Complex aperiodic mixed mode oscillations induced by crisis and transient chaos in a nonlinear system with slow parametric excitation

Chen, Zhenyang; Chen, Fangqi

doi:10.1007/s11071-020-05500-1

Cited by 18 publications

(7 citation statements)

References 56 publications

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“…The methods presented here can support the construction of snapshot phase portraits in engineering applications, too. These range from ships experiencing wave induced time varying restoring moments [79], through multiple scale dynamics [9], to investigating the effects of a moving mass on structures [76], or the switching between different working regimes [23], or milling stability with time dependent parameters [17]. Tipping events are sudden changes accompanying parameter drifts, and our method is useful to apply in such cases (see, e.g., [47,72]), too, including the avoidance of instability via fast enough rates of the parameter drift [6,85].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…It is therefore worth selecting a more specific ensemble, concentrated on one of the basic components of the phase space [38]. Being one-dimensional objects and simpler than chaotic seas, let us take the tori of system (9) with α = 0, i.e., the tori of the stationary phase space, and evolve them under (9) with α = 0. To illustrate a basic effect, we select 4 tori, lying in different regions of the original phase space (light green curves of Fig.…”

Section: The Choice Of Ensemblesmentioning

confidence: 99%

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Climate change in mechanical systems: the snapshot view of parallel dynamical evolutions

2021

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We argue that typical mechanical systems subjected to a monotonous parameter drift whose timescale is comparable to that of the internal dynamics can be considered to undergo their own climate change. Because of their chaotic dynamics, there are many permitted states at any instant, and their time dependence can be followed—in analogy with the real climate—by monitoring parallel dynamical evolutions originating from different initial conditions. To this end an ensemble view is needed, enabling one to compute ensemble averages characterizing the instantaneous state of the system. We illustrate this on the examples of (i) driven dissipative and (ii) Hamiltonian systems and of (iii) non-driven dissipative ones. We show that in order to find the most transparent view, attention should be paid to the choice of the initial ensemble. While the choice of this ensemble is arbitrary in the case of driven dissipative systems (i), in the Hamiltonian case (ii) either KAM tori or chaotic seas should be taken, and in the third class (iii) the best choice is the KAM tori of the dissipation-free limit. In all cases, the time evolution of the chosen ensemble on snapshots illustrates nicely the geometrical changes occurring in the phase space, including the strengthening, weakening or disappearance of chaos. Furthermore, we show that a Smale horseshoe (a chaotic saddle) that is changing in time is present in all cases. Its disappearance is a geometrical sign of the vanishing of chaos. The so-called ensemble-averaged pairwise distance is found to provide an easily accessible quantitative measure for the strength of chaos in the ensemble. Its slope can be considered as an instantaneous Lyapunov exponent whose zero value signals the vanishing of chaos. Paradigmatic low-dimensional bistable systems are used as illustrative examples whose driving in (i, ii) is chosen to decay in time in order to maintain an analogy with case (iii) where the total energy decreases all the time.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Section: The Choice Of Ensemblesmentioning

confidence: 99%

Climate change in mechanical systems: the snapshot view of parallel dynamical evolutions

2021

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“…In the study of a multi-type quasi-periodic system with memristors, Licai Liu et al observed the transient behavior from a weak chaotic state to a quasi-periodic limit cycle [55]. The transient nonlinear behaviors found in these reports are representative, and similar reports on transients are abundant [56]- [74]. However, these studies have only found that the chaotic system has transient transition behavior under certain specific parameters.…”

Section: Introductionmentioning

confidence: 99%

Multiple Transient Transitions Behavior Analysis of a Double Memristor’s Hidden System and Its Circuit

Liu

Shi

et al. 2020

IEEE Access

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In this paper, a low-dimensional hidden nonlinear system is constructed by replacing two linear resistors of a simple integrating circuit with two active memristors. The system was analyzed in detail by using the Lyapunov exponent, 0-1 Test, Poincaré map, phase diagram, power spectral density diagram, timedomain waveform, and chaotic characteristic diagram. The results show that the system can oscillate by itself under zero initial conditions, and there are various transient transition behaviors, such as from chaos to limit cycle, chaos to quasi-periodic, quasi-periodic to another quasi-periodic transition, quasi-periodic to periodic transition. Besides, these transient processes themselves include 2, 3, 4, and 5 different states respectively, showing multiple transient transitions behavior, which are not reported in the related literature. It is also found that the initial states of these transition states have rich symmetrical attractors and the stable state is multistability. In addition, the global entropy analysis method is adopted to test the universal existence of transient behaviors, which demonstrates that the system has rich nonlinear characteristics. Finally, the memristive circuit verifies the system's physical feasibility and enriches the application of memristors in circuit theory. INDEX TERMS Active memristor, hidden attractor, multiple transient transitions, chaos, simple integrating circuit.

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“…Although chaos has a noise-like appearance, it is deterministic in nature, whereas noise is random [2]. Chaos is a term used to describe nonlinear systems that exhibit chaotic behaviors [3]. Given that chaotic systems follow the butterfly effect, which states that even a small flapping of a butterfly's wings can generate a hurricane, they are extremely sensitive to ICs [4].…”

Section: Introductionmentioning

confidence: 99%

Chaotic Dynamics in the 2D System of Nonsmooth Ordinary Differential Equations

Rahman

Jasim

Al‐Yasir

2022

ijcsm

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Over the last decade, the chaotic behaviors of dynamical systems have been extensively explored. Recently, discovering or developing a 2D system of ordinary differential equations (ODEs) capable of exhibiting chaotic dynamical behaviors is an attractive research topic. In this study, a chaotic system with a 2D system of nonsmooth ODEs has been developed. This system is can exhibit chaotic dynamical behaviors. Its main dynamical behaviors, including time-series trajectories, phase portraits of attractors, and equilibria and their stability, have been investigated. The developed system has been verified by an excessive variety of fascinating chaotic behaviors, such as chaotic attractor, symmetry, sensitivity to initial conditions (ICs), fractal dimension, autocorrelation, power spectrum, Lyapunov exponent, and bifurcation diagram. Analytical and numerical simulations are used to study the dynamical behaviors of such a system. The developed system has extreme sensitivity to ICs, a fractal dimension of more than 1.8 and less than 2.05, an autocorrelation fluctuating randomly about an average of zero, a broadband power spectrum, and one positive Lyapunov exponent. The obtained numerical simulation results have proven the capability of the developed 2D system for exciting chaotic dynamical behaviors

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Complex aperiodic mixed mode oscillations induced by crisis and transient chaos in a nonlinear system with slow parametric excitation

Cited by 18 publications

References 56 publications

Climate change in mechanical systems: the snapshot view of parallel dynamical evolutions

Climate change in mechanical systems: the snapshot view of parallel dynamical evolutions

Multiple Transient Transitions Behavior Analysis of a Double Memristor’s Hidden System and Its Circuit

Chaotic Dynamics in the 2D System of Nonsmooth Ordinary Differential Equations

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