The primary resonance of van der Pol oscillator under fractional-order delayed negative feedback and forced excitation is studied. Firstly, the approximate analytical solution is obtained based on the averaging method, and it could be found that the fractionalorder delayed feedback has not only the property of delayed velocity feedback but also that of delayed displacement feedback. Moreover, the amplitude-frequency equation for the steady-state solution is established, and its stability conditions are also obtained. Then, the results of the approximate analytical solution and numerical integration are compared and analyzed. The agreement between the two methods is very high, so that the correctness and accuracy of the approximate analytical solution are verified. Finally, the effects of all the parameters in the fractional-order delayed feedback on the amplitude-frequency curves are analyzed. It could be concluded that fractional-order delayed feedback has important influences on the dynamical behavior of van der Pol oscillator, which is very significant to the optimization and control of a similar system.
This paper focuses on the bounds of the Lyapunov exponents for fractional differential systems, where the fractional derivatives are Riemann–Liouville and Caputo fractional derivatives with the exponential kernel. First, the essential properties of fractional integral and derivatives with the exponential kernel are given. Then the continuous dependence of solutions on the initial value problems of some particular parameters is studied. On these bases, the bounds of Lyapunov exponents are estimated. Finally, the theoretical results are illustrated by numerical simulations.
<abstract><p>In this paper, we study the existence, uniqueness, and stability of the solution of the fractional differential system with the generalized fractional derivative. First, the solution of the generalized fractional differential system is obtained by the transformation method. Based on the fixed point theorems, we establish the existing and unique theories of the solution. Furthermore, the sufficient criteria of local stabilities of one-dimensional, two-dimensional, and $ n $ -dimensional linear generalized fractional differential systems are dealt with. In addition, the linearization and stability theorems of the nonlinear generalized fractional differential systems are discussed. Finally, we take the generalized fractional Chen system as an example to illustrate the correctness of the theoretical analysis.</p></abstract>
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