Journal of Applied Mathematics

Journal of Applied Mathematics / 2018 / Article

Research Article | Open Access

Volume 2018 |Article ID 9414835 |

Ahmad Ruzitalab, Mohammad Hadi Farahi, Gholamhossien Erjaee, "Partial Contraction Analysis of Coupled Fractional Order Systems", Journal of Applied Mathematics, vol. 2018, Article ID 9414835, 9 pages, 2018.

Partial Contraction Analysis of Coupled Fractional Order Systems

Academic Editor: Jose L. Gracia
Received18 Dec 2017
Revised14 May 2018
Accepted04 Jun 2018
Published02 Jul 2018


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.

1. Introduction

Since the Dutch researcher, Christiaan Huygens, initiated the study of synchronization phenomenon in the century [1], many researchers of various fields, such as mathematics [24], 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 [1012]; 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 [1315]. 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 Section 2, 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 be contracting, 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. The partial 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 systemwhere and are state vector and vector function, respectively. Suppose that is continuously differentiable, (1) gives the relationwhere is a virtual displacement between two neighboring trajectories of system (1). The rate of change of squared distance between two trajectories is defined as

Theorem 1. If matrix 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 of the state space is called a contraction (semicontraction) region, if the matrix is uniformly negative definite (negative semidefinite) in that region.

Note that, by matrix , being uniformly negative definite means that the symmetric part of matrix G (that is, ) is negative definite; in other words,and we mean a region, an open connected set.

Consider now, an FOS:where and are like that of the integer order system, and indicates the fractional derivative of order . In this article, the Riemann-Liouville fractional operator was used as the main derivation tool which for order ,  ,  , is defined as [23]where the operator is defined on byfor and is called the Riemann-Liouville fractional integral operator of order . For simplicity, the left superscript and subscript were omitted and was assumed to be the Riemann-Liouville -order fractional derivative operator which was assumed to exist and be continuous.

Theorem 3. Assume that . Moreover, let and . Then,

Proof. See [24]: Page 29.

Note that the condition on implies a certain degree of smoothness and the fact that, as , sufficiently fast.

Suppose that the conditions of Theorem 3 hold. By applying to both sides of (5), one can writeTherefore, the contraction condition for system (5) is stated as follows [23].

Theorem 4. If matrix 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 of the state space is called contraction (semicontraction), if the matrix is uniformly negative definite (negative semidefinite) in that region. By convention, and system (5) are called contracting function and contracting system, respectively.

It is worth mentioning that, if , then global exponential convergence is guaranteed.

Remark 6. Consider the linear time-invariant (LTI) FOS:Applying to both sides of (10), one can writeConsidering two neighboring trajectories of the above equation and the virtual displacement between them yieldsThe rate of change of the squared distance between two neighboring trajectories of the system (10) is given byConsideringwe haveSo, LTI system (10) is contracting (semicontracting) if in 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 FOSand its auxiliary systemwhich 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 are and the solution with the specific property. Given that the virtual system is contracting, the solution 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 of , , which are contracting with a common trajectory , is contracting.

Proof. Consider the convex combinationwhich has a common trajectory (for instance, a common equilibrium), where and . Consider the contracting auxiliary systemwith two particular solutions x and . Therefore, all trajectories of the system converge to the trajectory .

Remark 10. The notion of a virtual contracting system can be applied to control problems. For example, consider a nonlinear fractional order control system:and assume that it is desired to reach the state , using the control input such thatNow, it is enough to have a contracting auxiliary system:with two particular solutions x and to guarantee the convergence of x to .

3. Coupled Fractional Order Systems

Let . Consider a pair of identical systemswhich are paired together in a unidirectional (one-way) coupling way, with coupling force .

Theorem 11. If the function in (24) is contracting, two systems (23) and (24) will be synchronized.

Proof. A particular solution of second system is , and the second system as virtual system is contracting. Therefore, and converge to each other and will be synchronized.

Example 12. Consider two coupled identical fractional order financial systems [26]:andwithTherefore,andIn this example, , and denote the interest rate, the investment demand, and the price index, respectively. The positive constants , , and are the saving amount, the cost per investment, and the demand elasticity of commercial markets, respectively (see [26]).
The systemis linear. With parameters ,  , and , the coefficient matrix in is uniformly negative definite, so is contracting and the states of two coupled fractional order financial systems will synchronize (Figure 1).

