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On stability, persistence, and Hopf bifurcation in fractional order dynamical systems. (English) Zbl 1175.37084

Motivation for the investigations of fractional order dynamical systems is explained by rich applications in mathematical biology [A. M. Nakhushev, Equations of mathematical biology. Moskva: Vysshaya Shkola (1995; Zbl 0991.35500)]. Therefore it is essential to study bifurcation, stability and chaos in fractional order dynamical systems. Since persistence and seasonability are important concepts in biology, it is relevant to study the persistence in biologically motivated nonautonomous fractional equations.

In this article after some relations between complex adaptive systems and fractional mathematics sufficient conditions for the persistence of some biologically inspired fractional nonautonomous equations are derived. Then Poincaré-Andronov-Hopf bifurcation in fractional order systems is studied. Also the local stability questions in functional equations are considered.

MSC:
37N25Dynamical systems in biology
92B99Mathematical biology
37J40Perturbations, normal forms, small divisors, KAM theory, Arnol’d diffusion
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