Analysis of fractional order Bonhoeffer–van der Pol oscillator

This study aims to investigate the harmonic resonance of third-order forced van der Pol oscillator with fractional-order derivative using the asymptotic method. The approximately analytical solution for the system is first determined, and the amplitude–frequency equation of the oscillator is established. The stability condition of the harmonic solution is then obtained by means of Lyapunov theory. A comparison between the traditional integer-order of forced van der Pol oscillator and the considered fractional-order one follows the numerical simulation. Finally, the numerical results are analyzed to show the influences of the parameters in the fractional-order derivative on the steady-state amplitude, the amplitude–frequency curves, and the system stability.

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Bifurcation and stability analysis of commensurate fractional-order van der Pol oscillator with time-delayed feedback

Indian Journal of Physics ◽

10.1007/s12648-019-01589-2 ◽

2019 ◽

Vol 94 (10) ◽

pp. 1615-1624

Author(s):

Jufeng Chen ◽

Yongjun Shen ◽

Xianghong Li ◽

Shaopu Yang ◽

Shaofang Wen

Keyword(s):

Stability Analysis ◽

Fractional Order ◽

Delayed Feedback ◽

Van Der Pol Oscillator ◽

Van Der Pol ◽

Bifurcation And Stability ◽

Time Delayed Feedback

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NEIMARK BIFURCATIONS OF A GENERALIZED DUFFING–VAN DER POL OSCILLATOR WITH NONLINEAR FRACTIONAL ORDER DAMPING

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127413501770 ◽

2013 ◽

Vol 23 (11) ◽

pp. 1350177 ◽

Cited By ~ 1

Author(s):

A. Y. T. LEUNG ◽

H. X. YANG ◽

P. ZHU

Keyword(s):

Fractional Order ◽

Harmonic Balance ◽

Harmonic Balance Method ◽

Viscoelastic Behavior ◽

Higher Order ◽

Balance Method ◽

Van Der Pol Oscillator ◽

Homotopy Continuation ◽

Van Der Pol ◽

Steady State Responses

A generalized Duffing–van der Pol oscillator with nonlinear fractional order damping is introduced and investigated by the residue harmonic homotopy. The cubic displacement involved in fractional operator is used to describe the higher-order viscoelastic behavior of materials and of aerodynamic damping. The residue harmonic balance method is employed to analytically generate higher-order approximations for the steady state responses of an autonomous system. Nonlinear dynamic behaviors of the harmonically forced oscillator are further explored by the harmonic balance method along with the polynomial homotopy continuation technique. A parametric investigation is carried out to analyze the effects of fractional order of damping and the effect of the magnitude of imposed excitation on the system using amplitude-frequency curves. Jump avoidance conditions are addressed. Neimark bifurcations are captured to delineate regions of instability. The existence of even harmonics in the Fourier expansions implies symmetry-breaking bifurcation in certain combinations of system parameters. Numerical simulations are given by comparing with analytical solutions for validation purpose. We find that all Neimark bifurcation points in the response diagram always exist along a straight line.

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Chaotic Control in a Fractional-Order Modified Van Der Pol Oscillator

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.474-476.83 ◽

2011 ◽

Vol 474-476 ◽

pp. 83-88

Author(s):

Xin Gao

Keyword(s):

Fractional Order ◽

Feedback Controller ◽

Van Der Pol Oscillator ◽

Linear Feedback ◽

Fractional Order Systems ◽

Van Der Pol ◽

Chaotic Control ◽

Linear Feedback Controller

The dynamics of fractional-order systems have attracted increasing attention in recent years. In this paper, we study the chaotic behaviors in a fractional-order modified van der Pol oscillator. We find that chaos exists in the fractional-order modified van der Pol oscillator with order less than 3. In addition, the lowest order we find for chaos to exist in such system is 2.4. Finally, a simple, but effective, linear feedback controller is also designed to stabilize the fractional order chaotic van der Pol oscillator.

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