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Synchronization of Coupled Oscillator Dynamics
4th International Conference on Advances in Control and Optimization of Dynamical Systems 4th Conference on 4th International International Conference on Advances Advances in in Control Control and and Optimization of Dynamical Systems 4th International Conference on Advances in Control and February 1-5, of 2016. NIT Tiruchirappalli, India Optimization Dynamical Systems Available online at www.sciencedirect.com Optimization of Dynamical Systems February 1-5, 2016. NIT Tiruchirappalli, India Optimization of Dynamical Systems February 1-5, 2016. NIT Tiruchirappalli, India February February 1-5, 1-5, 2016. 2016. NIT NIT Tiruchirappalli, Tiruchirappalli, India India
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IFAC-PapersOnLine 49-1 (2016) 320–325 Synchronization of Coupled Synchronization of Coupled Synchronization of Coupled Synchronization of Coupled Dynamics Dynamics Dynamics Dynamics∗∗ ∗
Oscillator Oscillator Oscillator Oscillator
∗∗∗ Shyam Indra Shyam Krishan Krishan Joshi Joshi ∗∗∗ Shaunak Shaunak Sen Sen ∗∗ Indra Narayan Narayan Kar Kar ∗∗∗ ∗∗ ∗∗ ∗∗∗ Shyam Krishan Joshi ∗ Shaunak Sen ∗∗ Indra Narayan Kar ∗∗∗ Shyam Krishan Joshi Shaunak Sen Indra Narayan Kar ∗∗∗ ∗ ∗ Electrical Engineering Department, Indian Institute of Technology, ∗ Electrical Engineering Department, Indian Institute of Technology, ∗ Engineering Department, Indianee.iitd.ac.in). Institute of Technology, India(e-mail: eez128367@ ∗ ElectricalDelhi, India(e-mail: eez128367@ Engineering Department, Indianee.iitd.ac.in). Institute of Technology, ∗∗ElectricalDelhi, India(e-mail: eez128367@ ee.iitd.ac.in). Engineering Department, Indian Institute ∗∗ ElectricalDelhi, Electrical Engineering Department, Indian Institute of of Technology, Technology, Delhi, India(e-mail: eez128367@ ee.iitd.ac.in). ∗∗ ∗∗ Electrical Engineering Department, Indian Institute of Technology, Delhi, India(e-mail: shaunak.sen@ ee.iitd.ac.in). ∗∗ Delhi, India(e-mail: shaunak.sen@ Engineering Department, Indian ee.iitd.ac.in). Institute of Technology, ∗∗∗Electrical Delhi, India(e-mail: shaunak.sen@ Engineering Department, Indian Institute ∗∗∗ Electrical Electrical Engineering Department, Indianee.iitd.ac.in). Institute of of Technology, Technology, Delhi, India(e-mail: shaunak.sen@ ee.iitd.ac.in). ∗∗∗ ∗∗∗ Electrical Engineering Department, Indian Institute of Technology, Delhi, India(e-mail: ink@ ee.iitd.ac.in). ∗∗∗ Delhi, India(e-mail: ink@ ee.iitd.ac.in). Electrical Engineering Department, Indian Institute of Technology, Delhi, India(e-mail: ink@ ee.iitd.ac.in). Delhi, India(e-mail: ink@ ee.iitd.ac.in). Abstract: Synchronization Abstract: Synchronization is is most most significant significant phenomena phenomena to to study study the the collective collective behaviour behaviour Abstract: Synchronization is most significant phenomena to study theand collective behaviour of coupled oscillators. Synchronization is said to occur if phase locking consensus among of coupled oscillators. Synchronization is said to occur if phase locking and consensus among Abstract: Synchronization is most significant phenomena to study the collective behaviour of coupled oscillators. Synchronization is said to occur if phase locking and consensus corresponding states of coupled dynamical systems is achieved. In general, how to achieve exact corresponding states ofSynchronization coupled dynamical systems is achieved. general,and howconsensus to achieveamong exact of coupled oscillators. is said to occur if phaseInlocking among corresponding states of coupled dynamical systems is achieved. In general, how to achieve exact conditions for synchronization is unclear. Therefore, it becomes essential to derive conditions conditions for states synchronization unclear. Therefore, it becomes to derive conditions corresponding of coupled isdynamical systems is achieved. In essential general, how to achieve exact conditions for synchronization is unclear. Therefore, it becomes essential to derive conditions on these coupled dynamical systems which lead to synchronization. Here, we aim to derive on these coupled dynamical systems which lead to synchronization. Here, aimconditions to derive conditions for synchronization is unclear. Therefore, it becomes essential to we derive on these coupled dynamical systems which lead to benchmark synchronization. Here, which we aim derive sufficient conditions for of oscillators are linearly sufficient conditions for synchronization synchronization of selected selected oscillators areto on these coupled dynamical systems which lead to benchmark synchronization. Here, which we aim tolinearly derive sufficient conditions for synchronization of selected benchmark oscillators which are linearly coupled. We have used Lyapunov approach to obtain sufficiency condition for synchronization coupled. have used approach obtainbenchmark sufficiency oscillators condition for synchronization sufficient We conditions forLyapunov synchronization of to selected which are linearly coupled. have used Lyapunov approach to obtain condition synchronization of coupled oscillators. We study of der and Fitzhugh Nagumo of coupledWe oscillators. We study synchronization synchronization of Van Van sufficiency der Pol Pol oscillators oscillators andfor Fitzhugh Nagumo coupled. We have used Lyapunov approach to obtain sufficiency condition for synchronization of coupled oscillators. We study synchronization of Van der Pol oscillators and Fitzhugh oscillators with all-to-all connectivity, for both uniformly and non-uniformly linearly coupled oscillators with all-to-all connectivity, for both ofuniformly andoscillators non-uniformly linearly Nagumo coupled of coupled oscillators. We study synchronization Van der Pol and Fitzhugh Nagumo oscillators with These all-to-all connectivity, for numerically both uniformly and non-uniformly linearly coupled configurations. results have been simulated for both the above types of configurations. These results have been numerically simulated for both the above types of oscillators with all-to-all connectivity, for both uniformly and non-uniformly linearly coupled configurations. These results have been numerically simulated for both the above types of oscillators. oscillators. configurations. These results have been numerically simulated for both the above types of oscillators. oscillators. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Van Van der der Pol, Pol, Fitzhugh Fitzhugh Nagumo(FHN), Nagumo(FHN), coupled coupled oscillators, oscillators, neuronal neuronal membrane membrane potential v and recovery variable w . i i Keywords: Van der Pol, Fitzhugh Nagumo(FHN), coupled oscillators, potential andder recovery variable Nagumo(FHN), wi . Keywords:viVan Pol, Fitzhugh coupled oscillators, neuronal neuronal membrane membrane potential vii and recovery variable wii . potential vi and recovery variable wi . 