# Partial Contraction Analysis of Coupled Fractional Order Systems

Authors: Ahmad Ruzitalab; Mohammad Hadi Farahi; Gholamhossien Erjaee
Journal: Journal of Applied Mathematics
 (2018)

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## Abstract

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.


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## Body

## 1. Introduction

Since the Dutch researcher, Christiaan Huygens, initiated the study of synchronization phenomenon in the17th century [1], many researchers of various fields, such as mathematics [2–4], robotics [5], electronics [6], neuroscience, and biology [7, 8], have investigated coupled oscillators and the stability of synchrony between coupled systems. Due to the significance of stability in the control theory [9] and synchronization phenomena, many techniques have been proposed to examine this property [10–12]; these include the contraction theory which is a more recent tool for analyzing the stability and convergence behavior of nonlinear systems in state space form [13–15]. In his dissertation, Soon-Jo Chung focused on the synchronization of multiple dynamical systems using the contraction theory, with applications in the cooperative control of multiagent systems and synchronization of interconnected dynamics such as tethered formation flight. He used contraction theory to prove that a nonlinear control law stabilizing a single-tethered spacecraft can also stabilize arbitrarily large circular arrays of tethered spacecraft, as well as a three-spacecraft inline configuration [16].Partial contraction method is employed to investigate the dynamics of coupled nonlinear systems, based on contraction analysis. Partial contraction theory extends contraction theory to include convergence of specific properties and it gives a general device for investigating the stability of systems. It is particularly suitable for the study of synchronization behaviors. In his thesis, using partial contraction theory, Wei Wang studied the spontaneous synchronization behavior of nonlinear networked systems [17].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 [18] and [19], respectively. Because of its significant role in engineering and modern sciences [20, 21], the study of the stability of fractional order systems (FOSs) [22] and synchronization of FOSs has attracted much attention [6, 8]. In this article,partial contraction method was developed to investigate the dynamics of coupled FOSs, with emphasis on the study of FOSs synchronization.The remainder of this paper is organized as follows: in Section2, after a summary of the contraction theory, the partial contraction theory was introduced for FOSs. In Section 3, the theory was clarified by studying the synchronization of FOSs and then the analysis was generalized to study synchronization in networks of arbitrary number of nonlinear FOSs. Examples are given to illustrate the concept.


## 2. Contraction and Partial Contraction Analysis of Fractional Order Systems

Basically, a nonlinear time-varying dynamic system is said to becontracting, if the initial conditions are exponentially forgotten, that is, if the dynamics of the system does not depend on the initial conditions and all trajectories converge to their nominal movement exponentially. Thepartial contraction theory extends the applications of contraction theory to a network of FOSs. In the extension, convergence to the specific properties of the systems is considered.


