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

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 |