Qualitative dynamical analysis of chaotic plasma perturbations model

Abstract In this work, an analytical framework to understand nonlinear dynamics of plasma perturbations model is introduced. In particular, we analyze the model presented by Constantinescu et al. [20] which consists of three coupled ODEs and contains three parameters. The basic dynamical properties of the system are first investigated by the ways of bifurcation diagrams, phase portraits and Lyapunov exponents. Then, the normal form technique and perturbation methods are applied so as to the different types of bifurcations that exist in the model are investigated. It is proved that pitcfork, Bogdanov–Takens, Andronov–Hopf bifurcations, degenerate Hopf and homoclinic bifurcation can occur in phase space of the model. Also, the model can exhibit quasiperiodicity and chaotic behavior. Numerical simulations confirm our theoretical analytical results.

[1]  James D. Meiss,et al.  Differential dynamical systems , 2007, Mathematical modeling and computation.

[2]  Pei Yu,et al.  Normal Forms, Melnikov Functions and Bifurcations of Limit Cycles , 2012 .

[3]  A. M. A. El-Sayed,et al.  Further nonlinear dynamical analysis of simple jerk system with multiple attractors , 2017 .

[4]  Y. Y. Chen,et al.  Homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators by the hyperbolic perturbation method , 2009 .

[5]  Xing-yuan Wang,et al.  Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control , 2009 .

[6]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[7]  Yaolin Jiang,et al.  Computation of universal unfolding of the double-zero bifurcation in -symmetric systems by a homological method , 2013 .

[8]  Guanrong Chen,et al.  From Chaos To Order Methodologies, Perspectives and Applications , 1998 .

[9]  Jinhu Lu,et al.  A New Chaotic Attractor Coined , 2002, Int. J. Bifurc. Chaos.

[10]  Nikolay V. Kuznetsov,et al.  On differences and similarities in the analysis of Lorenz, Chen, and Lu systems , 2014, Appl. Math. Comput..

[11]  O. Rössler.  An equation for continuous chaos , 1976 .

[12]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[13]  Peter Stavroulakis,et al.  Chaos Applications in Telecommunications , 2005 .

[14]  K. Y. Sze,et al.  A hyperbolic Lindstedt–Poincaré method for homoclinic motion of a kind of strongly nonlinear autonomous oscillators , 2009 .

[15]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[16]  H. Zohm,et al.  RMP ELM suppression analysis by means of a low-dimensional model system for quasi-periodic plasma perturbations , 2011 .

[17]  Alejandro J. Rodríguez-Luis,et al.  A bifurcation analysis of a simple electronic circuit , 2005 .

[18]  Chuandong Li,et al.  Hopf bifurcation and chaos in macroeconomic models with policy lag , 2005 .

[19]  Ljupco Kocarev,et al.  Chaos-Based Cryptography - Theory, Algorithms and Applications , 2011, Chaos-Based Cryptography.

[20]  Gennady A. Leonov,et al.  Existence criterion of homoclinic trajectories in the Glukhovsky–Dolzhansky system , 2015 .

[21]  Fuchen Zhang,et al.  Further Results on Ultimate Bound on the Trajectories of the Lorenz System , 2016 .

[22]  Denis de Carvalho Braga,et al.  Lyapunov Coefficients for Degenerate Hopf Bifurcations , 2007, 0709.3949.

[23]  L. Chua.  Memristor-The missing circuit element , 1971 .

[24]  Ahmed M. A. El-Sayed,et al.  Dynamical behavior, chaos control and synchronization of a memristor-based ADVP circuit , 2013, Commun. Nonlinear Sci. Numer. Simul..

[25]  Fuchen Zhang,et al.  Bounds for a new chaotic system and its application in chaos synchronization , 2011 .

[26]  Y. Kuznetsov.  Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.

[27]  Julien Clinton Sprott,et al.  Some simple chaotic jerk functions , 1997 .