Chaotic Behavior and Chaos Control for a Class of Complex Partial Differential Equations

Abstract

Systems of complex partial differential equations, which include the famous nonlinear Schrödinger, complex Ginzburg–Landau and Nagumo equations, as examples, are important from a practical point of view. These equations appear in many important fields of physics. The goal of this paper is to concentrate on this class of complex partial differential equations and study the fixed points and their stability analytically, the chaotic behavior and chaos control of their unstable periodic solutions. The presence of chaotic behavior in this class is verified by the existence of positive maximal Lyapunov exponent.The problem of chaos control is treated by applying the method of Pyragas. Some conditions on the parameters of the systems are obtained analytically under which the fixed points are stable (or unstable).

Simple driven chaotic oscillators with complex variables

“…[6][7][8][9][10] These systems have wide applications in physics, in areas as diverse as fluids, quantum mechanics, superconductivity, plasma physics, optical systems, astrophysics, and high-energy accelerators. 3,[11][12][13][14][15] Motivated by this more recent trend in research, and in line with previous work searching for simple systems of a given form that exhibit chaotic behavior, [16][17][18][19][20] this work began with a search for simple driven chaotic oscillators with a complex variable z, of the form ż + f͑z͒ = e i⍀t . This form is equivalent, with z = x + iy and f͑z͒ = u͑x , y͒ + iv͑x , y͒, where u and v are real functions, to the driven two-dimensional system, ẋ = − u͑x,y͒ + cos ⍀t, ͑1͒ ẏ = − v͑x,y͒ + sin ⍀t, which can be recast as the autonomous three-dimensional system, ẋ = − u͑x,y͒ + cos ⍀t, ẏ = − v͑x,y͒ + sin ⍀t, ͑2͒ ṫ = 1.…”

Adaptive Identifier for Uncertain Complex Nonlinear Systems Based on Continuous Neural Networks

“…In this field, several theoretical studies have been developed (involving nonlinear oscillators, often with periodic forcing) [31]- [33]. There are a number of applications in physics, in areas as diverse as fluids, quantum mechanics, superconductivity, plasma physics, optical systems, astrophysics, and high-energy accelerators, that can be described by this class of systems [29], [34], [35].…”

Analysis of chaotic and hyperchaotic conservative complex nonlinear systems

“…In recent years chaotic and hyperchaotic dissipative complex nonlinear systems have been proposed and studied in the literature [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]18]. However, there also are many interesting cases of conservative complex nonlinear systems have not yet been as actively explored.…”