Abstract
In this paper, bifurcations and synchronization of a fractional-order Bloch system are studied. Firstly, the bifurcations with the variation of every order and the system parameter for the system are discussed. The rich dynamics in the fractional-order Bloch system including chaos, period, limit cycles, period-doubling, and tangent bifurcations are found. Furthermore, based on the stability theory of fractional-order systems, the adaptive synchronization for the system with unknown parameters is realized by designing appropriate controllers. Numerical simulations are carried out to demonstrate the effectiveness and flexibility of the controllers.
Highlights
Nowadays, fractional calculus is a hot topic in the research field
It is well known that the initial conditions for the fractional differential equations with Caputo derivatives take on the same form as those for the integer-order ones, which is very suitable for practical problems
As the initial conditions for the fractional differential equations with Caputo derivatives take on the same form as those for the integer-order ones, we will use the Caputo definition for the fractional derivatives in this paper
Summary
Fractional calculus is a hot topic in the research field. It is well known that fractional calculus has an long history with classical calculus. In [17], for the fractional-order Bloch system, the chaotic dynamics including the chaotic attractors in different system parameters sets, bifurcations with the derivative order in commensurate-order case, were analyzed. Rich dynamics such as period-doubling and subharmonic cascade routes to chaos were found for the system in the commensurate-order case. Based on these results, we want to know the bifurcations of the fractional-order Bloch system with the variation of every order in incommensurate-order case as well as every system parameter.
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