1. oscillators 1. INTRODUCTION INTRODUCTION oscillators we we consider consider the the celebrated celebrated Kuramoto Kuramoto model, model, 1. INTRODUCTION oscillators we consider the celebrated model, having n number of coupled oscillators. having n number of coupled oscillators. Kuramoto 1. INTRODUCTION oscillators we consider the celebrated Kuramoto model, having n number of coupled oscillators. Synchronization of coupled oscillator dynamics is a funSynchronization of coupled oscillator dynamics is a fun- having n number of coupled oscillators. n Synchronization coupled oscillator dynamicsin isnature, a fundamental problem which finds wide n K damental problemof finds wide application application Synchronization ofwhich coupled oscillator dynamicsinisnature, a funK n θθ˙˙i = ω + (1) n sin(θj − θi ) damental problem which finds wide application in nature, science and engineering, Dorfler et.al (2014). It is therefore i n sin(θj − θi ) K = ω + (1) science and engineering, et.alapplication (2014). It isintherefore i i damental problem whichDorfler finds wide nature, n ˙ K θ = ω + (1) n science and engineering, Dorfler et.al (2014). It is therefore j=1 sin(θjj − θii ) understanding the mutual interactions of coupled oscillaii ii ˙ understanding the mutual interactions of coupled oscillaθi = ωi + n j=1 sin(θj − θi ) (1) science and engineering, Dorfler et.al (2014). It is therefore j=1 understanding the mutual interactions of coupled oscillators and to obtain consensus and phase locking among n j=1 i ∈ 1, .., n tors and to obtain consensus and phase locking oscillaamong j=1 i ∈ 1, .., n understanding the mutual interactions of coupled tors and to obtain and phase locking among corresponding states of which lead their i ∈ frequency 1, .., n corresponding states consensus of these these oscillators oscillators which lead to to their Where, θi is phase, ωi is natural tors and to obtain consensus and phase locking among and i ∈ frequency 1, .., n and K K is is corresponding states of these oscillators which lead to their Where, θi is phase, ωi is natural synchronization is a key challenge, Strogatz (2000). Thus, synchronization is a key challenge, Strogatz corresponding states of these oscillators which(2000). lead toThus, their coupling is phase, ω is natural frequency and K is Where, θ gain of the oscillator. i i i i coupling gain of the oscillator. synchronization is a key challenge, Strogatz (2000). Thus, understanding how individual rhythms of oscillators adjust is phase, ω is natural frequency and K is Where, θ i i understanding how rhythms of oscillators synchronization is aindividual key challenge, Strogatz (2000). adjust Thus, Chopra gain (2009) of the oscillator. have Chopra et.al et.al (2009) have demonstrated demonstrated that that Kuramoto Kuramoto understanding how individual rhythms of oscillators adjust coupling with each other such that the coupled oscillators oscillate coupling gain of the oscillator. with each other such that the coupled oscillators oscillate understanding how individual rhythms of oscillators adjust Chopra (2009) with haveall demonstrated that Kuramoto model oscillators, to but model of ofet.al oscillators, to all all connectivity connectivity but differdifferwith common each other such thatis the oscillators oscillate frequency, yet a which the et.al (2009) with havealldemonstrated that Kuramoto frequency, yet coupled a problem problem which needs needs the Chopra with common each other such thatisthe coupled oscillators oscillate model of oscillators, with all to all connectivity but different natural frequencies, locally exponentially synchronize. ent natural frequencies, locally exponentially synchronize. with common frequency, is yet a problem which needs the attention of researchers. Further the synchronization of model of oscillators, with all to all connectivity but differattention of researchers. the synchronization of ent with common frequency, isFurther yet a problem which needs the naturalet.al frequencies, locally exponentially synchronize. In Dorfler (2011), the manner in which power network In Dorfler et.al (2011), the manner in which power network attention of researchers. Further the synchronization of coupled oscillators become even more difficult task when ent natural frequencies, locally exponentially synchronize. coupled oscillators becomeFurther even more task when attention of researchers. the difficult synchronization of In Dorfler et.al (2011), manner in of which power network model is to order model coupled oscillators model is related related to first firstthe order model coupled oscillators coupled become states. even more task when it has occur The synchronization of Dorfler et.al (2011), the manner in of which power network it has to to oscillators occur on on multiple multiple The difficult synchronization of In coupled oscillators become states. even more difficult task when model is related to first order model of coupled oscillators has been explained. The authors have shown equivalence has been explained. The authors have shown equivalence it has to occur on multiple states. The synchronization of coupled oscillators therefore requires the necessary mathmodel is related to first order model of coupled oscillators coupled thereforestates. requires necessary mathit has tooscillators occur on multiple Thethe synchronization of between beenswing explained. Theof authors have shown equivalence equation power models and between equation power network network models and nonnoncoupled therefore requiresanalytical the necessary math- has ematical explanation using approach. beenswing explained. Theofauthors have shown equivalence ematicaloscillators explanation using modern modern approach. coupled oscillators therefore requiresanalytical the necessary math- has between swing equation of power network models and nonuniform Kuramoto model by using singular perturbation uniform Kuramoto model by using singular perturbation ematical explanation using modern analytical approach. between swing equation of power network models and nonematical explanation modern analytical approach. modelpure by using singular perturbation The dynamics of is by approach. In algebraic conditions have The coupled coupled dynamicsusing of oscillators oscillators is described described by ordiordi- uniform approach.Kuramoto In this this work, work, algebraic conditions have uniform Kuramoto modelpure by using singular perturbation The dynamics of oscillators is described by ordi- approach. In that this relates work, pure algebraic conditions have nary differential equation which of been Kuramoto model and nary coupled differential equation which comprises comprises of oscillator’s oscillator’s been derived derived Kuramoto model and transient transient The coupled dynamics of oscillators is described by ordi- approach. In that this relates work, pure algebraic conditions have nary differential equation which comprises of oscillator’s been derived that relates Kuramoto model and transient dynamic state with an additional weak coupling term. In stability of power networks under synchronization condidynamic state with an additional weak coupling term. In been stability of power under synchronization condinary differential equation which comprises of oscillator’s derived that networks relates Kuramoto model and transient dynamic state withsingle an additional weak coupling term. In stability of therefore, power networks under synchronization Kuramoto (1975), state model been tions. It understanding synchronization pheKuramoto (1975), state dynamical dynamical model has has been It is is understanding synchronizationcondiphedynamic state withsingle an additional weak coupling term. In tions. stability of therefore, power networks under synchronization condiKuramoto (1975), single state dynamical model has been tions. It is therefore, understanding synchronization pheconsidered and all to all coupling between oscillators is via nomenon in second order coupled oscillators by analysing considered (1975), and all to all coupling between oscillators via tions. nomenon second order coupled oscillators by analysing Kuramoto single state dynamical model has isbeen It isintherefore, understanding synchronization pheconsidered and all to all coupling between oscillators is via nomenon in second order coupled oscillators by analysing single state. Where as in Bharath et.al (2013), multi state both the states is unclear and deserves to be investigated. single state.and Where Bharathbetween et.al (2013), multi is state both the states is unclear and deserves to be by investigated. considered all toasallincoupling oscillators via nomenon in second order coupled oscillators analysing single state.model Whereof asoscillators in Bharathhas et.al (2013), multi state states unclear deserves to be investigated. dynamical been considered but Here, we to derive the sufficient condition on dynamical been considered but both Here, the we aim aim to is theand sufficient condition on coupling coupling single state.model Whereofasoscillators in Bharathhas et.al (2013), multi state both the states isderive