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Contraction theory regards the convergence between two arbitrary system trajectories. In this article we have introduced partial contraction theory as an extension of contraction theory to analyze coupled identical fractional order systems. It can, also, be applied to study the synchronization phenomenon in networks of various structures and with arbitrary number of systems. We have used partial contraction theory to derive exact and global results on synchronization and antisynchronization of fractional order systems.

Since the Dutch researcher, Christiaan Huygens, initiated the study of synchronization phenomenon in the

In the last two decades, scientists have used fractional differential equations to model several physical phenomena. For the recent history of fractional calculus and state space representation, reference can be made to [

The remainder of this paper is organized as follows: in Section

Basically, a nonlinear time-varying dynamic system is said to be

The basic definition of contraction theory for FOSs is summarized and the details can be found in [

First, consider the integer order system

If matrix

See [

For the given system (

Note that, by matrix

Consider now, an FOS:

Assume that

See [

Note that the condition on

Suppose that the conditions of Theorem

If matrix

See [

For the given FOS (

It is worth mentioning that, if

Consider the linear time-invariant (LTI) FOS:

The partial contraction theory was first introduced during the study of network synchronization [

Consider an FOS

Two particular solutions of the virtual y-system are

The original FOS (

A convex combination of

Consider the convex combination

The notion of a virtual contracting system can be applied to control problems. For example, consider a nonlinear fractional order control system:

Let

If the function

A particular solution of second system is

Consider two coupled identical fractional order financial systems [

Therefore,

and

The system

Synchronization of states of two financial FOSs with parameters

The extension of Theorem

Consider two identical systems which are paired together in a bidirectional (two-way) coupling method of the form

From the coupled system, we have

Researchers have given varying definitions to synchronization under different contexts. In this study, synchronization or

Now, one can state the following theorem which is more general than Theorems

Consider two systems coupled in an arbitrary manner. If there is a contraction function,

Let

(1) Theorems

(2) For a network of

(3) For a network containing

The forced Duffing FOS [

Therefore,

Phase synchronization of three Duffing systems for initial points

Complete synchronization of three Duffing systems for initial points

Divergence of three coupled Duffing systems for

If the vector function

From (

Two unforced Duffing systems

Antisynchronization of two unforced Duffing systems and the plot of errors

The stable limit-cycle of

We used partial contraction method to express and prove the conditions to reach synchronization and antisynchronization of two FOSs. In comparison with previous methods which were based on linearization, the results here are exact and global. We also used the partial contraction method to study networks with various structure and arbitrary number of systems.

The authors declare that they have no conflicts of interest.

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.