Qualitative behaviors of the continuous-time chaotic dynamical systems describing the interaction of waves in plasma

In this paper, a new Lorenz-like chaotic system describing the interaction of three resonantly coupled waves in plasma is studied. Explicit ultimate boundedness and global attraction domain are derived according to stability theory of dynamical systems. The innovation of the paper is that this paper not only proves this chaotic system is globally bounded for the parameters of this system but also gives a family of mathematical expressions of global exponential attractive sets for this system with respect to the parameters of this system. Furthermore, the exponential rate of the trajectories is also obtained. Finally, numerical localization of attractor is presented.

[1]  Nikolay V. Kuznetsov,et al.  Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor , 2014 .

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

[3]  Xing-yuan Wang,et al.  Dynamic analysis of the fractional-order Liu system and its synchronization. , 2007, Chaos.

[4]  Da Lin,et al.  The dynamical analysis of a new chaotic system and simulation , 2014 .

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

[6]  Gennady A. Leonov,et al.  General existence conditions of homoclinic trajectories in dissipative systems. Lorenz, Shimizu–Morioka, Lu and Chen systems , 2012 .

[7]  E. M. Elsayed,et al.  Solutions of rational difference systems of order two , 2012, Math. Comput. Model..

[8]  G. A. Leonov,et al.  Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations , 2014, 1410.2016.

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

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

[11]  G. Leonov,et al.  Attraktorlokalisierung des Lorenz-Systems , 1987 .

[12]  Nikolay V. Kuznetsov,et al.  Hidden attractor in smooth Chua systems , 2012 .

[13]  Nikolay V. Kuznetsov,et al.  Hidden attractors in Dynamical Systems. From Hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman Problems to Hidden Chaotic Attractor in Chua Circuits , 2013, Int. J. Bifurc. Chaos.

[14]  Mikhail I. Rabinovich,et al.  Stochastic self-oscillations and turbulence , 1978 .

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

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

[17]  G. Leonov,et al.  Localization of hidden Chuaʼs attractors , 2011 .

[18]  Pei Yu,et al.  A constructive proof on the existence of globally exponentially attractive set and positive invariant set of general Lorenz family , 2009 .

[19]  Xing-Yuan Wang,et al.  A new image encryption algorithm based on non-adjacent coupled map lattices , 2015, Appl. Soft Comput..

[20]  Gennady A. Leonov,et al.  The tricomi problem for the Shimizu-Morioka dynamical system , 2012 .

[21]  Gennady A. Leonov,et al.  Bounds for attractors and the existence of homoclinic orbits in the lorenz system , 2001 .

[22]  Xing-yuan Wang,et al.  Projective synchronization of fractional order chaotic system based on linear separation , 2008 .

[23]  Pan Zheng,et al.  New results of the ultimate bound on the trajectories of the family of the Lorenz systems , 2015 .

[24]  Xiaofeng Liao,et al.  On the global boundedness of the Lü system , 2016, Appl. Math. Comput..

[25]  T. N. Mokaev,et al.  Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion Homoclinic orbits, and self-excited and hidden attractors , 2015 .

[26]  Xingyuan Wang,et al.  A hyperchaos generated from Lorenz system , 2008 .

[27]  Fuchen Zhang,et al.  Dynamical behaviors of the chaotic Brushless DC motors model , 2016, Complex..

[28]  Nikolay V. Kuznetsov,et al.  Algorithms for finding hidden oscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chua’s circuits , 2011 .

[29]  Gennady A. Leonov,et al.  Necessary and sufficient conditions of the existence of homoclinic trajectories and cascade of bifurcations in Lorenz-like systems: birth of strange attractor and 9 homoclinic bifurcations , 2016 .

[30]  Zhang Huaguang,et al.  A new hyperchaotic system and its circuit implementation , 2010 .

[31]  Gennady A. Leonov,et al.  Lyapunov's direct method in the estimation of the Hausdorff dimension of attractors , 1992 .

[32]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and an Introduction to Chaos , 2003 .

[33]  Xing-Yuan Wang,et al.  A symmetric image encryption algorithm based on mixed linear-nonlinear coupled map lattice , 2014, Inf. Sci..

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