unclear and deserves to be investigated. dynamical model of oscillators has been considered but Here, we aim to derive the sufficient condition onoscillators coupling the coupling is achieved via single state despite that the gain K at which the multi state benchmark the coupling is achieved via single that but the gain the the multi state benchmark dynamical model of oscillators hasstate beendespite considered Here, K we at aimwhich to derive sufficient condition onoscillators coupling the coupling is achieved via single state despite that the gain K at which the multi state benchmark oscillators oscillators possessed the multiple states . The former apget synchronized. Specifically, using Lyapunov approach oscillators possessed the multiple The former usingbenchmark Lyapunov oscillators approach the coupling is achieved via singlestates state .despite that apthe get gainsynchronized. K at which Specifically, the multi state oscillators possessed the multiple states . The former ap- get Specifically, using Lyapunov approach proach to basic process it established when derivative of proach helps helps to understand understand basic synchronization synchronization process it is issynchronized. established that that when the the derivative of quadratic quadratic oscillators possessed the multiple states . The former ap- get synchronized. Specifically, using Lyapunov approach proach helpsdynamics, to understand basic synchronization process it is established that when the derivative of quadratic in aa simple while the later one gives practical Lyapunov function of difference of states becomes negative in simple dynamics, while the later one gives practical Lyapunov function of difference of states becomes negative proach helps to understand basic synchronization process it is established that when the derivative of quadratic in a simple dynamics, while the later one gives practical Lyapunov function of difference of states becomes negative approach for dealing with problems of coupling in multidefinite, it marks synchronization of coupled oscillator approach for dealing with problems of coupling in multidefinite, it marks synchronization of coupled oscillator in a simple dynamics, while the later one gives practical Lyapunov function of difference of states becomes negative approach for dealing with problems of coupling in multidefinite, it marks synchronization of coupled oscillator state oscillators. dynamics. Finally, we have computed the critical coupling state oscillators. dynamics.itFinally, have computedofthe criticaloscillator coupling approach for dealing with problems of coupling in multi- definite, marks we synchronization coupled state we have computed the critical coupling As example, gain sufficient for of couAs an anoscillators. example, to to understand understand synchronization synchronization in in coupled coupled dynamics. gain that that is isFinally, sufficient for synchronization synchronization of linearly linearly coustate oscillators. dynamics. Finally, we have computed the critical coupling As an example, to understand synchronization in coupled gain that is sufficient for synchronization of linearly couAs an example, to understand synchronization in coupled gain that is sufficient for synchronization of linearly cou-
Copyright © 2016, 2016 IFAC 320Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 320 Copyright © 2016 IFAC 320 Copyright © 2016 IFAC 320 Peer review under responsibility of International Federation of Automatic Copyright © 2016 IFAC 320Control. 10.1016/j.ifacol.2016.03.073
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pled Van der Pol and Fitzhugh Nagumo oscillators. It is important to point out here that, the sufficiency conditions obtained using Lyapunov approach for the two models are analogous. For the case of coupled Fitzhugh Nagumo oscillators the sufficiency condition also matches with that obtained using partial contraction analysis by Wang et.al (2004), in a similar model of oscillator. These results should help to develop a framework for synchronization of second order coupled oscillators. In section 2, we describe selected benchmark oscillators. In section 3, we give the formal definition of synchronization of coupled oscillators. In section 4 we calculate sufficient coupling gain K for synchronization of linearly coupled oscillators using Lyapunov approach. In section 5, the results are validated through simulations and summarized in section 6.
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2.2 Fitzhugh Nagumo neural oscillator Fitzhugh Nagumo neural oscillator is a well known computational model of a biological neuron, Izhikevich (2004). It is also a second order model. Its dynamic equation is given by, 1 (4) v˙i = c[vi − wi − vi3 ] + I 3 (5) w˙ i = d[a + bvi − wi ] Here, vi represents the membrane potential, wi is the called as recovery variable which represents the channel dynamics, I = 2 is the excitation current and a = 2, b = 1, c = 1 and d = 0.1 are the constants. The oscillatory response of Fitzhugh Nagumo oscillator with initial conditions (-1,2) is shown in Fig.2.
2. BENCHMARK OSCILLATOR MODELS To illustrate the synchronization of coupled oscillators, we reproduce in this section two benchmark systems. 2.1 Van der Pol oscillator Van der Pol oscillator is a second order model of oscillator which find large applications in modelling oscillation in electronics and in real world problems, Strogatz (1994). The dynamical equation of Van der Pol oscillator can be given as, 1 x˙i = µ(xi − yi − x3i ) 3 1 y˙i = xi µ
(2) (3)
Here, xi and yi are state variables and µ is a parameter. The oscillatory response of Van der Pol oscillator for µ = 1 with initial conditions (−1, 2) is shown in Fig.1.
Fig. 2. Oscillations in Fitzhugh Nagumo oscillator 2.3 Linear coupling The dynamical equation for N coupled Van der Pol oscillator be given as, N 1 x˙i = µ(xi − yi − x3i ) + K (xp − xi ) (6) 3 p=1 1 xi , ∀ i, j = (1, . . . ,N) (7) µ The dynamical equation for N coupled Fitzhugh Nagumo oscillator is given by, N 1 (vs − vi ) (8) v˙i = c[vi − wi − vi3 ] + I + K 3 s=1 y˙i =
w˙ i = d[a + bvi − wi ],
∀ i, j = (1, . . . ,N)
(9)
3. DEFINITION OF SYNCHRONIZATION 3.1 Definition 1 A set of coupled oscillators are said to be synchronized if the difference between corresponding states become constant asymptotically.
Fig. 1. Oscillations in Van der Pol oscillator. 321
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For example, the coupled Van der Pol oscillators are said to synchronize if, |xj − xi | = constant as t → ∞ ∀ i, j = (1, . . . ,N) (10) ⇒ |x˙j − x˙i | = 0 |yj − yi | = constant as t → ∞ ∀ i, j = (1, . . . ,N) (11) ⇒ |y˙j − y˙i | = 0 The difference dynamics of corresponding states of coupled Van der Pol oscillators be given as, 1 (x˙j − x˙i ) = µ[(xj − xi ) − (yj − yi ) − (x3j − x3i )] N 3 N +K (xr − xj ) − (xp − xi ) (12) r=1
p=1
1 (13) (y˙j − y˙i ) = (xj − xi ) µ ⇒ (x˙j − x˙i ) = µ[(xj − xi ) − (yj − yi ) 1 (14) − (x3j − x3i )] + KN (xi − xj ) 3 1 (15) (y˙j − y˙i ) = (xj − xi ) µ Similarly, coupled Fitzhugh Nagumo neural oscillators are said to synchronize if, |vj − vi | = constant as t → ∞ ∀i, j = (1, . . . ,N) ⇒ |v˙j − v˙i | = 0 (16) |wj − wi | = constant as t → ∞ ∀i, j = (1, . . . ,N) ⇒ |w˙j − w˙ i | = 0 (17) Also, the difference dynamics of corresponding states of Fitzhugh Nagumo oscillator be given as, 1 (v˙j − v˙i ) = c[(vj − vi ) + (wi − wj ) − (vj3 − vi3 )] N 3 N +K (vq − vj ) − (vs − vi ) (18) q=1
s=1
(w˙j − w˙ i ) = d [b (vj − vi ) + (wi − wj )]
(19) 1 3 ⇒ (v˙j − v˙i ) = c[(vj − vi ) + (wi − wj ) − (vj − vi3 )] 3 (20) + KN (vi − vj ) (w˙j − w˙ i ) = d [b (vj − vi ) + (wi − wj )] (21) 4. SYNCHRONIZATION ANALYSIS USING LYAPUNOV APPROACH
In this section, we have focused on the Lyapunov approach as described in Chopra et.al (2009) to compute coupling gain K, sufficient for synchronization of linearly coupled Van der Pol and Fitzhugh Nagumo oscillators. 4.1 Coupled Van der Pol oscillator
early coupled Van der Pol oscillators synchronize asymptotically, if coupling gain K ≥ µ/N .
