The Real-Valued Maxwell-Bloch Equations with Controls: From a Hamilton-Poisson System to a Chaotic One

Applying parametric controls to the 3D real-valued Maxwell–Bloch equations, we obtain a Hamilton–Poisson system, a dissipative system with chaotic behavior, and a transitional system between the aforementioned states, which is a conservative system that has only one constant of motion. In the Hamiltonian case, we present some connections of the energy-Casimir mapping with the equilibrium states and the existence of the homoclinic orbits. We study the stability of the equilibrium points of the transitional system and the dissipative system. Furthermore, we point out the chaotic behavior of the introduced system.

[1]  J. P. Lasalle.  Some Extensions of Liapunov's Second Method , 1960 .

[2]  T. Bînzar,et al.  A Rikitake type system with one control , 2013 .

[3]  Julien Clinton Sprott,et al.  A dynamical system with a strange attractor and invariant tori , 2014 .

[4]  Gregory L. Baker,et al.  Chaotic Dynamics: An Introduction , 1990 .

[5]  J. Marsden,et al.  Introduction to mechanics and symmetry , 1994 .

[6]  Guanrong Chen,et al.  Bifurcation Analysis of Chen's equation , 2000, Int. J. Bifurc. Chaos.

[7]  Guanrong Chen,et al.  From Chaos to Order - Perspectives and Methodologies in Controlling Chaotic Nonlinear Dynamical Systems , 1993 .

[8]  Darryl D. Holm,et al.  Multiple lie-poisson structures, reductions, and geometric phases for the Maxwell-Bloch travelling wave equations , 1992 .

[9]  J. Yorke,et al.  Chaotic behavior of multidimensional difference equations , 1979 .

[10]  Jaume Llibre,et al.  Heteroclinic, Homoclinic and Closed Orbits in the Chen System , 2016, Int. J. Bifurc. Chaos.

[11]  G. Tigan.  Degenerate with respect to parameters fold-Hopf bifurcations , 2016 .

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

[13]  On some dynamical and geometrical properties of the Maxwell-Bloch equations with a quadratic control , 2013, 1402.5942.

[14]  Guanrong Chen,et al.  A New Chaotic System and its Generation , 2003, Int. J. Bifurc. Chaos.

[15]  Lee Sun-Jin.  From Chaos to Order , 2011 .

[16]  Cees Diks,et al.  E&F Chaos: A User Friendly Software Package for Nonlinear Economic Dynamics , 2008 .

[17]  Guanrong Chen,et al.  YET ANOTHER CHAOTIC ATTRACTOR , 1999 .

[18]  Tudor Bînzar,et al.  ON A HAMILTONIAN VERSION OF CONTROLS DYNAMIC FOR A DRIFT-FREE LEFT INVARIANT CONTROL SYSTEM ON G4 , 2012 .

[19]  Daizhan Cheng,et al.  A New Chaotic System and Beyond: the Generalized Lorenz-like System , 2004, Int. J. Bifurc. Chaos.

[20]  Anania Aron,et al.  On a Hamiltonian Version of the Rikitake System , 2009, SIAM J. Appl. Dyn. Syst..

[21]  Julien Clinton Sprott,et al.  A Proposed Standard for the Publication of New Chaotic Systems , 2011, Int. J. Bifurc. Chaos.

[22]  Periodic orbits in the case of a zero eigenvalue , 2007, 0705.2361.

[23]  Cristian Lazureanu,et al.  A Rikitake Type System with quadratic Control , 2012, Int. J. Bifurc. Chaos.

[24]  Hermann Haken,et al.  At least one Lyapunov exponent vanishes if the trajectory of an attractor does not contain a fixed point , 1983 .

[25]  Julien Clinton Sprott,et al.  Coexistence of Point, periodic and Strange attractors , 2013, Int. J. Bifurc. Chaos.

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

[27]  Xiyin Liang,et al.  Force Analysis of Qi Chaotic System , 2016, Int. J. Bifurc. Chaos.

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

[29]  A. Rucklidge.  Chaos in models of double convection , 1992, Journal of Fluid Mechanics.