## Fractional Rayleigh-Duffing-like system and its synchronization.(English)Zbl 1268.34089

Summary: This paper presents a periodically driven Rayleigh-Duffing-like system with the function $$x|x|$$. It is proven via the Melnikov function method that the quadratic function $$x|x|$$ induces Smale horseshoes to the Rayleigh-Duffing-like system. The Rayleigh-Duffing-like oscillator with fractional order is also discussed, and results of computer simulation demonstrate the chaotic dynamic behaviors of the system. Furthermore, two fractional Rayleigh-Duffing-like systems are synchronized by active control technology, the method based on state observer and nonlinear feedback method. Numerical results validate the effectiveness and applicability of the proposed synchronization schemes.

### MSC:

 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations 34D06 Synchronization of solutions to ordinary differential equations 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 34A08 Fractional ordinary differential equations 34E10 Perturbations, asymptotics of solutions to ordinary differential equations

### Software:

Sprott's Software
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### References:

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