Proof: Let the positive definite Lyapunov function Sij be given as, 1 2 (xj − xi ) (22) 2 along the trajectories of (14) is
Sij = The derivative of Sij S˙ij = (xj − xi ) (x˙j − x˙i )
(23) S˙ij = (xj − xi ) × 1 3 3 µ (xj − xi ) − (yj − yi ) − (xj − xi ) + KN (xi − xj ) 3 (24) 2 ˙ Sij = µ (xj − xi ) + µ (xj − xi ) (yi − yj ) µ 2 2 − (xj − xi ) x2j + x2i + xi xj − KN (xj − xi ) 3 (25) Similarly, let the positive definite Lyapunov function Pij be given as, 1 2 (yj − yi ) (26) 2 The derivative of this function along the trajectories of (15) is, P˙ij = (yj − yi ) (y˙j − y˙i ) (27) Pij =
1 P˙ij = (yj − yi ) (xj − xi ) µ
(28)
Let, Lij = Sij + (µ)2 Pij 2 ⇒ L˙ij = (µ − KN ) (xj − xi ) 2 x2j (xi + xj ) µ x2i 2 + + − (xj − xi ) 3 2 2 2
(29)
(30)
In (30), L˙ij would be negative semi definite if, µ (31) K≥ N This ensures, (xj − xi )→ 0 and the boundedness of(yj − yi ). Consider set A = (yi − yj ) ∈ R ∀i, j|L˙ ij = 0 and
let B be the largest invariant set contained in A where (xj − xi ) = 0, (yj − yi ) = 0, (x˙ j − x˙ i ) = 0 and (y˙ j − y˙ i ) = 0 as per (14) and (15). Using Lasalle’s invariance principle, all trajectories (x˙ j − x˙ i ), (y˙ j − y˙ i ) ∀i, j converge to B as t → ∞. Hence the oscillators synchronize asymptotically. For all the coupled oscillators the components of Lij i.e. Sij and Pij can be computed and accordingly synchronization of linearly coupled Van der Pol oscillators can be shown. 4.2 Coupled Fitzhugh Nagumo neural oscillator
Here, we compute coupling gain K, sufficient for synchronization of linearly coupled Van der Pol oscillators using Lyapunov analysis. We assume the variables xi and yi of all the oscillators are chosen arbitrarily from set of real and we impose no particular distribution on them. The main outcome of this section is stated as below. Theorem 4.1: Consider system dynamics (14) and (15). Let at t =0, (xi , xj , yi , yj ∈ R) ∀i, j = (1, ..., N ), then lin322
Here, we develop a lower bound on coupling gain K, sufficient for synchronization of linearly coupled Fitzhugh Nagumo neural oscillators using Lyapunov analysis. We assume that all the neuronal potentials vi and recovery variables wi of all the oscillators are chosen arbitrarily from set of real and we impose no particular distribution on them. The main outcome of this section is stated as below.
IFAC ACODS 2016 February 1-5, 2016. NIT Tiruchirappalli, India Shyam Krishan Joshi et al. / IFAC-PapersOnLine 49-1 (2016) 320–325
Theorem 4.2: Consider system dynamics (20) and (21). Let at t = 0, (vi , vj , wi , wj ∈ R) ∀i, j = (1, ..., N ), then linearly coupled Fitzhugh Nagumo neural oscillators synchronize asymptotically, if coupling gain K ≥ c/N . Proof: Let the positive definite Lyapunov function Sij be given as, 1 2 Sij = (vj − vi ) (32) 2 The derivative of Sij along the trajectories of (20) is, S˙ij = (vj − vi ) (v˙j − v˙i ) (33)
S˙ij = (vj − vi ) × 1 c (vj − vi ) + (wi − wj ) − (vj3 − vi3 ) + KN (vi − vj ) 3 (34) 2 ˙ Sij = c (vj − vi ) + c (vj − vi ) (wi − wj ) c 2 2 (35) − (vj − vi ) vj2 + vi2 + vi vj − KN (vj − vi ) 3 Similarly, let the positive definite Lyapunov function Pij be given as,
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linearly coupled Fitzhugh Nagumo neural oscillators using theorem 4.2 corroborates with that computed using theorem 6 (example 4.4) based on partial contraction analysis in Wang et.al (2004) for a similar model of coupled oscillators (See Appendix A). Remark 2: This work can further be extended to obtain sufficiency conditions on coupling gain for synchronization of non-uniformly linearly coupled benchmark oscillator systems described above. For the case of non-uniformly linearly coupled Van der Pol oscillators the sufficient coupling gain should be such that Kij > µ/N and for non-uniformly linearly coupled Fitzhugh Nagumo neural oscillators Kij > c/N ∀i, j = (1, ...., N ). 5. SIMULATION RESULTS
1 2 (wj − wi ) (36) 2 The derivative of this function along the trajectories of (21) is, Pij =
P˙ij = (wj − wi ) (w˙j − w˙ i ) P˙ij = (wj − wi ) d [b (vj − vi ) + (wi − wj )] 2 P˙ij = d b (vj − vi ) (wj − wi ) − (wj − wi ) Let,
2 P˙ij = db (vj − vi ) (wj − wi ) − d (wj − wi )
(37) (38) (39) (40)
c (41) db 2 ⇒ L˙ij = (c − KN ) (vj − vi ) 2 vj2 (vi + vj ) c c vi2 2 2 + + − (wj − wi ) − (vj − vi ) 3 2 2 2 b (42) Since all the terms in (42) except the term containing (cKN) are purely negative, L˙ij would be negative definite if, c (43) K≥ N Accordingly, the linearly coupled Fitzhugh Nagumo neural oscillators synchronize asymptotically as t → ∞. For all the coupled oscillators the components of Lij i.e. Sij and Pij can be computed and accordingly synchronization of linearly coupled FHN oscillator can be shown. Lij = Sij + Pij
4.3 Remarks Remark 1: The Lyapunov approach based condition on sufficient coupling gain for synchronization of linearly coupled Van der Pol oscillator and linearly coupled Fitzhugh Nagumo neural oscillators using theorem 4.1 and 4.2 have been derived. It is important to point out here that the sufficient coupling gain calculated for synchronization of 323
Fig. 3. Synchronization of three linearly coupled Van der Pol oscillators for µ=1, N=3 and coupling gain K=0.33 introduced at t=50 secs. We have simulated linearly coupled Van der Pol oscillators as defined in (6)and (7) using ode45 in Matlab. It was observed that, linearly coupled Van der Pol oscillators synchronizes asymptotically for K ≥ µ/N . Here, we have taken µ = 1 and accordingly coupling gain K = 0.33 and K = 0.25 introduced at t=50 second for N=3 and N=4 cases respectively, (Fig.3. and Fig.4.). Also, we have simulated linearly coupled Fitzhugh Nagumo oscillators as defined in (8) and (9) using ode45 in Matlab. It was observed that, for K ≥ c/N the linearly coupled Fitzhugh Nagumo oscillators synchronizes asymptotically. Here, a = 2, b = 1, c = 1, d = 0.1 and I = 2 accordingly coupling gain K = 0.33 and K = 0.25 introduced at t= 100 seconds for N=3 and N=4 cases respectively, (Fig.5. and Fig.6.).The initial conditions were [(-1,-4),(3,-2),(6,2)] and [(-1,-4),(3,-2),(6,2),(-3,1)] for both the oscillator models. Fig. 7. and Fig.8 shows simulations of non-uniformly linearly coupled Van der Pol and Fitzhugh Nagumo neural oscillators respectively for N=4, K12 = 0.3, K23 = 0.4, K34 = 0.5 and K41 = 0.6, K42 = 0.7 and K31 = 0.8 with
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Fig. 4. Synchronization of Four linearly coupled Van der Pol oscillators for µ=1, N=4 and coupling gain K=0.25 introduced at t=50 secs.
Fig. 6. Synchronization of Four linearly coupled Fitzhugh Nagumo neural oscillators for c=1, N=4 and coupling gain K=0.25 introduced at t=100 secs.
Fig. 5. Synchronization of three linearly coupled Fitzhugh Nagumo neural oscillators for c=1, N=3 and coupling gain K=0.33 introduced at t=100 secs. initial conditions and parameters remaining the same as described above.
