Modeling nonlinear dissipative chemical dynamics by a forced modified Van der Pol-Duffing oscillator with asymmetric potential: Chaotic behaviors predictions

Abstract This paper addresses the issues of nonlinear chemical dynamics modeled by a modified Van der Pol-Duffing oscillator with asymmetric potential. The Melnikov method is utilized to analytically determine the domains boundaries where Melnikov’s chaos appears in chemical oscillations. Routes to chaos are investigated through bifurcations structures, Lyapunov exponent, phase portraits and Poincare section. The effects of parameters in general and in particular the effect of the constraint parameter β which shows the difference between a nonlinear chemical dynamics order two differential equation and ordinary Van der Pol-Duffing equation are analyzed. Results of analytical investigations are validated and complemented by numerical simulations.

[1]  C. H. Miwadinou,et al.  Nonlinear dynamics of a $$\varvec{\phi ^6}-$$ϕ6- modified Duffing oscillator: resonant oscillations and transition to chaos , 2017 .

[2]  P. Cai,et al.  Hopf bifurcation and chaos control in a new chaotic system via hybrid control strategy , 2017 .

[3]  Dynamics of a biological system with time-delayed noise , 2012 .

[4]  J J Zebrowski,et al.  Nonlinear oscillator model reproducing various phenomena in the dynamics of the conduction system of the heart. , 2007, Chaos.

[5]  C. H. Miwadinou,et al.  Effect of Nonlinear Dissipation on the Basin Boundaries of a Driven Two-Well Modified Rayleigh-Duffing Oscillator , 2015, Int. J. Bifurc. Chaos.

[6]  C. H. Miwadinou,et al.  Active Control of the Parametric Resonance in the Modified Rayleigh-Duffing Oscillator , 2013, 1303.0536.

[7]  J. Boissonade,et al.  Transitions from bistability to limit cycle oscillations. Theoretical analysis and experimental evidence in an open chemical system , 1980 .

[8]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[9]  Paul Woafo,et al.  On the dynamics of Rayleigh beams resting on fractional-order viscoelastic Pasternak foundations subjected to moving loads , 2016 .

[10]  Michael Menzinger,et al.  Stirring Effects and Phase-Dependent Inhomogeneity in Chemical Oscillations: The Belousov-Zhabotinsky Reaction in a CSTR , 1997 .

[11]  Kenneth Showalter,et al.  Nonlinear Chemical Dynamics: Oscillations, Patterns, and Chaos , 1996 .

[12]  Raymond Kapral,et al.  Slow manifold structure and the emergence of mixed-mode oscillations , 1997, chao-dyn/9706029.

[13]  Paul Woafo,et al.  Appearance of horseshoes chaos on a buckled beam controlled by disseminated couple forces , 2011 .

[14]  R. J. Field,et al.  Oscillations and Traveling Waves in Chemical Systems , 1985 .

[15]  Britton Chance,et al.  Biological and biochemical oscillators , 1973 .

[16]  A. Nayfeh.  Introduction To Perturbation Techniques , 1981 .

[17]  K. Tornheim.  Are Metabolic Oscillations Responsible for Normal Oscillatory Insulin Secretion? , 1997, Diabetes.

[18]  Regular and chaotic behaviors of plasma oscillations modeled by a modified Duffing equation , 2008 .

[19]  S. Sarwar,et al.  Stability analysis, dynamical behavior and analytical solutions of nonlinear fractional differential system arising in chemical reaction , 2017 .

[20]  C. H. Miwadinou,et al.  Melnikov Chaos in a Modified Rayleigh-Duffing Oscillator with ϕ6 Potential , 2015, Int. J. Bifurc. Chaos.

[21]  Irving R. Epstein,et al.  Oscillations and waves in metal-ion-catalyzed bromate oscillating reactions in highly oxidized states , 1993 .

[22]  G. Gentile,et al.  Stable dynamics in forced systems with sufficiently high/low forcing frequency. , 2015, Chaos.

[23]  C. H. Miwadinou,et al.  Analysis of Multiresonance and Chaotic Behavior of the Polarization in Materials Modeled by a Duffing Equation with Multifrequency Excitations , 2014 .

[24]  J. B. Chabi Orou,et al.  Nonlinear dynamics of plasma oscillations modeled by an anharmonic oscillator , 2008 .

[25]  Asymmetric Duffing Equation and the Appearance of “Chaos” , 1987 .

[26]  S. Rajasekar,et al.  Analytical estimates of the effect of amplitude modulated signal in nonlinearly damped Duffing-vander Pol oscillator , 2017 .

[27]  K. Sriram,et al.  Complex dynamics in the Oregonator model with linear delayed feedback. , 2008, Chaos.

[28]  G. Litak,et al.  Melnikov's criteria, parametric control of chaos, and stationary chaos occurrence in systems with asymmetric potential subjected to multiscale type excitation. , 2011, Chaos.

[29]  N. Berman,et al.  Oscillations of lactate released from islets of Langerhans: evidence for oscillatory glycolysis in beta-cells. , 1992, The American journal of physiology.

[30]  W. Peltier,et al.  Deterministic chaos in the Belousov-Zhabotinsky reaction: Experiments and simulations. , 1993, Chaos.

[31]  Vadim S. Anishchenko,et al.  Modeling chemical reactions by forced limit-cycle oscillator: synchronization phenomena and transition to chaos , 2003 .

[32]  P Woafo,et al.  Analysis of tristable energy harvesting system having fractional order viscoelastic material. , 2015, Chaos.

[33]  J. B. Chabi Orou,et al.  Nonlinear dynamics and strange attractors in the biological system , 2007 .

[34]  J. Reich,et al.  Energy metabolism of the cell : a theoretical treatise , 1981 .

[35]  C. Hayashi,et al.  Nonlinear oscillations in physical systems , 1987 .