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Estimating the ultimate bound and positively invariant set for the hyperchaotic Lorenz-Haken system. (English) Zbl 1197.37034
Summary: To estimate the ultimate bound and positively invariant set for a dynamic system is an important but quite challenging task in general. In this paper, we attempt to investigate the ultimate bound and positively invariant set for the hyperchaotic Lorenz-Haken system using a technique combining the generalized Lyapunov function theory and optimization. For the Lorenz-Haken system, we derive a four-dimensional ellipsoidal ultimate bound and positively invariant set. Furthermore, the two-dimensional parabolic ultimate bound with respect to $x-z$ is established. Finally, numerical results to estimate the ultimate bound are also presented for verification. The results obtained in this paper are important and useful in control, synchronization of hyperchaos and their applications. Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

##### MSC:
 37D45 Strange attractors, chaotic dynamics 34C11 Qualitative theory of solutions of ODE: growth, boundedness 34C14 Symmetries, invariants (ODE) 34D35 Stability of manifolds of solutions of ODE
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##### References:
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