Fig. 7. Synchronization of Four non-uniformly linearly coupled Van der Pol oscillators for µ = 1, N=4 and coupling gains K12 = 0.3, K23 = 0.4, K34 = 0.5,K41 = 0.6, K42 = 0.7 and K31 = 0.8 introduced at t=100 secs.
6. CONCLUSION
that the Lyapunov approach gives sufficient condition for synchronization of both the above types of oscillators.
The non-linear analysis tool of Lyapunov approach provide a good mathematical formalism to understand synchronization of coupled oscillator dynamics. In this paper, we have carried out synchronization studies on second order oscillator models. Here, we apply this method on linearly coupled Van der Pol and Fitzhugh Nagumo oscillators. We have computed sufficient coupling gain using Lyapunov approach. It is important to point out here 324
Efforts have been made to obtain numerical simulation results when both types of oscillators viz. Van der Pol and Fitzhugh Nagumo, are non-uniformly linearly coupled for N=4 case. Prospective directions of research may include applications of non-linear approaches on coupled oscillators with more than two states. Thus, finding the sufficiency conditions for their synchronization. Further the effect of noise,
IFAC ACODS 2016 February 1-5, 2016. NIT Tiruchirappalli, India Shyam Krishan Joshi et al. / IFAC-PapersOnLine 49-1 (2016) 320–325
325
ramoto oscillators. SIAM Journal on Control and Optimization, volume 50, 1616–1642. Wang, W and Slotine, J.J.(2004) On partial contraction analysis for coupled nonlinear oscillators. Biological Cybernetics, volume 1, 92–96. Izhikevich, E.M.(2004) Simple model of spiking neurons. IEEE Transaction on Neural Networks, volume 14, 1569-1572. Strogatz, S.H.(1994) Nonlinear dynamics of chaos with applications to physics, biology, chemistry and engineering. Addison Wesley Publishing company, edition 1. Appendix A. ILLUSTRATION OF SYNCHRONIZATION CONDITION USING PARTIAL CONTRACTION ANALYSIS.
Fig. 8. Synchronization of Four non-uniformly linearly coupled Fitzhugh Nagumo neural oscillators for c=1, N=4 and coupling gains K12 = 0.3, K23 = 0.4, K34 = 0.5,K41 = 0.6, K42 = 0.7 and K31 = 0.8 introduced at t=100 secs. time delays and perturbations in synchronous regimes of the coupled oscillators can be explored. Developing mathematical formalisms for multi state synchronization of coupled dynamics of oscillators would be instrumental in explaining the coupled behaviour of large systems having coupled oscillators. Also, such studies helps to understand the rhythm adjustment phenomena in multi state dynamical systems. These results should help in formulating basis to understand the mutual interactions among coupled oscillators which finally lead to synchronization. Together, it will aid to develop stability studies of large dynamical systems wherein oscillators are coupled. Specifically, these results can be used to analyse and ensure synchronization among coupled non-linear oscillators.
The dynamical model which we are considering is the special case of the model worked out in Wang et.al (2004). And, the sufficiency condition of synchronization on coupling gain which we are obtaining using Lyapunov analysis i.e. K ≥ c/N as found in theorem 4.2 corroborates with one obtained through Partial Contraction analysis in example 4.4 in Wang et.al (2004). The statement of example 4.4 is explained as below. The authors have considered a diffusion coupled network with n-identical Fitzhugh Nagumo neurons given by, N 1 kij (vj − vi ) (A.1) v˙i = c[vi − wi − vi3 ] + I + 3 j∈Ni
1 w˙ i = − [vi − a + bwi ] i = (1, ...., n) (A.2) c Where a,b and c are strictly positive constants. The whole network synchronizes exponentially if, b λ2 (A.3) Tknij > c (ij)∈N
Where,
Tknij
REFERENCES Dorfler, F and Bullo, F.(2014) Synchronization in complex network of phase oscillators: A Survey. Automatica, volume 50, 1539–1564. Strogatz, S.H.(2000) From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators. Physica D, volume 2, 1–20. Kuramoto, Y.(1975) Self-entrainment of a population of coupled non-linear oscillators. In international symposium on mathematical problems in theoretical physics. Lecture notes in physics, volume 39, 420–422. Bharath, R and Slotine, J.J.(2013) Nonlinear observer design and synchronization analysis for classical models of neural oscillators arXiv:1310.0479v1. Chopra, N and Spong, M.(2009) On exponential synchronization of Kuramoto oscillators. Transactions on Automatic Control, volume 54, 353–357. Dorfler, F and Bullo, F.(2011) Synchronization and transient stability in power networks and non-uniform Ku325
.. .. . . . · · · K · · · −K · · · .. . . .. = . . . · · · −K · · · K · · · .. .. . . . . . n×n ..
(A.4)
All the elements in Tkni j are zero except those displayed above at four intersections of ith and j th columns. One has Tkni j =0 if K=0. Also, N = ∪ni=1 Ni denote set of active links in the network. The authors have proved using Partial Contraction analysis that the coupling gain sufficient to give synchronization among linearly coupled FHN model should be such that K ≥ c/N . Our findings which are based on Lyapunov analysis and also show analogous results. This strengthens and substantiates the outcomes which we have arrived at for synchronization of linearly coupled FHN neural oscillators.
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IFAC-PapersOnLine 49-1 (2016) 320–325 Synchronization of Coupled Synchronization of Coupled Synchronization of Coupled Synchronization of Coupled Dynamics Dynamics Dynamics Dynamics∗∗ ∗
Oscillator Oscillator Oscillator Oscillator
∗∗∗ Shyam Indra Shyam Krishan Krishan Joshi Joshi ∗∗∗ Shaunak Shaunak Sen Sen ∗∗ Indra Narayan Narayan Kar Kar ∗∗∗ ∗∗ ∗∗ ∗∗∗ Shyam Krishan Joshi ∗ Shaunak Sen ∗∗ Indra Narayan Kar ∗∗∗ Shyam Krishan Joshi Shaunak Sen Indra Narayan Kar ∗∗∗ ∗ ∗ Electrical Engineering Department, Indian Institute of Technology, ∗ Electrical Engineering Department, Indian Institute of Technology, ∗ Engineering Department, Indianee.iitd.ac.in). Institute of Technology, India(e-mail: eez128367@ ∗ ElectricalDelhi, India(e-mail: eez128367@ Engineering Department, Indianee.iitd.ac.in). Institute of Technology, ∗∗ElectricalDelhi, India(e-mail: eez128367@ ee.iitd.ac.in). Engineering Department, Indian Institute ∗∗ ElectricalDelhi, Electrical Engineering Department, Indian Institute of of Technology, Technology, Delhi, India(e-mail: eez128367@ ee.iitd.ac.in). ∗∗ ∗∗ Electrical Engineering Department, Indian Institute of Technology, Delhi, India(e-mail: shaunak.sen@ ee.iitd.ac.in). ∗∗ Delhi, India(e-mail: shaunak.sen@ Engineering Department, Indian ee.iitd.ac.in). Institute of Technology, ∗∗∗Electrical Delhi, India(e-mail: shaunak.sen@ Engineering Department, Indian Institute ∗∗∗ Electrical Electrical Engineering Department, Indianee.iitd.ac.in). Institute of of Technology, Technology, Delhi, India(e-mail: shaunak.sen@ ee.iitd.ac.in). ∗∗∗ ∗∗∗ Electrical Engineering Department, Indian Institute of Technology, Delhi, India(e-mail: ink@ ee.iitd.ac.in). ∗∗∗ Delhi, India(e-mail: ink@ ee.iitd.ac.in). Electrical Engineering Department, Indian Institute of Technology, Delhi, India(e-mail: ink@ ee.iitd.ac.in). Delhi, India(e-mail: ink@ ee.iitd.ac.in). Abstract: Synchronization Abstract: Synchronization is is most most significant significant phenomena phenomena to to study study the the collective collective behaviour behaviour Abstract: Synchronization is most significant phenomena to study theand collective behaviour of coupled oscillators. Synchronization is said to occur if phase locking consensus among of coupled oscillators. Synchronization is said to occur if phase locking and consensus among Abstract: Synchronization is most significant phenomena to study the collective behaviour of coupled oscillators. Synchronization is said to occur if phase locking and consensus corresponding states of coupled dynamical systems is achieved. In general, how to achieve exact corresponding states ofSynchronization coupled dynamical systems is achieved. general,and howconsensus to achieveamong exact of coupled oscillators. is said to occur if phaseInlocking among corresponding states of coupled dynamical systems is achieved. In general, how to achieve exact conditions for synchronization is unclear. Therefore, it becomes essential to derive conditions conditions for states synchronization unclear. Therefore, it becomes to derive conditions corresponding of coupled isdynamical systems is achieved. In essential general, how to achieve exact conditions for synchronization is unclear. Therefore, it becomes essential to derive