omega =3.486$ , the system has the first period-doubling bifurcation and the corresponding Floquet multipliers of the system are ${\lambda }_1=-1$ and ${\lambda }_2=-0.486$ , where ${\lambda }_1=-1$ goes through $(-1,0)$ on the unit circle, and the absolute value of another eigenvalue $|{\lambda }_2|=0.486<1$ is still inside the unit circle. As $\omega$ increases to $3.712$ , the system has the second period-doubling bifurcation which converts the attractor from period-2 to period-4. For $\omega =3.797$ , the period-4 attractor evolves into a period-8 attractor. After a series of period-doubling bifurcations, the system evolves into chaos. When $\omega$ is greater than $3.868$ , the system again returns to periodic motion with a period-8 attractor [see Fig. 2(a) ]. In addition, there are period-adding bifurcations in the unperturbed system [see Fig. 2(b) ].

B. The torus-doubling of the system with quasiperiodic excitation

For system (3), we take $\xi =(\sqrt{5}-1)/2$ , since $\xi$ is the inverse of the golden ratio, the dynamics along the $\theta$ -axis is ergodic, and the trajectory for every initial condition uniformly covers the $\theta$ -axis. We take the frequency $\omega$ as the control parameter, varying between $3.45$ and 3.6. The amplitude $\epsilon =0.3$ , the other parameter values, and the initial values are the same as in Sec. III A. The maximum Lyapunov exponent varying with the parameter $\omega$ is shown in Fig. 3 , where $1\mathrm{T},2\mathrm{T}$ , and $4\mathrm{T}$ correspond to torus, doubled torus, and torus of period-4, respectively. In system (3), SNAs do exist between quasiperiodic motion and chaotic motion. With the aid of the maximum Lyapunov exponent, we can distinguish the types of attractors in the given parameter interval. When the parameter $\omega \in [3.45,3.562]$ , the system exhibits quasiperiodic motion. We select the interval $(3.562,3.565)$ as the candidate interval for the existence of SNAs. In Sec. IV, we will further verify that this interval is corresponding to SNAs through the Lyapunov exponent, phase diagrams, and phase sensitivity property. When the parameter $\omega \in [3.565,3.6]$ , the system exhibits chaotic motion, namely, the maximum Lyapunov exponent of the system is positive.

IV. STRANGE NONCHAOTIC ATTRACTORS

Section:

In this section, we use quantitative and qualitative methods to verify the existence of SNAs in system (3) and describe four scenarios for the birth of the SNAs from quasiperiodic behaviors.

We first introduce the power spectrum (Fourier amplitude spectrum), which is an effective tool to distinguish various types of attractors. In general, a dynamical system can manifest two types of power spectrum, namely, continuous and discrete spectrum. Discrete spectrum corresponds to the occurrence of oscillations at specific frequencies such as that of periodic or quasiperiodic oscillations. In contrast, the broadband nature of the frequency spectrum in the case of chaotic oscillations points toward continuous spectrum. For a special case like SNA, the spectrum exhibits a combination of both continuous and discrete components, known as a singular continuous spectrum (cf. Ref. 37 37. A. S. Pikovsky and U. Feudel, "Correlations and spectra of strange non-chaotic attractors," J. Phys. A 27, 5209–5219 (1994). https://doi.org/10.1088/0305-4470/27/15/020 ). We take the Fourier transform of the process $\left\{x_n\right\}$ ,

$$X(\mathrm{\Omega },T)=\sum _{n=1}^T{x}_n{\mathrm{e}}^{i2\pi n\mathrm{\Omega }}.$$

(5)

Hence, the power spectrum of the attractor is defined as ³⁸ 38. A. H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics (John Wiley & Sons, Inc., New York, 1995).

$$P_{\mathrm{\Omega }}=\underset{T\to \mathrm{\infty }}{lim}|X(\mathrm{\Omega },T)/T{|}^2,$$

(6)

where $\mathrm{\Omega }$ is proportional to the frequency ratio of the two excitation forces in the system.