### 2.1. Contraction Theory

The basic definition of contraction theory for FOSs is summarized and the details can be found in [23].First, consider the integer order system(1)x˙=fx,twhere x and f are n×1 state vector and vector function, respectively. Suppose that fx,t is continuously differentiable, (1) gives the relation(2)δx˙=∂f∂xx,tδxwhere δx is a virtual displacement between two neighboring trajectories of system (1). The rate of change of squared distance (δx)Tδx between two trajectories is defined as(3)dδxTδxdt=2δxTδx˙=2δxT∂f∂xδx.Theorem 1.
If matrix∂f/∂x is uniformly negative definite, all the solution trajectories of system (1) will converge exponentially to a single trajectory, irrespective of their initial conditions.Proof.
See [13].Definition 2.
For the given system (1), a region Ω⊆Rn of the state space is called a contraction (semicontraction) region, if the matrix ∂f/∂x is uniformly negative definite (negative semidefinite) in that region.Note that, by matrixG, being uniformly negative definite means that the symmetric part of matrix G (that is, 1/2(G+GT)) is negative definite; in other words,(4)12G+GT≤-βI<0,∃β>0,∀x,∀t≥0,and we mean aregion, an open connected set.Consider now, an FOS:(5)Dtαxt=fxtwhere x and f are like that of the integer order system, and Dtα indicates the fractional derivative of order α∈R+. In this article, the Riemann-Liouville fractional operator was used as the main derivation tool which for order α>0,  m-1<α<m,  m∈N, is defined as [23](6)DαaRLfx=DmJam-αfx,where the operator Jaα is defined on L1[a,b] by(7)Jaαfx=1Γα∫axfτx-τ1-αdτ,for a≤x≤b and is called the Riemann-Liouville fractional integral operator of order α. For simplicity, the left superscript RL and subscript a were omitted and Dtα was assumed to be the Riemann-Liouville α-order fractional derivative operator which was assumed to exist and be continuous.Theorem 3.
Assume thatα1,α2≥0. Moreover, let ϕ∈L1[a,b] and xt=Jaα1+α2ϕ. Then,(8)Dα1Dα2xt=Dα1+α2xt.Proof.
See [24]: Page 29.Note that the condition onxt implies a certain degree of smoothness and the fact that, as t→a, xt→0 sufficiently fast.Suppose that the conditions of Theorem3 hold. By applying Dt1-α to both sides of (5), one can write(9)x˙=Dt1-αfxtTherefore, the contraction condition for system (5) is stated as follows [23].Theorem 4.
If matrixG=DxαDt1-αfxtDxαx-1 is uniformly negative definite, all the solution trajectories of system (5) converge to a single trajectory, with exponential rate, irrespective of the initial conditions.Proof.
See [23].Definition 5.
For the given FOS (5), a region Ω⊆Rn of the state space is called contraction (semicontraction), if the matrix DxαDt1-αfxtDxαx-1 is uniformly negative definite (negative semidefinite) in that region. By convention, fxt and system (5) are called contracting function and contracting system, respectively.It is worth mentioning that, ifΩ=Rn, then global exponential convergence is guaranteed.Remark 6.
Consider the linear time-invariant (LTI) FOS:(10)Dtαx=Ax.Applying Dt1-α to both sides of (10), one can write(11)x˙=Dt1-αAx.Considering two neighboring trajectories of the above equation and the virtual displacement δx between them yields(12)δx˙=∂Dt1-αAx∂xδx.The rate of change of the squared distance (δx)Tδx between two neighboring trajectories of the system (10) is given by(13)dδxTδxdt=2δxTδx˙=2δxT∂Dt1-αAx∂xδx.Considering(14)Dt1-αAx=1Γ1-1-αddt∫0tAxτt-τ1-αdτ,we have(15)∂Dt1-αAx∂x=1Γαddt∫0tAt-τ1-αdτ=Atα-1ΓαSo, LTI system (10) is contracting (semicontracting) if Atα-1/Γ(α) in [t0,∞) is uniformly negative definite (negative semidefinite).


### 2.2. Partial Contraction Theory

The partial contraction theory was first introduced during the study of network synchronization [25]; since then, its applicability and flexibility have been proven in many fields. Now, the partial contraction theory for the FOSs, which is the base of this work, is introduced.Theorem 7.
Consider an FOS(16)Dtαx=fx,x,tand its auxiliary system(17)Dtαy=fy,x,twhich is contracting with respect to y. Then trajectories of the original x-system verify each smooth specific property which auxiliary system verifies.Proof.
Two particular solutions of the virtual y-system areyt=x(t) and the solution with the specific property. Given that the virtual system is contracting, the solution x(t) converges to the solution with the specific property.Definition 8.
The original FOS (16) is said to be partially contracting.Corollary 9.
A convex combination ofDtαx=fix,t, i=1,…,n, which are contracting with a common trajectory x0(t), is contracting.Proof.
Consider the convex combination(18)Dtαx=∑iαix,tfix,twhich has a common trajectory x0(t) (for instance, a common equilibrium), where αix,t≥0 and ∑iαix,t=1. Consider the contracting auxiliary system(19)Dtαy=∑iαix,tfiy,twith two particular solutions x(t) and x0(t). Therefore, all trajectories of the system converge to the trajectory x0(t).Remark 10.
The notion of a virtual contracting system can be applied to control problems. For example, consider a nonlinear fractional order control system:(20)Dtαx=fx,x,u,tand assume that it is desired to reach the state xd(t), using the control input u(x,xd,t) such that(21)Dtαxd=fxd,x,u,t.Now, it is enough to have a contracting auxiliary system:(22)Dtαy=fy,x,u,twith two particular solutions x(t) and xd(t) to guarantee the convergence of x to xd.