conditions on these coupled dynamical systems which lead to synchronization. Here, we aim to derive on these coupled dynamical systems which lead to synchronization. Here, aimconditions to derive conditions for synchronization is unclear. Therefore, it becomes essential to we derive on these coupled dynamical systems which lead to benchmark synchronization. Here, which we aim derive sufficient conditions for of oscillators are linearly sufficient conditions for synchronization synchronization of selected selected oscillators areto on these coupled dynamical systems which lead to benchmark synchronization. Here, which we aim tolinearly derive sufficient conditions for synchronization of selected benchmark oscillators which are linearly coupled. We have used Lyapunov approach to obtain sufficiency condition for synchronization coupled. have used approach obtainbenchmark sufficiency oscillators condition for synchronization sufficient We conditions forLyapunov synchronization of to selected which are linearly coupled. have used Lyapunov approach to obtain condition synchronization of coupled oscillators. We study of der and Fitzhugh Nagumo of coupledWe oscillators. We study synchronization synchronization of Van Van sufficiency der Pol Pol oscillators oscillators andfor Fitzhugh Nagumo coupled. We have used Lyapunov approach to obtain sufficiency condition for synchronization of coupled oscillators. We study synchronization of Van der Pol oscillators and Fitzhugh oscillators with all-to-all connectivity, for both uniformly and non-uniformly linearly coupled oscillators with all-to-all connectivity, for both ofuniformly andoscillators non-uniformly linearly Nagumo coupled of coupled oscillators. We study synchronization Van der Pol and Fitzhugh Nagumo oscillators with These all-to-all connectivity, for numerically both uniformly and non-uniformly linearly coupled configurations. results have been simulated for both the above types of configurations. These results have been numerically simulated for both the above types of oscillators with all-to-all connectivity, for both uniformly and non-uniformly linearly coupled configurations. These results have been numerically simulated for both the above types of oscillators. oscillators. configurations. These results have been numerically simulated for both the above types of oscillators. oscillators. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Van Van der der Pol, Pol, Fitzhugh Fitzhugh Nagumo(FHN), Nagumo(FHN), coupled coupled oscillators, oscillators, neuronal neuronal membrane membrane potential v and recovery variable w . i i Keywords: Van der Pol, Fitzhugh Nagumo(FHN), coupled oscillators, potential andder recovery variable Nagumo(FHN), wi . Keywords:viVan Pol, Fitzhugh coupled oscillators, neuronal neuronal membrane membrane potential vii and recovery variable wii . potential vi and recovery variable wi . 1. oscillators 1. INTRODUCTION INTRODUCTION oscillators we we consider consider the the celebrated celebrated Kuramoto Kuramoto model, model, 1. INTRODUCTION oscillators we consider the celebrated model, having n number of coupled oscillators. having n number of coupled oscillators. Kuramoto 1. INTRODUCTION oscillators we consider the celebrated Kuramoto model, having n number of coupled oscillators. Synchronization of coupled oscillator dynamics is a funSynchronization of coupled oscillator dynamics is a fun- having n number of coupled oscillators. n Synchronization coupled oscillator dynamicsin isnature, a fundamental problem which finds wide n K damental problemof finds wide application application Synchronization ofwhich coupled oscillator dynamicsinisnature, a funK n θθ˙˙i = ω + (1) n sin(θj − θi ) damental problem which finds wide application in nature, science and engineering, Dorfler et.al (2014). It is therefore i n sin(θj − θi ) K = ω + (1) science and engineering, et.alapplication (2014). It isintherefore i i damental problem whichDorfler finds wide nature, n ˙ K θ = ω + (1) n science and engineering, Dorfler et.al (2014). It is therefore j=1 sin(θjj − θii ) understanding the mutual interactions of coupled oscillaii ii ˙ understanding the mutual interactions of coupled oscillaθi = ωi + n j=1 sin(θj − θi ) (1) science and engineering, Dorfler et.al (2014). It is therefore j=1 understanding the mutual interactions of coupled oscillators and to obtain consensus and phase locking among n j=1 i ∈ 1, .., n tors and to obtain consensus and phase locking oscillaamong j=1 i ∈ 1, .., n understanding the mutual interactions of coupled tors and to obtain and phase locking among corresponding states of which lead their i ∈ frequency 1, .., n corresponding states consensus of these these oscillators oscillators which lead to to their Where, θi is phase, ωi is natural tors and to obtain consensus and phase locking among and i ∈ frequency 1, .., n and K K is is corresponding states of these oscillators which lead to their Where, θi is phase, ωi is natural synchronization is a key challenge, Strogatz (2000). Thus, synchronization is a key challenge, Strogatz corresponding states of these oscillators which(2000). lead toThus, their coupling is phase, ω is natural frequency and K is Where, θ gain of the oscillator. i i i i coupling gain of the oscillator. synchronization is a key challenge, Strogatz (2000). Thus, understanding how individual rhythms of oscillators adjust is phase, ω is natural frequency and K is Where, θ i i understanding how rhythms of oscillators synchronization is aindividual key challenge, Strogatz (2000). adjust Thus, Chopra gain (2009) of the oscillator. have Chopra et.al et.al (2009) have demonstrated demonstrated that that Kuramoto Kuramoto understanding how individual rhythms of oscillators adjust coupling with each other such that the coupled oscillators oscillate coupling gain of the oscillator. with each other such that the coupled oscillators oscillate understanding how individual rhythms of oscillators adjust Chopra (2009) with haveall demonstrated that Kuramoto model oscillators, to but model of ofet.al oscillators, to all all connectivity connectivity but differdifferwith common each other such thatis the oscillators oscillate frequency, yet a which the et.al (2009) with havealldemonstrated that Kuramoto frequency, yet coupled a problem problem which needs needs the Chopra with common each other such thatisthe coupled oscillators oscillate model of oscillators, with all to all connectivity but different natural frequencies, locally exponentially synchronize. ent natural frequencies, locally exponentially synchronize. with common frequency, is yet a problem which needs the attention of researchers. Further the synchronization of model of oscillators, with all to all connectivity but differattention of researchers. the synchronization of ent with common frequency, isFurther yet a problem which needs the naturalet.al frequencies, locally exponentially synchronize. In Dorfler (2011), the manner in which power network In Dorfler et.al (2011), the manner in which power network attention of researchers. Further the synchronization of coupled oscillators become even more difficult task when ent natural frequencies, locally exponentially synchronize. coupled oscillators becomeFurther even more task when attention of researchers. the difficult synchronization of In Dorfler et.al (2011), manner in of which power network model is to order model coupled oscillators model is related related to first firstthe order model coupled oscillators coupled become states. even more task when it has occur The synchronization of Dorfler et.al (2011), the manner in of which power network it has to to oscillators occur on on multiple multiple The difficult synchronization of In coupled oscillators become states. even more difficult task when model is related to first order model of coupled oscillators has been explained. The authors have shown equivalence has been explained. The authors have shown equivalence it has to occur on multiple states. The synchronization of coupled oscillators therefore requires the necessary mathmodel is related to first order model of coupled oscillators coupled thereforestates. requires necessary mathit has tooscillators occur on multiple Thethe synchronization of between beenswing explained. Theof authors have shown equivalence equation power models and between equation power network network models and nonnoncoupled therefore requiresanalytical the necessary math- has ematical explanation using approach. beenswing explained. Theofauthors have