Remark 13. The extension of Theorem 11 for a network of FOSs with an open chain structurehas 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 formIn such a system, if is contracting, then and are synchronized.

Proof. From the coupled system, we havedefineand consider the following auxiliary system:which has two particular solutions, and . Given that is contracting, the auxiliary system is contracting. Therefore, and , as two solutions of the auxiliary system, converge together.

Researchers have given varying definitions to synchronization under different contexts. In this study, synchronization or complete synchronization is that the difference of states of synchronized systems converges to zero; that is, . In the case of phase 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 or antiphase synchronization was defined as .

Now, one can state the following theorem which is more general than Theorems 11 and 14.

Theorem 15 (synchronization). Consider two systems coupled in an arbitrary manner. If there is a contraction function, , such thatthen and will be synchronized.

Proof. Let and be two trajectories of the coupled systems. Defineand, therefore,Suppose the auxiliary system is as follows:Due 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, and .

Remark 16. (1) Theorems 11 and 14 are two special cases of Theorem 15. In fact, in Theorem 11, , and in Theorem 14  .
(2) For a network of systems with a complete graph structuresynchronization condition is like the condition of Theorem 14.
(3) For a network containing oscillators with all-to-all symmetry, that is, a network in which each system is coupled to all the others,if is contracting, synchronization of the whole network is guaranteed.

Example 17. The forced Duffing FOS [27] is as follows:which is chaotic for the parameter values ,  , and . Using a coupling function to make a network of three identical Duffing FOSs yieldsandwhere is the coupling strength; the values and were chosen.
Therefore,Choosing , , the coefficient matrix is negative semidefinite; therefore, 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 , , the coefficient matrix is uniformly negative definite and 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 and , the coefficient matrix is positive definite and is not contracting and, as shown in Figure 4, two coupled systems diverge.

Theorem 18. If the vector function inis contracting and , then will converge to zero. Moreover, for each initial condition, other than zero, there will be antisynchrony between and , if the systemhas a stable limit-cycle.

Proof. From (46) and oddness of , we haveand, therefore, from Theorem 15, it is concluded that and converge to each other exponentially; in other words, and reach antisynchrony.

Example 19. Two unforced Duffing systemsandwith parameters ,  , negative coupling strength (inhibitory coupling instead of excitatory coupling), and fractional order , are antisynchronized, becauseis odd in ; moreover the system yields a stable limit-cycle; therefore, for nonzero initial conditions, and will oscillate and reach antisynchrony. As shown in Figure 5, the unforced Duffing systems with given parameters and two arbitrary initial points, and  , reach antisynchrony and summation of states (error of antisynchronization) converges to zero. Figure 6 represents the stable limit-cycle of for unforced Duffing system with three arbitrary initial points ,  ,  and .

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.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