## 2.1. Contraction Theory

The basic definition of contraction theory for FOSs is summarized and the details can be found in [23].First, consider the integer order system(1)x˙=fx,twhere x and f are n×1 state vector and vector function, respectively. Suppose that fx,t is continuously differentiable, (1) gives the relation(2)δx˙=∂f∂xx,tδxwhere δx is a virtual displacement between two neighboring trajectories of system (1). The rate of change of squared distance (δx)Tδx between two trajectories is defined as(3)dδxTδxdt=2δxTδx˙=2δxT∂f∂xδx.Theorem 1.
If matrix∂f/∂x is uniformly negative definite, all the solution trajectories of system (1) will converge exponentially to a single trajectory, irrespective of their initial conditions.Proof.
See [13].Definition 2.
For the given system (1), a region Ω⊆Rn of the state space is called a contraction (semicontraction) region, if the matrix ∂f/∂x is uniformly negative definite (negative semidefinite) in that region.Note that, by matrixG, being uniformly negative definite means that the symmetric part of matrix G (that is, 1/2(G+GT)) is negative definite; in other words,(4)12G+GT≤-βI<0,∃β>0,∀x,∀t≥0,and we mean aregion, an open connected set.Consider now, an FOS:(5)Dtαxt=fxtwhere x and f are like that of the integer order system, and Dtα indicates the fractional derivative of order α∈R+. In this article, the Riemann-Liouville fractional operator was used as the main derivation tool which for order α>0,  m-1<α<m,  m∈N, is defined as [23](6)DαaRLfx=DmJam-αfx,where the operator Jaα is defined on L1[a,b] by(7)Jaαfx=1Γα∫axfτx-τ1-αdτ,for a≤x≤b and is called the Riemann-Liouville fractional integral operator of order α. For simplicity, the left superscript RL and subscript a were omitted and Dtα was assumed to be the Riemann-Liouville α-order fractional derivative operator which was assumed to exist and be continuous.Theorem 3.
Assume thatα1,α2≥0. Moreover, let ϕ∈L1[a,b] and xt=Jaα1+α2ϕ. Then,(8)Dα1Dα2xt=Dα1+α2xt.Proof.
See [24]: Page 29.Note that the condition onxt implies a certain degree of smoothness and the fact that, as t→a, xt→0 sufficiently fast.Suppose that the conditions of Theorem3 hold. By applying Dt1-α to both sides of (5), one can write(9)x˙=Dt1-αfxtTherefore, the contraction condition for system (5) is stated as follows [23].Theorem 4.
If matrixG=DxαDt1-αfxtDxαx-1 is uniformly negative definite, all the solution trajectories of system (5) converge to a single trajectory, with exponential rate, irrespective of the initial conditions.Proof.
See [23].Definition 5.
For the given FOS (5), a region Ω⊆Rn of the state space is called contraction (semicontraction), if the matrix DxαDt1-αfxtDxαx-1 is uniformly negative definite (negative semidefinite) in that region. By convention, fxt and system (5) are called contracting function and contracting system, respectively.It is worth mentioning that, ifΩ=Rn, then global exponential convergence is guaranteed.Remark 6.
Consider the linear time-invariant (LTI) FOS:(10)Dtαx=Ax.Applying Dt1-α to both sides of (10), one can write(11)x˙=Dt1-αAx.Considering two neighboring trajectories of the above equation and the virtual displacement δx between them yields(12)δx˙=∂Dt1-αAx∂xδx.The rate of change of the squared distance (δx)Tδx between two neighboring trajectories of the system (10) is given by(13)dδxTδxdt=2δxTδx˙=2δxT∂Dt1-αAx∂xδx.Considering(14)Dt1-αAx=1Γ1-1-αddt∫0tAxτt-τ1-αdτ,we have(15)∂Dt1-αAx∂x=1Γαddt∫0tAt-τ1-αdτ=Atα-1ΓαSo, LTI system (10) is contracting (semicontracting) if Atα-1/Γ(α) in [t0,∞) is uniformly negative definite (negative semidefinite).