shown equivalence ematicaloscillators explanation using modern modern approach. coupled oscillators therefore requiresanalytical the necessary math- has between swing equation of power network models and nonuniform Kuramoto model by using singular perturbation uniform Kuramoto model by using singular perturbation ematical explanation using modern analytical approach. between swing equation of power network models and nonematical explanation modern analytical approach. modelpure by using singular perturbation The dynamics of is by approach. In algebraic conditions have The coupled coupled dynamicsusing of oscillators oscillators is described described by ordiordi- uniform approach.Kuramoto In this this work, work, algebraic conditions have uniform Kuramoto modelpure by using singular perturbation The dynamics of oscillators is described by ordi- approach. In that this relates work, pure algebraic conditions have nary differential equation which of been Kuramoto model and nary coupled differential equation which comprises comprises of oscillator’s oscillator’s been derived derived Kuramoto model and transient transient The coupled dynamics of oscillators is described by ordi- approach. In that this relates work, pure algebraic conditions have nary differential equation which comprises of oscillator’s been derived that relates Kuramoto model and transient dynamic state with an additional weak coupling term. In stability of power networks under synchronization condidynamic state with an additional weak coupling term. In been stability of power under synchronization condinary differential equation which comprises of oscillator’s derived that networks relates Kuramoto model and transient dynamic state withsingle an additional weak coupling term. In stability of therefore, power networks under synchronization Kuramoto (1975), state model been tions. It understanding synchronization pheKuramoto (1975), state dynamical dynamical model has has been It is is understanding synchronizationcondiphedynamic state withsingle an additional weak coupling term. In tions. stability of therefore, power networks under synchronization condiKuramoto (1975), single state dynamical model has been tions. It is therefore, understanding synchronization pheconsidered and all to all coupling between oscillators is via nomenon in second order coupled oscillators by analysing considered (1975), and all to all coupling between oscillators via tions. nomenon second order coupled oscillators by analysing Kuramoto single state dynamical model has isbeen It isintherefore, understanding synchronization pheconsidered and all to all coupling between oscillators is via nomenon in second order coupled oscillators by analysing single state. Where as in Bharath et.al (2013), multi state both the states is unclear and deserves to be investigated. single state.and Where Bharathbetween et.al (2013), multi is state both the states is unclear and deserves to be by investigated. considered all toasallincoupling oscillators via nomenon in second order coupled oscillators analysing single state.model Whereof asoscillators in Bharathhas et.al (2013), multi state states unclear deserves to be investigated. dynamical been considered but Here, we to derive the sufficient condition on dynamical been considered but both Here, the we aim aim to is theand sufficient condition on coupling coupling single state.model Whereofasoscillators in Bharathhas et.al (2013), multi state both the states isderive unclear and deserves to be investigated. dynamical model of oscillators has been considered but Here, we aim to derive the sufficient condition onoscillators coupling the coupling is achieved via single state despite that the gain K at which the multi state benchmark the coupling is achieved via single that but the gain the the multi state benchmark dynamical model of oscillators hasstate beendespite considered Here, K we at aimwhich to derive sufficient condition onoscillators coupling the coupling is achieved via single state despite that the gain K at which the multi state benchmark oscillators oscillators possessed the multiple states . The former apget synchronized. Specifically, using Lyapunov approach oscillators possessed the multiple The former usingbenchmark Lyapunov oscillators approach the coupling is achieved via singlestates state .despite that apthe get gainsynchronized. K at which Specifically, the multi state oscillators possessed the multiple states . The former ap- get Specifically, using Lyapunov approach proach to basic process it established when derivative of proach helps helps to understand understand basic synchronization synchronization process it is issynchronized. established that that when the the derivative of quadratic quadratic oscillators possessed the multiple states . The former ap- get synchronized. Specifically, using Lyapunov approach proach helpsdynamics, to understand basic synchronization process it is established that when the derivative of quadratic in aa simple while the later one gives practical Lyapunov function of difference of states becomes negative in simple dynamics, while the later one gives practical Lyapunov function of difference of states becomes negative proach helps to understand basic synchronization process it is established that when the derivative of quadratic in a simple dynamics, while the later one gives practical Lyapunov function of difference of states becomes negative approach for dealing with problems of coupling in multidefinite, it marks synchronization of coupled oscillator approach for dealing with problems of coupling in multidefinite, it marks synchronization of coupled oscillator in a simple dynamics, while the later one gives practical Lyapunov function of difference of states becomes negative approach for dealing with problems of coupling in multidefinite, it marks synchronization of coupled oscillator state oscillators. dynamics. Finally, we have computed the critical coupling state oscillators. dynamics.itFinally, have computedofthe criticaloscillator coupling approach for dealing with problems of coupling in multi- definite, marks we synchronization coupled state we have computed the critical coupling As example, gain sufficient for of couAs an anoscillators. example, to to understand understand synchronization synchronization in in coupled coupled dynamics. gain that that is isFinally, sufficient for synchronization synchronization of linearly linearly coustate oscillators. dynamics. Finally, we have computed the critical coupling As an example, to understand synchronization in coupled gain that is sufficient for synchronization of linearly couAs an example, to understand synchronization in coupled gain that is sufficient for synchronization of linearly cou-
Copyright © 2016, 2016 IFAC 320Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 320 Copyright © 2016 IFAC 320 Copyright © 2016 IFAC 320 Peer review under responsibility of International Federation of Automatic Copyright © 2016 IFAC 320Control. 10.1016/j.ifacol.2016.03.073
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pled Van der Pol and Fitzhugh Nagumo oscillators. It is important to point out here that, the sufficiency conditions obtained using Lyapunov approach for the two models are analogous. For the case of coupled Fitzhugh Nagumo oscillators the sufficiency condition also matches with that obtained using partial contraction analysis by Wang et.al (2004), in a similar model of oscillator. These results should help to develop a framework for synchronization of second order coupled oscillators. In section 2, we describe selected benchmark oscillators. In section 3, we give the formal definition of synchronization of coupled oscillators. In section 4 we calculate sufficient coupling gain K for synchronization of linearly coupled oscillators using Lyapunov approach. In section 5, the results are validated through simulations and summarized in section 6.
321
2.2 Fitzhugh Nagumo neural oscillator Fitzhugh Nagumo neural oscillator is a well known computational model of a biological neuron, Izhikevich (2004). It is also a second order model. Its dynamic equation is given by, 1 (4) v˙i = c[vi − wi − vi3 ] + I 3 (5) w˙ i = d[a + bvi − wi ] Here, vi represents the membrane potential, wi is the called as recovery variable which represents the channel dynamics, I = 2 is the excitation current and a = 2, b = 1, c = 1 and d = 0.1 are the constants. The oscillatory response of Fitzhugh Nagumo oscillator with initial conditions (-1,2) is shown in Fig.2.
