Mixed mode oscillations induced by bi-stability and fractal basins in the FGP plate under slow parametric and resonant external excitations

Chen, Zhenyang; Chen, Fangqi

doi:10.1016/j.chaos.2020.109814

Cited by 13 publications

(6 citation statements)

References 47 publications

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“…We observe that the system shows the possibility of the development of periodic and chaotic bursting that appeared due to a sharp explosion in the equilibrium points of the system, known as a pulse-shaped explosion. Although the emergence of periodic compound bursting patterns via pulse-shaped explosion is reported, earlier [37,44,45], the generic nature of the pulse-shaped explosion induced bursting pattern is demonstrated in the chosen model. Also, interestingly, we provide evidence for the development of rare but recurrent chaotic large bursting along with regular small oscillations, which approved all the conditions of the EE phenomenon.…”

Section: Introductionmentioning

confidence: 85%

“…Two different types of regular bursting patterns, that is, bursting types of point-point type and cycle-cycle type, have been observed in the Rayleigh system via pulseshaped explosion [37]. In addition to that, the pulse-shaped explosion can also induce different bursting patterns in other dynamical systems as well [44,45]. Even though considerable work has been done in recent times regarding the pulse-shaped explosion-induced bursting dynamics, this area still requires further investigation.…”

Section: Introductionmentioning

confidence: 99%

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Extreme bursting events via pulse-shaped explosion in mixed Rayleigh-Lienard nonlinear oscillator

Kaviya,

Suresh,

Chandrasekar

2022

Preprint

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We study the dynamics of a parametrically and externally driven Rayleigh-Liénard hybrid model and report the emergence of extreme bursting events due to a novel pulse-shaped explosion mechanism. The system exhibit complex periodic and chaotic bursting patterns amid small oscillations as a function of excitation frequencies. In particular, the advent of rare and recurrent chaotic bursts that emerged for certain parameter regions is characterized as extreme events. We have identified that the appearance of a sharp pulse-like transition that occurred in the equilibrium points of the system is the underlying mechanism for the development of bursting events. Further, the controlling aspect of extreme events is attempted by incorporating a linear damping term, and we show that for sufficiently strong damping strength, the extreme events are eliminated from the system, and only periodic bursting is feasible.

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Section: Introductionmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

Extreme bursting events via pulse-shaped explosion in mixed Rayleigh-Lienard nonlinear oscillator

Kaviya,

Suresh,

Chandrasekar

2022

Preprint

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“…Generally, the slow-fast systems are presented in two forms during practical applications, i.e. the autonomous one with state variables of different time scale and the non-autonomous one containing slow excitations [20]. A typical autonomous slow-fast dynamical system with two scales can be divided into two subsystems, i.e., the fast subsystems(FS) and the slow subsystems(SS), expressed in the standard form [21] ẋ = f(x, y, µ µ µ), (Fast Subsystem) ẏ = εg(x, y, µ µ µ), (Slow Subsystem) (1) where x ∈ R M , y ∈ R N , µ µ µ ∈ R K , while 0 < ε ≪ 1 describes the ratio between the fast and slow scales.…”

Section: Introductionmentioning

confidence: 99%

Novel bursting patterns and the bifurcation mechanism in a piecewise smooth Chua’s circuit with two scales

Zhang

Peng

2022

Nonlinear Dyn

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The aim of this paper is to investigate the influence of the coupling of two scales on the dynamics of a piecewise smooth dynamical system. A relatively simple model with two switching boundaries is taken as an example by introducing a nonlinear piecewise resistor and a harmonically changed electric source into a typical Chua's circuit. Taking suitable values of the parameters, four different types of bursting oscillations are observed corresponding to different values of the exciting amplitude. Regarding the periodic excitation as a slow-varying parameter, equilibrium branches of the fast subsystem as well as the related bifurcations, such as fold bifurcation, Hopf bifurcation, period doubling bifurcation, nonsmooth Hopf bifurcation and nonsmooth fold limit cycle bifurcation, are explored with theoretical and numerical methods. With the help of the overlap of the transformed phase portrait and the equilibrium branches, the mechanism of the bursting oscillations can be analyzed in detail. It is found that for relatively small exciting amplitude, since the trajectory is governed by a smooth subsystem, only conventional bifurcations take place, leading to the transitions between the spiking states and quiescent states. However, with an increase of the exciting amplitude so that the trajectory passes across the switching boundaries, nonsmooth bifurcations occurring at the boundaries may involve the structures of attractors, leading to complicated bursting oscillations. Further increasing the exciting amplitude, the number of the spiking states decreases although more bifurcations take place, which can be explained by the delay effect of bifurcation.

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“…Dynamics with multi-time scales and inherent nonlinear characteristics are well-known to display a variety of complex oscillations and often produce undesirable oscillation patterns in practical electronic engineering fields (Chen and Chen, 2020; Sun and Wei, 2019; Sun et al , 2020). Thus, a great deal of interests have been paid to develop effective algorithms to generate different types of MMOs (Kuehn, 2010; Guay, 2016).…”

Section: Introductionmentioning

confidence: 99%

Canard-induced mixed mode oscillations as a mechanism for the Bonhoeffer-van der Pol circuit under parametric perturbation

Zhang

Chen

et al. 2021

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Purpose This paper aims to investigate the singular Hopf bifurcation and mixed mode oscillations (MMOs) in the perturbed Bonhoeffer-van der Pol (BVP) circuit. There is a singular periodic orbit constructed by the switching between the stable focus and large amplitude relaxation cycles. Using a generalized fast/slow analysis, the authors show the generation mechanism of two distinct kinds of MMOs. Design/methodology/approach The parametric modulation can be used to generate complicated dynamics. The BVP circuit is constructed as an example for second-order differential equation with periodic perturbation. Then the authors draw the bifurcation parameter diagram in terms of a containing two attractive regions, i.e. the stable relaxation cycle and the stable focus. The transition mechanism and characteristic features are investigated intensively by one-fast/two-slow analysis combined with bifurcation theory. Findings Periodic perturbation can suppress nonlinear circuit dynamic to a singular periodic orbit. The combination of these small oscillations with the large amplitude oscillations that occur due to canard cycles yields such MMOs. The results connect the theory of the singular Hopf bifurcation enabling easier calculations of where the oscillations occur. Originality/value By treating the perturbation as the second slow variable, the authors obtain that the MMOs are due to the canards in a supercritical case or in a subcritical case. This study can reveal the transition mechanism for multi-time scale characteristics in perturbed circuit. The information gained from such results can be extended to periodically perturbed circuits.

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Mixed mode oscillations induced by bi-stability and fractal basins in the FGP plate under slow parametric and resonant external excitations

Cited by 13 publications

References 47 publications

Extreme bursting events via pulse-shaped explosion in mixed Rayleigh-Lienard nonlinear oscillator

Extreme bursting events via pulse-shaped explosion in mixed Rayleigh-Lienard nonlinear oscillator

Novel bursting patterns and the bifurcation mechanism in a piecewise smooth Chua’s circuit with two scales

Canard-induced mixed mode oscillations as a mechanism for the Bonhoeffer-van der Pol circuit under parametric perturbation

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