## 2.2. Partial Contraction Theory

The partial contraction theory was first introduced during the study of network synchronization [25]; since then, its applicability and flexibility have been proven in many fields. Now, the partial contraction theory for the FOSs, which is the base of this work, is introduced.Theorem 7.
Consider an FOS(16)Dtαx=fx,x,tand its auxiliary system(17)Dtαy=fy,x,twhich is contracting with respect to y. Then trajectories of the original x-system verify each smooth specific property which auxiliary system verifies.Proof.
Two particular solutions of the virtual y-system areyt=x(t) and the solution with the specific property. Given that the virtual system is contracting, the solution x(t) converges to the solution with the specific property.Definition 8.
The original FOS (16) is said to be partially contracting.Corollary 9.
A convex combination ofDtαx=fix,t, i=1,…,n, which are contracting with a common trajectory x0(t), is contracting.Proof.
Consider the convex combination(18)Dtαx=∑iαix,tfix,twhich has a common trajectory x0(t) (for instance, a common equilibrium), where αix,t≥0 and ∑iαix,t=1. Consider the contracting auxiliary system(19)Dtαy=∑iαix,tfiy,twith two particular solutions x(t) and x0(t). Therefore, all trajectories of the system converge to the trajectory x0(t).Remark 10.
The notion of a virtual contracting system can be applied to control problems. For example, consider a nonlinear fractional order control system:(20)Dtαx=fx,x,u,tand assume that it is desired to reach the state xd(t), using the control input u(x,xd,t) such that(21)Dtαxd=fxd,x,u,t.Now, it is enough to have a contracting auxiliary system:(22)Dtαy=fy,x,u,twith two particular solutions x(t) and xd(t) to guarantee the convergence of x to xd.


