Bifurcations Induced in a Bistable Oscillator via Joint Noises and Time Delay

In this paper, noise-induced and delay-induced bifurcations in a bistable Duffing–van der Pol (DVP) oscillator under time delay and joint noises are discussed theoretically and numerically. Based on the qualitative changes of the plane phase, delay-induced bifurcations are investigated in the deterministic case. However, in the stochastic case, the response of the system is a stochastic non-Markovian process owing to the existence of noise and time delay. Then, methods have been employed to derive the stationary probability density function (PDF) of the amplitude of the response. Accordingly, stochastic P-bifurcations can be observed with the variations in the qualitative behavior of the stationary PDF for amplitude. Furthermore, results from both theoretical analyses and numerical simulations best demonstrate the appearance of noise-induced and delay-induced bifurcations, which are in good agreement.

[1]  Sandip Banerjee,et al.  Delay-induced model for tumor-immune interaction and control of malignant tumor growth , 2008, Biosyst..

[2]  Qintao Gan,et al.  Hopf bifurcation analysis of a BAM neural network with multiple time delays and diffusion , 2015, Appl. Math. Comput..

[3]  C Masoller.  Distribution of residence times of time-delayed bistable systems driven by noise. , 2003, Physical review letters.

[4]  Shuzhi Sam Ge,et al.  Exponential ultimate boundedness of nonlinear stochastic difference systems with time-varying delays , 2015, Int. J. Control.

[5]  Wei Guo,et al.  Transitions induced by time delays and cross-correlated sine-Wiener noises in a tumor–immune system interplay , 2012 .

[6]  Yong Xu,et al.  Stochastic averaging for a class of single degree of freedom systems with combined Gaussian noises , 2014 .

[7]  Kwok-wai Chung,et al.  Effects of time delayed position feedback on a van der Pol–Duffing oscillator , 2003 .

[8]  Honeycutt.  Stochastic Runge-Kutta algorithms. II. Colored noise. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[9]  Colin H. Hansen,et al.  Dynamics of two delay coupled van der Pol oscillators , 2006 .

[10]  J Kurths,et al.  Stochastic bifurcations and coherencelike resonance in a self-sustained bistable noisy oscillator. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  Jian Xu,et al.  Delay-Induced bifurcations in a nonautonomous System with Delayed Velocity Feedbacks , 2004, Int. J. Bifurc. Chaos.

[12]  A. Pisarchik,et al.  Noise-induced attractor annihilation in the delayed feedback logistic map , 2013 .

[13]  Qun Liu.  The effects of time-dependent delays on global stability of stochastic Lotka–Volterra competitive model , 2015 .

[14]  Ping Bi,et al.  Periodic and chaotic oscillations in a tumor and immune system interaction model with three delays. , 2014, Chaos.

[15]  D. Mei,et al.  Effects of time delay on symmetric two-species competition subject to noise. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Noise-and delay-induced phase transitions of the dimer–monomer surface reaction model , 2012 .

[17]  Philipp Hövel,et al.  Transient behavior in systems with time-delayed feedback. , 2009, Chaos.

[18]  Tonghua Zhang,et al.  Bifurcation analysis of a mathematical model for genetic regulatory network with time delays , 2015, Appl. Math. Comput..

[19]  Haiyan Hu,et al.  Global Dynamics of a Duffing oscillator with Delayed Displacement Feedback , 2004, Int. J. Bifurc. Chaos.

[20]  Daniel Jost,et al.  Bifurcation in epigenetics: implications in development, proliferation, and diseases. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  G. Er,et al.  Probabilistic solution of nonlinear oscillators excited by combined Gaussian and Poisson white noises , 2011 .

[22]  Seiya Kasai,et al.  Concept, analysis, and demonstration of a novel delay network exhibiting stochastic resonance induced by external noise , 2015, Digit. Signal Process..

[23]  Xiaoqin Luo,et al.  Diversity-induced resonance with two different kinds of time delays , 2011 .

[24]  L. Tsimring,et al.  Noise-induced dynamics in bistable systems with delay. , 2001, Physical review letters.

[25]  Zhongjin Guo,et al.  Steady state bifurcation of a periodically excited system under delayed feedback controls , 2012 .

[26]  T. Kofané,et al.  Deterministic and stochastic bifurcations in the Hindmarsh-Rose neuronal model. , 2013, Chaos.

[27]  Jian Peng,et al.  Synchronization and bifurcation in Limit Cycle oscillators with Delayed Couplings , 2011, Int. J. Bifurc. Chaos.

[28]  H. Poincaré.  Sur l'équilibre d'une masse fluide animée d'un mouvement de rotation , 1885, Bulletin astronomique.

[29]  Lefever,et al.  Sensitivity of a Hopf bifurcation to multiplicative colored noise. , 1986, Physical review letters.

[30]  Bin Deng,et al.  Delay-induced synchronization transitions in modular scale-free neuronal networks with hybrid electrical and chemical synapses , 2014 .

[31]  Haijun Jiang,et al.  Hopf bifurcation analysis for a model of single genetic negative feedback autoregulatory system with delay , 2013, Neurocomputing.

[32]  Huiqing Zhang,et al.  Stochastic bifurcations in a bistable Duffing-Van der Pol oscillator with colored noise. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[33]  J. Touboul.  Mean-field equations for stochastic firing-rate neural fields with delays: Derivation and noise-induced transitions , 2012 .

[34]  Irina A. Bashkirtseva,et al.  Noise-induced Chaos and Backward Stochastic bifurcations in the Lorenz Model , 2013, Int. J. Bifurc. Chaos.

[35]  Zhongkui Sun,et al.  Delay-induced stochastic bifurcations in a bistable system under white noise. , 2015, Chaos.

[36]  L S Tsimring,et al.  Dynamics of an ensemble of noisy bistable elements with global time delayed coupling. , 2003, Physical review letters.