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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.

Nowadays, fractional calculus is a hot topic in the research field. It is well known that fractional calculus has an equally long history with classical calculus. It did not attract enough attention for the absence of geometrical interpretation and applications at the initial stage of development. As the development of technology and science continues, fractional calculus has been applied in many fields, such as control theory, dynamics, mathematics, mechanics, and physics [

As the research of fractional calculus moves along, many nonlinear systems with fractional orders are proposed and investigated. The chaos and bifurcations which are observed in integer-order systems are also found in fractional-order ones, such as fractional versions of Duffing system, Lorenz system, and Chen system [

Motivated by the above discussed, in this paper, the bifurcations with the derivative order in incommensurate-order case and system parameters are studied firstly. A series of period-doubling bifurcations and tangent bifurcations are obtained by numerical simulations. Meanwhile, different chaotic and periodic attractors are also observed. Furthermore, based on the stability theory of fractional-order systems, the adaptive synchronization of the fraction-order systems with uncertain parameters is realized by designing appropriate controllers. Numerical simulations are carried out to demonstrate the effectiveness and flexibility of the controllers.

The paper is organized as follows. In Section

There are many definitions for the general fractional derivative. The three most frequently used ones are the Grunwald-Letnikov definition and the Riemann-Liouville and the Caputo definitions. 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. Therefore, we will use the Caputo definition for the fractional derivatives in this paper.

The Caputo fractional derivative is defined as follows:

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.

In the following, we will give the definitions of commensurate-order and incommensurate-order fractional-order systems [

For a fractional-order system, which can be described by

Compared with the numerical algorithm for solving an ordinary differential equation, the numerical solution of a fractional differential equation is not easy to obtain. There are two approximation methods which can frequently be used for numerical computation on chaos and bifurcations with fractional differential equations. One is an improved version of Adams-Bashforth-Moulton algorithm based on the predictor-correctors scheme [

Simulation of fractional-order systems using the time domain methods is complicated and, due to long memory characteristics of these systems, requires a very long simulation time but on the other hand, it is more accurate [

The Bloch system is usually used to describe an ensemble of spins. The integer-order and fractional-order Bloch systems were studied in [

The fractional-order Bloch system can be described as follows:

The chaotic attractor of system (

When the system parameters are fixed as

The bifurcation and the corresponding LLE diagrams of system (

The bifurcation diagram

The corresponding LLE

The phase portraits and time series diagrams of system (

The phase portrait when

The time series diagram when

The phase portrait when

The time series diagram when

The phase portrait when

The time series diagram when

The phase portrait when

The time series diagram when

When the system parameters are fixed as the same values, and the orders

The bifurcation and the corresponding LLE diagrams of system (

The bifurcation diagram

The corresponding LLE

When the other parameters are taken as

The bifurcation and the corresponding LLE diagrams of system (

The bifurcation diagram

The corresponding LLE

The phase portraits and time series diagrams of system (

The phase portrait when

The time series diagram when

The phase portrait when

The time series diagram when

The phase portrait when

The time series diagram when

The phase portrait when

The time series diagram when

When

The bifurcation and the corresponding LLE diagrams of system (

The bifurcation diagram

The corresponding LLE

The bifurcation and the corresponding LLE diagrams of system (

The bifurcation diagram for

The corresponding LLE for

The bifurcation diagram for

The corresponding LLE for

The bifurcation diagram for

The corresponding LLE for

In this section, the adaptive synchronization for system (

For simplicity, system (

Adaptive synchronization between systems (

Firstly, controllers (

In numerical simulations, the real values of the uncertain parameters are

The synchronization results of the numerical simulation.

The synchronization error curves

The identification curves of the unknown parameters

In this paper, bifurcations and synchronization of a fractional-order Bloch system have been studied. Firstly, bifurcations of the system as every order is varied are obtained by the numerical simulations. Period-doubling and tangent bifurcations can be observed in the system. Meanwhile, bifurcations of the system with the variation of every system parameter are also determined by numerical computation. Besides the period-doubling and tangent bifurcations, limit cycles coexisting are also found for the fractional-order Bloch system. From these results, it can be seen clearly that the derivative orders are also the important parameters which affect the dynamics of the fractional-order Bloch system. The bifurcations of the system in such a parameter set are demonstrated in detail. Finally, 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.

The authors declare that they have no conflicts of interest.

This work is supported by the National Natural Science Foundation of China (NSFC) under Grant nos. 11332008 and 11672218.