## 3. Coupled Fractional Order Systems

Letx1,x2∈Rn. Consider a pair of identical systems(23)Dtαx1=fx1,t(24)Dtαx2=fx2,t+ux1-ux2which are paired together in a unidirectional (one-way) coupling way, with coupling force u(x1)-u(x2).Theorem 11.
If the function(f-u)(x2) in (24) is contracting, two systems (23) and (24) will be synchronized.Proof.
A particular solution of second system isx2t=x1(t), and the second system as virtual system is contracting. Therefore, x1 and x2 converge to each other and will be synchronized.Example 12.
Consider two coupled identical fractional order financial systems [26]:(25)Dtαx1=fx1and(26)Dtαx2=fx2+ux1-ux2with(27)x1=x1y1z1,x2=x2y2z2,fx=z+y-ax1-by-x2-x-cz,ux=xy-x2-x.
Therefore,(28)fx1=Dtαx1=z1+y1-ax1Dtαy1=1-by1-x12Dtαz1=-x1-cz1
and(29)fx2+ux1-ux2=Dtαx2=z2+y2-ax2+x1y1-x2y2Dtαy2=1-by2-x22-x12+x22Dtαz2=-x2-cz2-x1+x2.In this example, x,y, and z denote the interest rate, the investment demand, and the price index, respectively. The positive constants a, b, and c are the saving amount, the cost per investment, and the demand elasticity of commercial markets, respectively (see [26]).
The system(30)f-ux=z-ax1-by-cz=-a010-b000-cxyz+010is linear. With parameters a=1,  b=0.1, and c=1, the coefficient matrix A in f-u is uniformly negative definite, so f-u is contracting and the states of two coupled fractional order financial systems will synchronize (Figure 1).Figure 1
Synchronization of states of two financial FOSs with parametersa=1, b=0.1, c=1, and α=0.87.Remark 13.
The extension of Theorem11 for a network of n FOSs with an open chain structure(31)Dtαx1=fx1,tDtαx2=fx2,t+ux1-ux2⋮Dtαxn=fxn,t+uxn-1-uxnhas the same synchronization condition as that for systems (23) and (24).Theorem 14.
Consider two identical systems which are paired together in a bidirectional (two-way) coupling method of the form(32)Dtαx1=fx1,t+ux2-ux1Dtαx2=fx2,t+ux1-ux2In such a system, if f-2u is contracting, then x1 and x2 are synchronized.Proof.
From the coupled system, we have(33)Dtαx1-fx1,t-ux2+ux1=Dtαx2-fx2,t-ux1+ux2;define(34)gx1,x2,t=Dtαx1-fx1,t=Dtαx2-fx2,t-2ux1+2ux2and consider the following auxiliary system:(35)Dtαy=fy,t+gx1,x2,t+2ux1-2uywhich has two particular solutions, y=x1(t) and y=x2(t). Given that f-2u is contracting, the auxiliary system is contracting. Therefore, x1(t) and x2(t), as two solutions of the auxiliary system, converge together.Researchers have given varying definitions to synchronization under different contexts. In this study, synchronization orcomplete synchronization is that the difference of states of synchronized systems converges to zero; that is, x1=x2. In the case ofphase synchronization, the difference between various states of synchronized systems converges to a constant vector or even a periodic state instead of the zero. Similarly, antisynchronization orantiphase synchronization was defined as x1=-x2.Now, one can state the following theorem which is more general than Theorems11 and 14.Theorem 15 (synchronization).
Consider two systems coupled in an arbitrary manner. If there is a contraction function,h(x,t), such that(36)Dtαx1-hx1,t=Dtαx2-hx2,t,then x1 and x2 will be synchronized.Proof.
Letx1(t) and x2(t) be two trajectories of the coupled systems. Define(37)gx1,x2,t=Dtαx1-hx1,t=Dtαx2-hx2,t,and, therefore,(38)Dtαx1=hx1,t+gx1,x2,tDtαx2=hx2,t+gx1,x2,tSuppose the auxiliary system is as follows:(39)Dtαy=hy,t+gx1,x2,tDue to the contraction of function h, the auxiliary system is contracting. Therefore, solutions of auxiliary system converge together exponentially, and it is true for the solutions, y=x1(t) and y=x2(t).Remark 16.
(1) Theorems11 and 14 are two special cases of Theorem 15. In fact, in Theorem 11, hx,t=f-ux,t, and in Theorem 14  hx,t=f-2ux,t.
(2) For a network ofn systems with a complete graph structure(40)Dtαx1=fx1,t+∑i=2nuxi-ux1Dtαx2=fx2,t+∑i=1i≠2nuxi-ux2⋮Dtαxn=fxn,t+∑i=1n-1uxi-uxnsynchronization condition is like the condition of Theorem 14.
