Chaotic and hyperchaotic attractors of a complex nonlinear system. (English) Zbl 1131.37036

Mahmoud, Gamal M.; Al-Kashif, M. A.; Farghaly, A. A.

J. Phys. A, Math. Theor. 41, No. 5, Article ID 055104, 12 p. (2008).

Summary: We introduce a complex nonlinear hyperchaotic system which is a five-dimensional system of nonlinear autonomous differential equations. This system exhibits both chaotic and hyperchaotic behavior and its dynamics is very rich. Based on the Lyapunov exponents, the parameter values at which this system has chaotic, hyperchaotic attractors, periodic and quasi-periodic solutions and solutions that approach fixed points are calculated. The stability analysis of these fixed points is carried out. The fractional Lyapunov dimension of both chaotic and hyperchaotic attractors is calculated. Some figures are presented to show our results. Hyperchaos synchronization is studied analytically as well as numerically, and excellent agreement is found.

MSC:

37D45	Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37C10	Dynamics induced by flows and semiflows
34C28	Complex behavior and chaotic systems of ordinary differential equations
34C25	Periodic solutions to ordinary differential equations
34C08	Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
34D45	Attractors of solutions to ordinary differential equations
28A78	Hausdorff and packing measures
93D15	Stabilization of systems by feedback

Keywords:

five-dimensional system of nonlinear autonomous differential equations; Lyapunov exponents; periodic and quasi-periodic solutions; fixed points; stability analysis; hyperchaos synchronization