A. The torus collision route

Torus collision is a generating mechanism of SNAs, which is closely related to period-doubling bifurcation. With the change of the control parameter $\omega$ , a stable $2^n\mathrm{T}$ quasiperiodic attractor appears by torus-doubling bifurcation. The parent torus becomes unstable in the quasiperiodically driven system (the pitchfork bifurcation can take place in the system without driving force, that is, the unstable periodic $2^n$ orbit can form the periodic $2^{n+1}$ orbit). Hence, the stable $2^n\mathrm{T}$ quasiperiodic attractor collides with its unstable parent torus to generate SNAs.

To clearly indicate the geometrically smooth or complicated structure of attractors, we can draw the phase diagrams of Poincaré section in the $({\theta }_n,x_n)$ -plane with $\theta$ $mod$ $2\pi$ , and in $(v_n,x_n)$ -plane, respectively. Figure 4 shows the evolution of typical attractors as $\omega$ varies and we can depict the creation of the SNAs by the torus-doubling. When $\omega =3.45$ , the attractor is quasiperiodic in the Poincaré sections, and the system exhibits an invariant torus in the $(v_n,x_n)$ -plane, namely, a $1\mathrm{T}$ quasiperiodic attractor occurs [see Figs. 4(a) and 4(b) ]. As $\omega$ increases, for example, $\omega =3.5$ , the attractor becomes a two-tori (2T) quasiperiodic attractor and the invariant curve becomes two smooth curves, which are created by a torus-doubling bifurcation from the period-1 repellor [see Figs. 4(c) and 4(d) ]. When $\omega$ passes through $3.548$ , the torus-doubling bifurcation occurs again, and the $2\mathrm{T}$ quasiperiodic attractor evolves into a $4\mathrm{T}$ quasiperiodic attractor [see Figs. 4(e) and 4(f) ]. In general, the torus-doubling continues indefinitely until a critical point is reached beyond which the system exhibits chaotic motion. However, as $\omega$ increases further to $3.5647$ , the $4\mathrm{T}$ quasiperiodic attractor collides with its unstable parent torus; hence, the attractor becomes extremely wrinkled, loses its smoothness, and results in a strange attractor. This is reflected in Figs. 4(g) and 4(h) , which indicate the apparent discontinuity in the Poincaré map and the doubling of the torus is interrupted. During this process, the maximum Lyapunov exponent remains negative $(\lambda \approx -0.00203)$ , which is shown in Fig. 3 . Therefore, we conclude that the system is in the strange nonchaotic state for $\omega =3.5647$ . Finally, the attractor deforms and the SNA becomes larger and forms a chaotic attractor with $\omega =3.567$ [see Figs. 4(i) and 4(j) ].

FIG. 4. For $\epsilon =0.3$ , the phase diagrams of attractors in the $({\theta }_n,x_n)$ -plane and $(v_n,x_n)$ -plane: (a) and (b) $\omega =3.45$ , (c) and (d) $\omega =3.5$ , (e) and (f) $\omega =3.551$ , (g) and (h) $\omega =3.5647$ , (i) and (j) $\omega =3.567$ .

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B. Check the strange properties of attractors

In Secs. I–III, we studied the evolution of the attractors with the help of phase diagram and power spectrum and verified the nonchaotic characteristics of SNAs by the maximum Lyapunov exponent. To further illustrate the strange properties of the attractor, we will use the phase sensitivity function, singular continuous spectrum, and rational approximation method.

1. Phase sensitivity property

If the attractor is regarded as a fractal curve, then its nondifferentiability can be detected by the phase sensitivity. This method is based on the sensitivity of the attractor to the phase of the external excitation. There are some special tangent bifurcation points in SNAs, where the derivative of these bifurcation points with respect to the phase is infinite, and the tangent is orthogonal to the $\theta$ -axis, showing that the attractor is nonsmooth. The derivative with respect to the external phase can be denoted as (cf. Ref. 3 3. A. S. Pikovsky and U. Feudel, "Characterizing strange nonchaotic attractors," Chaos 5, 253–260 (1995). https://doi.org/10.1063/1.166074 )

where $N$ is the number of iterations. If $S_i(N)$ tends to be infinite as $N\to \mathrm{\infty }$ , the attractor is nonsmooth, which means that the attractor is strange.

The phase sensitivity can be calculated from the time series in the attractors. For any small $\epsilon$ , we can find $n_0$ to satisfy the phase difference ${\epsilon }_0=|{\theta }_{n_0}-{\theta }_0|<\epsilon$ , and the $S_i(N)$ can be approximately expressed as

$$\begin{array}{rl}{S}_i(N)& =\frac{\mathrm{\partial }{f}_i}{\mathrm{\partial }\theta }\approx \sum _{k=1}^{N-n_0}\left|\frac{f_i(k+n_0)-f_i(k)}{\theta (k+n_0)-\theta (k)}\right|\\ & =\sum _{k=1}^{N-n_0}\left|\frac{f_i(k+n_0)-f_i(k)}{{\epsilon }_0}\right|(i=1,2),\end{array}$$

(8)

where $k+n_0⩽N$ , and $f_i(k)$ denotes the $k$ th iteration of $f_i$ . The maximum value of $S_i(N)$ after $N$ times iterations is denoted as

If the number of iterations increases, the value of ${\tau }_i^N$ increases gradually. Then, $S_i(N)$ tends to infinite as $N\to \mathrm{\infty }$ , which means that the attractor is strange.

We take two groups of parameter values to verify strange properties by phase sensitivity function. Let $n_0=4182$ , then $|\theta (k+n_0)-\theta (k)|\equiv 0.000672$ , when $\omega =3.5647$ , the value of ${\tau }_1^N$ tends to infinity as the number of iterations increases, which indicates that the attractor is strange. On the contrary, when $\omega =3.551$ , the value of ${\tau }_1^N$ does not increase with the increase of iterations, which indicates that the attractor is smooth (see Fig. 5 ).

FIG. 5. The phase sensitivity exponent ${\tau }_1^N$ vs $N$ , the 4T for $\omega =3.551$ (red line) and SNA for $\omega =3.5647$ (blue line).

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2. Singular continuous spectrum

The strange properties of the attractor can be examined by means of the singular continuous spectrum. For $\omega =3.5647$ , the spectrum shows singular continuity [see Fig. 6(a) ]. In general, $X(\mathrm{\Omega },T)$ and $T$ have a power-law relationship (see Ref. 39 39. S. M. Aravindh, A. Venkatesan, and M. Lakshmanan, "Strange nonchaotic attractors for computation," Phys. Rev. E 97, 052212 (2018). https://doi.org/10.1103/PhysRevE.97.052212 for details). Let

where $\rho$ is a scaling exponent. If an SNA exists, then $1<\rho <2$ . When the system exhibits periodic or quasiperiodic motion, the corresponding scaling exponent is $\rho =2$ . If the system exhibits chaotic motion, the scaling exponent is $\rho =1.$ For the SNA in Fig. 4(g) , the corresponding finite-time Fourier spectrum $X(\mathrm{\Omega },T)$ vs $T$ in logarithmic scale is exhibited in Fig. 6(b) , where we detect a relatively robust power-law behavior with $\rho =1.2564$ . In addition, $X(\mathrm{\Omega },T)$ defines a path in a complex plane $(\mathrm{Re}\mathrm{X},\mathrm{Im}\mathrm{X})$ when $T$ is regarded as time and a fractal self-similar path corresponds to $|X(\mathrm{\Omega },T){|}^2\sim T^{1.2564}$ [see Fig. 6(c) ].

FIG. 6. For $\omega =3.5647$ . (a) The singular continuous spectrum. (b) The finite-time Fourier spectrum $X(\mathrm{\Omega },T)$ vs $T$ in logarithmic scale. (c) The fractal walk in the complex plane $(ReX,ImX)$ .