(3) For a network containingn oscillators with all-to-all symmetry, that is, a network in which each system is coupled to all the others,(41)Dtαxi=fxi,t+∑j=1nuxj-uxii=1,2,…,nif f-nu is contracting, synchronization of the whole network is guaranteed.Example 17.
The forced Duffing FOS [27] is as follows:(42)Dtαx=yDtαy=ax-by-x3+ccos⁡2πdtwhich is chaotic for the parameter values a=b=0.35,  c=0.3, and d=0.2. Using a coupling function to make a network of three identical Duffing FOSs yields(43)Dtαx1=y1+x1-x1+sx2+x3-x1Dtαy1=ax1-by1-x13+ccos⁡2πdt+s-x23-x33+x13,Dtαx2=y2+x2-x2+sx1+x3-x2Dtαy2=ax2-by2-x23+ccos⁡2πdt+s-x13-x33+x23,and(44)Dtαx3=y3+x3-x3+sx1+x2-x3Dtαy3=ax3-by3-x33+ccos⁡2πdt+s-x13-x23+x33,where s is the coupling strength; the values s=0.5 and α=0.85 were chosen.
Therefore,(45)f-2u=y-xax-by=-11a-bxy.Choosing a=b=0.35, α=0.85, the coefficient matrix A=-11a-b is negative semidefinite; therefore, f-2u is semicontracting and phase synchronization occurs. As we can see in Figure 2 trajectories of the given systems have been synchronized with constant difference. If a=0.2, b=0.35, the coefficient matrix A is uniformly negative definite and f-2u is contracting, so, complete synchronization occurs. As we can see in Figure 3 trajectories of the given systems converge to each other and complete synchronization occurs. But, choosing a=0.4 and b=0.35, the coefficient matrix A is positive definite and f-2u is not contracting and, as shown in Figure 4, two coupled systems diverge.Figure 2
Phase synchronization of three Duffing systems for initial pointsx1,0,y1,0=1,0.5, x2,0,y2,0=-1,0.4, and x3,0,y3,0=2,-0.3 and parameters a=b=0.35, c=0.3, and d=0.2.Figure 3
Complete synchronization of three Duffing systems for initial pointsx1,0,y1,0=1,0.5, x2,0,y2,0=-1,0.4, and x3,0,y3,0=2,-0.3 and parameters a=0.2, b=0.35, c=0.3, and d=0.2.Figure 4
Divergence of three coupled Duffing systems fora=0.4, b=0.35, c=0.3, and d=0.2.Theorem 18.
If the vector functionhx,t in(46)Dtαx1=hx1,t+ux2,t-ux1,tDtαx2=hx2,t+ux1,t-ux2,tis contracting and h-x,t=-hx,t, then x1+x2 will converge to zero. Moreover, for each initial condition, other than zero, there will be antisynchrony between x1 and x2, if the system(47)Dtαz=hz,t-2uz,thas a stable limit-cycle.Proof.
From (46) and oddness of h, we have(48)Dtαx1-hx1,t=ux2,t-ux1,t=-Dtαx2+hx2,t=Dtα-x2-h-x2,tand, therefore, from Theorem 15, it is concluded that x1 and -x2 converge to each other exponentially; in other words, x1 and x2 reach antisynchrony.Example 19.
Two unforced Duffing systems(49)Dtαx1=y1+x1-x1+sx2-x1Dtαy1=ax1-by1-x13+s-x23+x13and(50)Dtαx2=y2+x2-x2+sx1-x2Dtαy2=ax2-by2-x23+s-x13+x23,with parameters a=-0.25,  b=0.35, negative coupling strength s=-0.5 (inhibitory coupling instead of excitatory coupling), and fractional order α=0.85, are antisynchronized, because(51)h=yax-by-x3=-11-0.25-0.35xy+x-x3is odd in x=(x,y); moreover the system h-2u yields a stable limit-cycle; therefore, for nonzero initial conditions, x1=(x1,y1) and x2=(x2,y2) will oscillate and reach antisynchrony. As shown in Figure 5, the unforced Duffing systems with given parameters and two arbitrary initial points, x1,0=[-0.2,0.5] and  x2,0=[-1,0.4], reach antisynchrony and summation of states (error of antisynchronization) converges to zero. Figure 6 represents the stable limit-cycle of h-2u for unforced Duffing system with three arbitrary initial points x1,0=[1,0.5],  x2,0=[-1,-0.5],  and x3,0=[2,3].Figure 5
Antisynchronization of two unforced Duffing systems and the plot of errorsx1+x2 and y1+y2.Figure 6
The stable limit-cycle ofh-2u for unforced Duffing system with three arbitrary initial points x1,0=[1,0.5], x2,0=[-1,-0.5], and x3,0=[2,3].


## 4. Conclusion

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.



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