On two-parameter bifurcation analysis of switched system composed of Duffing and van der Pol oscillators

Zhang, Chun; Bi, Qinsheng; Han, Xiujing; Zhang, Zhengdi

doi:10.1016/j.cnsns.2013.06.028

Cited by 9 publications

(2 citation statements)

References 23 publications

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“…We say that there is coexistence of attractors of the same nature if the system has the same dynamics but with different amplitudes otherwise we say that attractors of different natures coexist. It is necessary to note that the Lyapunov exponent of stable periodic solution is less than zero while the Lyapunov exponent of chaos is greater than zero and Lyapunov exponent equal to zero means that bifurcation occurs, corresponding to a quasi-periodic dynamic [31].…”

Section: Hysteresis Coexistence Of Attractors and Multistabilitymentioning

confidence: 99%

Effect of an amplitude modulated force on vibrational resonance, chaos, and multistability in a modified Van der Pol-Duffing oscillator

Miwadinou,

Hinvi,

Ainamon

et al. 2023

J. Nonlinear Sci. Appl.

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This paper deals with the effects of an amplitude modulated (AM) excitation on the nonlinear dynamics of reactions between four molecules. The computation of the fixed points of the autonomous nonlinear chemical system has been made in detail using Cardan's method. Routes to chaos have been investigated through bifurcations structures, Lyapunov exponent and phase portraits. The effects of the control force on chaotic motions have been strongly analyzed and the control efficiency is found in the cases g = 0 (unmodulated case), g = 0 with Ω = nω; n a natural number and Ω ω = p q ; p and q are simple positive integers. Vibrational Resonance (VR), hysteresis and coexistence of several attractors have been studied in details based on the relationship between the frequencies of the AM force. Results of analytical investigations are validated and complemented by numerical simulations.

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Section: Hysteresis Coexistence Of Attractors and Multistabilitymentioning

confidence: 99%

Effect of an amplitude modulated force on vibrational resonance, chaos, and multistability in a modified Van der Pol-Duffing oscillator

Miwadinou,

Hinvi,

Ainamon

et al. 2023

J. Nonlinear Sci. Appl.

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“…However, many problems such as dynamical behaviors, bifurcations associated with the switching conditions, and the mechanism of complexity with the variation in the parameters are seldom researched. Switched systems introduce many new characteristics, especially strong nonlinearity and singularity caused by the non-differentiability or discontinuity of vector fields [22,23]. Therefore, many dynamic characteristics of nonsmooth systems can not be treated by the ordinary smooth dynamic system theory, and special theories and methods need to be developed.…”

Section: Introductionmentioning

confidence: 99%

Grazing and Symmetry-Breaking Bifurcations Induced Oscillations in a Switched System Composed of Duffing and van der Pol Oscillators

2022

Self Cite

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By introducing a switching scheme related to the state and time, a typical switched model alternating between a Duffing oscillator and van der Pol oscillator is established to explore the typical dynamical behaviors as well as the mechanism of the switched system. Shooting methods to locate the limit cycle and specify bifurcation sets are described by defining an appropriate Poincaré map. Different types of multiple-Focus/Cycle and single-Focus/Cycle period oscillations in the system can be observed. Symmetry-breaking, period-doubling, and grazing bifurcation curves are obtained in the plane of bifurcation parameters, dividing the parameters plane into several regions corresponding to different kinds of oscillations. Meanwhile, based on the numerical simulation and bifurcation analysis, the mechanisms of several typical dynamical behaviors observed in different regions are presented.

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Nonlinear behaviors as well as the mechanism in a piecewise-linear dynamical system with two time scales

Chen

Kurths

et al. 2016

Nonlinear Dyn

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On two-parameter bifurcation analysis of switched system composed of Duffing and van der Pol oscillators

Cited by 9 publications

References 23 publications

Effect of an amplitude modulated force on vibrational resonance, chaos, and multistability in a modified Van der Pol-Duffing oscillator

Effect of an amplitude modulated force on vibrational resonance, chaos, and multistability in a modified Van der Pol-Duffing oscillator

Grazing and Symmetry-Breaking Bifurcations Induced Oscillations in a Switched System Composed of Duffing and van der Pol Oscillators

Nonlinear behaviors as well as the mechanism in a piecewise-linear dynamical system with two time scales

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