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3. The rational approximation

The rational approximation is also an important method to determine the strange properties of SNAs, which is based on the fact that all irrational numbers can be approximated by appropriate rational numbers (cf. Ref. 3 3. A. S. Pikovsky and U. Feudel, "Characterizing strange nonchaotic attractors," Chaos 5, 253–260 (1995). https://doi.org/10.1063/1.166074 ). When the approximation order is low, only some periodic points can be observed and the shape of SNAs cannot be presented. As the order of approximation increases, the number of periodic points tends to infinite, and the structure of the attractor is not a finite set of points and is not piecewise differentiable, which depicts the strangeness of the attractor. In the following, we use rational numbers to approximate the frequency $\xi$ , which is taken as $(\sqrt{5}-1)/2$ . The ratio ${\xi }_k$ of the Fibonacci sequences can be used to approximate $\xi$ , where ${\xi }_k=F_{k-1}/F_k,F_k=F_{k-1}+F_{k-2},F_1=1,F_2=1$ , and $\xi =\underset{k\to \mathrm{\infty }}{lim}{\xi }_k.$ Take the attractor in Fig. 4(j) as an example. Figures 7(a) –7(d) are the phase diagrams of the attractor in the $({\theta }_n,x_n)$ -plane for different approximate orders $k$ . In Fig. 7(a) , only some periodic points can be observed when ${\xi }_5=5/8$ . As $k$ gradually increases, the number of periodic points gradually increases, and the fractal characteristics of the attractor gradually appear [Figs. 7(b) and 7(c) ]. When ${\xi }_{19}=4181/6765$ , the number of periodic points becomes virtually uncountable, the attractor begins to appear nonsmoothly, and an SNA emerges [Fig. 7(d) ]. We find that the higher is the order of the approximation, the more significant is the degree of the fractal structure of the attractor.

FIG. 7. The rational approximations of the SNA for $\omega =3.5647$ in the $({\theta }_n,x_n)$ -plane. (a) ${\xi }_5=5/8$ , (b) ${\xi }_{12}=89/144$ , (c) ${\xi }_{17}=1597/2584$ , (d) ${\xi }_{19}=4181/6765$ .

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C. The fractal route

In fractal routes for the creation of SNAs, a torus attractor gets increasingly wrinkled and transforms into an SNA without the apparent mediation of any nearby unstable orbit. This generation mechanism of SNAs is independent of a bifurcation phenomenon.

We fix the parameter $\epsilon =3.1$ and let $\omega$ vary from $3.2$ to $3.44$ . The other parameters and the initial values remain the same as Sec. III A. The attractors for successively larger $\omega$ are shown in Fig. 8 . For $\omega =3.2$ , the attractor has one smooth branch, which indicates that the system is in a single torus state $(1\mathrm{T})$ [see Fig. 8(b) ]. As $\omega$ increases, the attractor does not undergo torus-doubling bifurcation as in the last scenario. Instead, the branch of the attractor becomes irregular as shown in Figs. 8(c) and 8(d) , but the attractor is continuous. In other words, the system is still in a torus state for $\omega =3.38$ . When $\omega =3.43$ , the attractor becomes extremely wrinkled, and ultimately result is a strange attractor [see Figs. 8(e) and 8(f) ]. Meanwhile, its maximum Lyapunov exponent is nonpositive [ $\lambda \approx -0.00849$ , see Fig. 9(a) ]. Therefore, it is an SNA. For $\omega =3.44$ , the SNA is transformed into an attractor with a fractal structure and positive maximum Lyapunov exponent $(\lambda \approx 0.00458)$ , and, thus, the attractor [Fig. 8(g) ] is chaotic.

FIG. 8. For $\epsilon =3.1$ , the phase diagrams of attractors in the $({\theta }_n,x_n)$ -plane and $(v_n,x_n)$ -plane: (a) and (b) $\omega =3.2$ , (c) and (d) $\omega =3.38$ , (e) and (f) $\omega =3.43$ , (g) and (h) $\omega =3.44$ .

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FIG. 9. For $\omega =3.43$ . (a) The maximum Lyapunov exponent. (b) The phase sensitivity exponent ${\tau }_1^N$ vs $N$ , the $1\mathrm{T}$ for $\omega =3.2$ (red line) and SNA for $\omega =3.43$ (blue line). (c) The finite-time Fourier spectrum $X(\mathrm{\Omega },T)$ vs $T$ in logarithmic scale. (d) The fractal walk in the complex plane $(ReX,ImX)$ .

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In the fractal route, it is sometimes difficult to decide when the transition to SNAs occurs since the extremely wrinkled invariant curve looks very similar to an SNA. However, we can give a strict criterion by the phase sensitivity function. Figure 9(b) shows the time evolution of the maximum derivative, where the red and blue curves correspond to attractors as shown in Figs. 8(a) and 8(e) , respectively. As the number of iterations increases, the derivative is not saturated after the transition to SNA, we infer that the derivative is not bounded. Therefore, the attractor shown in Fig. 8(e) is strange. To provide further evidence, the previous methods are also used to confirm the strange structure of SNA. We compute the scaling exponent $\rho$ and present a reasonable power-law relationship $|X(\mathrm{\Omega },T){|}^2\sim T^{1.4540}$ and this behavior is shown in Fig. 9(c) . Meanwhile, Fig. 9(d) demonstrates that the spectral trajectory in the complex plane $(ReX,ImX)$ should exhibit a fractal behavior, as required for SNAs. To sum up, the attractor is an SNA.

D. The bubbling route

In addition to the route studied above, we also identified an uncommon route in a piecewise linear oscillator, called bubbling route (cf. Refs. 25 25. D. V. Senthilkumar, V. Srinivasan, K. Thamilmaran, and M. Lakshmanan, "Birth of strange nonchaotic attractors through formation and merging of bubbles in a quasiperiodically forced Chua's oscillator," Phys. Rev. E 78, 066211 (2008). https://doi.org/10.1103/PhysRevE.78.066211 and 26 26. K. Suresh, A. Prasad, and K. Thamilmaran, "Bubbling route to strange nonchaotic attractor in a nonlinear series LCR circuit with a nonsinusoidal force," Phys. Lett. A 377, 612–621 (2013). https://doi.org/10.1016/j.physleta.2012.12.026 ) for the creation of SNAs. In this route, the torus quasiperiodic attractor forms local bubble-like structures in some regions and wrinkling of these under external force. However, the rest of the strands of the torus outside the bubbles remain largely unaffected. It is worth noting that this route is quite different from the well known fractal route, where the entire strands of the $n$ -period torus continuously deform and become extremely wrinkled as the control parameters change.

To illustrate the emergence of bubbling route to SNAs in system (3), we fix the parameter $\epsilon =1.5$ and let the frequency $\omega$ vary from $3.36$ to $3.37$ . The other parameters and the initial values remain the same as Sec. III A. For $\omega =3.363$ , a 1T quasiperiodic attractor [Figs. 10(a) and 10(b) ] occurs. As $\omega$ is increased to 3.3645, in the 1T quasiperiodic attractor appears bubble-like structures in the strand of the torus [see Figs. 10(c) and 10(d) ]. These bubble-like structures are localized, namely, they occur on partial regions of the torus while the remaining part of the torus strand remains unaffected. As $\omega$ increases further to $3.365$ , the bubbles deform and become extremely wrinkled, which lead to the formation of an SNA [see Figs. 10(e) and 10(f) ]. As $\omega$ continues to increase to $3.37$ , the SNA forms a 1-band chaotic attractor [Fig. 10(g) ]. This transition process from a quasiperiodic attractor to an SNA via the bubbling route can be further verified by the entire Lyapunov spectrum [Fig. 11(a) ], phase sensitivity function [Fig. 11(b) ], and singular continuous spectrum analysis [Fig. 11(c) ]: ${\log }_{10}|X(\mathrm{\Omega },T){|}^2$ vs ${\log }_{10}T$ , where $|X(\mathrm{\Omega },T){|}^2\sim T^{1.2286}$ , and fractal trajectory in the complex plane $(ReX,ImX)$ [Fig. 11(d) ].

FIG. 10. For $\epsilon =1.5$ , the phase diagrams of attractors in the $({\theta }_n,x_n)$ -plane and $(v_n,x_n)$ -plane: (a) and (b) $\omega =3.363$ , (c) and (d) $\omega =3.3645$ , (e) and (f) $\omega =3.365$ , (g) and (h) $\omega =3.37$ .

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FIG. 11. For $\omega =3.365$ . (a) The entire Lyapunov spectrum. (b) The phase sensitivity exponent ${\tau }_1^N$ vs $N$ , the $1\mathrm{T}$ for $\omega =3.363$ (red line) and SNA for $\omega =3.365$ (blue line). (c) The finite-time Fourier spectrum $X(\mathrm{\Omega },T)$ vs $T$ in logarithmic scale. (d) The fractal walk in the complex plane $(ReX,ImX)$ .

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The mechanism of the bubbling route is that the quasiperiodic orbit gets increasingly unstable in its transverse direction as the control parameter $\omega$ changes, resulting in the formation of the bubble-like structures. This is reflected in Fig. 10(e) , in partial regions of the torus.

E. The type-I intermittency route

Intermittency routes can be the generation mechanisms for SNAs. We describe SNAs in terms of $4T$ torus intermittency, which is related to the saddle-node bifurcation. Let $\epsilon =0.1659095$ and the frequency $\omega$ vary from $9.995$ to $10$ . For $\omega =9.995$ , the attractor is $4T$ quasiperiodic, as shown in Figs. 12(a) and 12(b) . When $\omega$ is increased to 10, the approximate shape of the quasiperiodic orbit of the system is still present, but there are many disordered points near the orbit, which are characteristics of intermittency, as shown in Figs. 12(c) and 12(d) . Here, the attractor is strange nonchaotic in Fig. 12(c) . The nonchaotic property is denoted by the maximum Lyapunov exponent $\lambda \approx -0.00161$ [Fig. 13(a) ]. The strange property of the attractor is characterized by the phase sensitivity [Fig. 13(b) ] and the singular continuous spectrum [Fig. 13(c) ].

FIG. 12. For $\epsilon =0.1659095$ , the phase diagrams of attractors in the $({\theta }_n,x_n)$ -plane and $(v_n,x_n)$ -plane: (a) and (b) $\omega =9.995,(c)and(d)\omega =10$ .

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FIG. 13. For $\omega =10.$ (a) The maximum Lyapunov exponent. (b) The phase sensitivity exponent ${\tau }_1^N$ vs $N$ , the $4\mathrm{T}$ for $\omega =9.995$ (red line) and SNA for $\omega =10$ (blue line). (c) The singular continuous spectrum.

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V. CONCLUSIONS

Section:

Most of the research on strange nonchaotic dynamics focuses on smooth systems in different fields. In this work, the dynamics of a single-degree-of-freedom nonlinear oscillator under quasiperiodic excitation is considered. The complicated and interesting strange nonchaotic dynamics phenomena are revealed via the phase diagram, maximum Lyapunov exponent, singular continuous spectrum, phase sensitivity function, rational approximation, and so on. We demonstrate that the first scenario is torus-doubling bifurcation followed by the transition from the torus attractors to SNAs in such class of nonsmooth systems. The second is that the torus does not undergo a period-doubling cascade as controlling parameters vary, instead, the torus becomes extremely wrinkled, loses smoothness, and finally becomes fractal. The third is that the quasiperiodic attractor gets increasingly unstable in its transverse direction as the control parameter changes, resulting in the formation of bubble-like structures in partial regions of the torus, which eventually evolves into an SNA. The last one is that in the neighborhood of a saddle-node bifurcation many disordered points abruptly appear during the transition from $4T$ torus to SNA. The results of this work offer ideas for the study of strange nonchaotic dynamics in nonsmooth systems as well as provide support for the design and optimization of devices in engineering fields.

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Source: https://aip.scitation.org/doi/10.1063/5.0096959

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