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


  1. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Unverisity Press, Cambridge, Mass, USA, 2001. View at: Publisher Site | MathSciNet
  2. G. H. Erjaee and S. Momani, “Phase synchronization in fractional differential chaotic systems,” Physics Letters A, vol. 372, no. 14, pp. 2350–2354, 2008. View at: Google Scholar
  3. G. H. Erjaee, “On analytical justification of phase synchronization in different chaotic systems,” Chaos, Solitons & Fractals, vol. 39, no. 3, pp. 1195–1202, 2009. View at: Publisher Site | Google Scholar
  4. M. Shahiri, R. Ghaderi, A. Ranjbar N., S. H. Hosseinnia, and S. Momani, “Chaotic fractional-order Coullet system: synchronization and control approach,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 3, pp. 665–674, 2010. View at: Publisher Site | Google Scholar | MathSciNet
  5. S.-J. Chung and J.-J. E. Slotine, “Cooperative robot control and synchronization of lagrangian systems,” in Proceedings of the 46th IEEE Conference on Decision and Control (CDC '07), pp. 2504–2509, New Orleans, La, USA, December 2007. View at: Publisher Site | Google Scholar
  6. S. Rinaldi, P. Ferrari, A. Flammini et al., Network Synchronization: An Introduction, Wiley Encyclopedia of Electrical and Electronics Engineering, 2017.
  7. F. Orsucci, R. Petrosino, G. Paoloni et al., “Prosody and synchronization in cognitive neuroscience,” EPJ Nonlinear Biomedical Physics, vol. 1, no. 1, 2013. View at: Publisher Site | Google Scholar
  8. L. Glass, “Synchronization and rhythmic processes in physiology,” Nature, vol. 410, no. 6825, pp. 277–284, 2001. View at: Publisher Site | Google Scholar
  9. A. Bacciotti and L. Rosier, Liapunov Functions and Stability in Control Theory, Springer Communications and Control Engineering, 2006.
  10. T. Abdeljawad and V. Gejji, “Lyapunov-Krasovskii stability theorem for fractional systems with delay,” Romanian Journal of Physics, vol. 56, no. 5-6, pp. 636–643, 2011. View at: Google Scholar
  11. D. Matignon, “Stability results for fractional differential equations with applications to control processing,” in Computational Engineering in Systems Applications, vol. 2, pp. 963–968, IMACS, IEEE-SMC, Lille, France, 1996. View at: Google Scholar
  12. J. Jouffroy and T. I. Fossen, “A tutorial on incremental stability analysis using contraction theory,” Modeling, Identification and Control, vol. 31, no. 3, pp. 93–106, 2010. View at: Publisher Site | Google Scholar
  13. W. Lohmiller and J. E. Slotine, “On contraction analysis for non-linear systems,” Automatica, vol. 34, no. 6, pp. 683–696, 1998. View at: Publisher Site | Google Scholar | MathSciNet
  14. W. Lohmiller and J.-J. E. Slotine, “Nonlinear process control using contraction theory,” AIChE Journal, vol. 46, no. 3, pp. 588–596, 2000. View at: Publisher Site | Google Scholar
  15. Q.-C. Pham, N. Tabareau, and J.-J. Slotine, “A contraction theory approach to stochastic incremental stability,” Institute of Electrical and Electronics Engineers Transactions on Automatic Control, vol. 54, no. 4, pp. 816–820, 2009. View at: Publisher Site | Google Scholar | MathSciNet
  16. S. J. Chung, “Nonlinear control and synchronization of multiple Lagrangian systems with application to tethered formation flight spacecraft. Dissertation, Massachusetts Institute of Technology, 2007”. View at: Google Scholar
  17. W. Wang, “Contraction and partial contraction: a study of synchronization in nonlinear networks, Dissertation, Massachusetts Institute of Technology, 2005”. View at: Google Scholar
  18. J. A. Tenreiro Machado, V. Kiryakova, and F. Mainardi, “A poster about the recent history of fractional calculus,” Fractional Calculus and Applied Analysis, vol. 13, no. 3, pp. 329–334, 2010. View at: Google Scholar | MathSciNet
  19. J. Sabatier, C. Farges, and J.-C. Trigeassou, “Fractional systems state space description: some wrong ideas and proposed solutions,” Journal of Vibration and Control, vol. 20, no. 7, pp. 1076–1084, 2014. View at: Publisher Site | Google Scholar | MathSciNet
  20. Y. Q. Chen and K. L. Moore, “Discretization schemes for fractional-order differentiators and integrators,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 49, no. 3, pp. 363–367, 2002. View at: Publisher Site | Google Scholar | MathSciNet
  21. J. Sabatier, O. P. Agrawal, and J. A. Machado, Advances in Fractional Calculus, Springer, Dordrecht, Netherlands, 2007.
  22. Li. CP and Zhang. F. R., “A survey on the stability of fractional differential equations,” The European Physical Journal-Special Topics, vol. 193, no. 1, pp. 27–47, 2011. View at: Google Scholar
  23. B. Bandyopadhyay and S. Kamal, Stabilization and Control of Fractional Order Systems: A Sliding Mode Approach, Springer International Publishing, Switzerland, 2015.
  24. K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer, 2010.
  25. W. Wang and J. E. Slotine, “On partial contraction analysis for coupled nonlinear oscillators,” Biological Cybernetics, vol. 92, no. 1, pp. 38–53, 2005. View at: Publisher Site | Google Scholar | MathSciNet
  26. W. C. Chen, “Nonlinear dynamics and chaos in a fractional-order financial system,” Chaos, Solitons & Fractals, vol. 36, no. 5, pp. 1305–1314, 2008. View at: Publisher Site | Google Scholar
  27. Z. Li, D. Chen, J. Zhu, and Y. Liu, “Nonlinear dynamics of fractional order Duffing system,” Chaos, Solitons & Fractals, vol. 81, pp. 111–116, 2015. View at: Publisher Site | Google Scholar

Copyright © 2018 Ahmad Ruzitalab et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles