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We investigate synchronizing fractional-order Volta chaotic systems with nonidentical orders in finite time. Firstly, the fractional chaotic system with the same structure and different orders is changed to the chaotic systems with identical orders and different structure according to the property of fractional differentiation. Secondly, based on the lemmas of fractional calculus, a controller is designed according to the changed fractional chaotic system to synchronize fractional chaotic with nonidentical order in finite time. Numerical simulations are performed to demonstrate the effectiveness of the method.

Fractional calculus, starting from some speculations of G. W. Leibniz (1695, 1697) and L. Euler (1730), has a history of over 300 years old. But its application to physics and engineering has attracted lots of attention only in the recent decades. Up until now, people have investigated and developed many methods to synchronize fractional dynamical systems with identical order, for example, one-way coupling method [

In general, the order of the drive fractional chaotic system may not be equal with the order of the response fractional chaotic system. It is necessary to study synchronizing fractional chaotic system with nonidentical order. In [

Generally, synchronizing error systems converge to zero in infinite time. But in some cases such as communication, the synchronizing errors are usually demanded to converge to zero in finite time. Finite-time synchronizing integer-order chaotic has been studied [

Motivated by the above discussion, we propose a new approach to design a controller to realize finite-time synchronizing fractional order chaotic system with nonidentical orders based on the properties of fractional differentiation. The theorem is easy to understand. Numerical simulations are used to verify the effectiveness of this approach.

The rest of the paper is organized as follows. In Section

There are three commonly used definitions of the fractional-order differential operator: Grunwald-Letnikov, Riemann-Liouville, and Caputo definitions. In this paper, we study the stability of fractional system based on the Caputo definition.

The Caputo definition of fractional order can be expressed as [

Suppose

Consider

For

In this section, we investigate finite-time synchronizing fractional-order chaotic Volta systems with nonidentical orders.

The fractional-order Volta’s system is depicted by

When system parameters

Define the system (

According to Lemma

Define the error variables as

If the controller satisfies

Substituting (

In this section, the simulation results are carried out to show the effectiveness of the designed controller.

Take

Time responses of state variables in the response system and the drive system.

Synchronization errors

In this paper, we study the finite-time synchronization of fractional chaotic systems with nonidentical orders. A new approach is proposed to design the controller. The approach is not only simple but also easy to understand, which can broad the approach of synchronizing fractional chaotic system. Numerical simulations show the effectiveness of the approach.