2. BENCHMARK OSCILLATOR MODELS To illustrate the synchronization of coupled oscillators, we reproduce in this section two benchmark systems. 2.1 Van der Pol oscillator Van der Pol oscillator is a second order model of oscillator which find large applications in modelling oscillation in electronics and in real world problems, Strogatz (1994). The dynamical equation of Van der Pol oscillator can be given as, 1 x˙i = µ(xi − yi − x3i ) 3 1 y˙i = xi µ
(2) (3)
Here, xi and yi are state variables and µ is a parameter. The oscillatory response of Van der Pol oscillator for µ = 1 with initial conditions (−1, 2) is shown in Fig.1.
Fig. 2. Oscillations in Fitzhugh Nagumo oscillator 2.3 Linear coupling The dynamical equation for N coupled Van der Pol oscillator be given as, N 1 x˙i = µ(xi − yi − x3i ) + K (xp − xi ) (6) 3 p=1 1 xi , ∀ i, j = (1, . . . ,N) (7) µ The dynamical equation for N coupled Fitzhugh Nagumo oscillator is given by, N 1 (vs − vi ) (8) v˙i = c[vi − wi − vi3 ] + I + K 3 s=1 y˙i =
w˙ i = d[a + bvi − wi ],
∀ i, j = (1, . . . ,N)
(9)
3. DEFINITION OF SYNCHRONIZATION 3.1 Definition 1 A set of coupled oscillators are said to be synchronized if the difference between corresponding states become constant asymptotically.
Fig. 1. Oscillations in Van der Pol oscillator. 321
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For example, the coupled Van der Pol oscillators are said to synchronize if, |xj − xi | = constant as t → ∞ ∀ i, j = (1, . . . ,N) (10) ⇒ |x˙j − x˙i | = 0 |yj − yi | = constant as t → ∞ ∀ i, j = (1, . . . ,N) (11) ⇒ |y˙j − y˙i | = 0 The difference dynamics of corresponding states of coupled Van der Pol oscillators be given as, 1 (x˙j − x˙i ) = µ[(xj − xi ) − (yj − yi ) − (x3j − x3i )] N 3 N +K (xr − xj ) − (xp − xi ) (12) r=1
p=1
1 (13) (y˙j − y˙i ) = (xj − xi ) µ ⇒ (x˙j − x˙i ) = µ[(xj − xi ) − (yj − yi ) 1 (14) − (x3j − x3i )] + KN (xi − xj ) 3 1 (15) (y˙j − y˙i ) = (xj − xi ) µ Similarly, coupled Fitzhugh Nagumo neural oscillators are said to synchronize if, |vj − vi | = constant as t → ∞ ∀i, j = (1, . . . ,N) ⇒ |v˙j − v˙i | = 0 (16) |wj − wi | = constant as t → ∞ ∀i, j = (1, . . . ,N) ⇒ |w˙j − w˙ i | = 0 (17) Also, the difference dynamics of corresponding states of Fitzhugh Nagumo oscillator be given as, 1 (v˙j − v˙i ) = c[(vj − vi ) + (wi − wj ) − (vj3 − vi3 )] N 3 N +K (vq − vj ) − (vs − vi ) (18) q=1
s=1
(w˙j − w˙ i ) = d [b (vj − vi ) + (wi − wj )]
(19) 1 3 ⇒ (v˙j − v˙i ) = c[(vj − vi ) + (wi − wj ) − (vj − vi3 )] 3 (20) + KN (vi − vj ) (w˙j − w˙ i ) = d [b (vj − vi ) + (wi − wj )] (21) 4. SYNCHRONIZATION ANALYSIS USING LYAPUNOV APPROACH
In this section, we have focused on the Lyapunov approach as described in Chopra et.al (2009) to compute coupling gain K, sufficient for synchronization of linearly coupled Van der Pol and Fitzhugh Nagumo oscillators. 4.1 Coupled Van der Pol oscillator
early coupled Van der Pol oscillators synchronize asymptotically, if coupling gain K ≥ µ/N .
Proof: Let the positive definite Lyapunov function Sij be given as, 1 2 (xj − xi ) (22) 2 along the trajectories of (14) is
Sij = The derivative of Sij S˙ij = (xj − xi ) (x˙j − x˙i )
(23) S˙ij = (xj − xi ) × 1 3 3 µ (xj − xi ) − (yj − yi ) − (xj − xi ) + KN (xi − xj ) 3 (24) 2 ˙ Sij = µ (xj − xi ) + µ (xj − xi ) (yi − yj ) µ 2 2 − (xj − xi ) x2j + x2i + xi xj − KN (xj − xi ) 3 (25) Similarly, let the positive definite Lyapunov function Pij be given as, 1 2 (yj − yi ) (26) 2 The derivative of this function along the trajectories of (15) is, P˙ij = (yj − yi ) (y˙j − y˙i ) (27) Pij =
1 P˙ij = (yj − yi ) (xj − xi ) µ
(28)
Let, Lij = Sij + (µ)2 Pij 2 ⇒ L˙ij = (µ − KN ) (xj − xi ) 2 x2j (xi + xj ) µ x2i 2 + + − (xj − xi ) 3 2 2 2
(29)
(30)
In (30), L˙ij would be negative semi definite if, µ (31) K≥ N This ensures, (xj − xi )→ 0 and the boundedness of(yj − yi ). Consider set A = (yi − yj ) ∈ R ∀i, j|L˙ ij = 0 and
let B be the largest invariant set contained in A where (xj − xi ) = 0, (yj − yi ) = 0, (x˙ j − x˙ i ) = 0 and (y˙ j − y˙ i ) = 0 as per (14) and (15). Using Lasalle’s invariance principle, all trajectories (x˙ j − x˙ i ), (y˙ j − y˙ i ) ∀i, j converge to B as t → ∞. Hence the oscillators synchronize asymptotically. For all the coupled oscillators the components of Lij i.e. Sij and Pij can be computed and accordingly synchronization of linearly coupled Van der Pol oscillators can be shown. 4.2 Coupled Fitzhugh Nagumo neural oscillator
Here, we compute coupling gain K, sufficient for synchronization of linearly coupled Van der Pol oscillators using Lyapunov analysis. We assume the variables xi and yi of all the oscillators are chosen arbitrarily from set of real and we impose no particular distribution on them. The main outcome of this section is stated as below. Theorem 4.1: Consider system dynamics (14) and (15). Let at t =0, (xi , xj , yi , yj ∈ R) ∀i, j = (1, ..., N ), then lin322
Here, we develop a lower bound on coupling gain K, sufficient for synchronization of linearly coupled Fitzhugh Nagumo neural oscillators using Lyapunov analysis. We assume that all the neuronal potentials vi and recovery variables wi of all the oscillators are chosen arbitrarily from set of real and we impose no particular distribution on them. The main outcome of this section is stated as below.
IFAC ACODS 2016 February 1-5, 2016. NIT Tiruchirappalli, India Shyam Krishan Joshi et al. / IFAC-PapersOnLine 49-1 (2016) 320–325
Theorem 4.2: Consider system dynamics (20) and (21). Let at t = 0, (vi , vj , wi , wj ∈ R) ∀i, j = (1, ..., N ), then linearly coupled Fitzhugh Nagumo neural oscillators synchronize asymptotically, if coupling gain K ≥ c/N . Proof: Let the positive definite Lyapunov function Sij be given as, 1 2 Sij = (vj − vi ) (32) 2 The derivative of Sij along the trajectories of (20) is, S˙ij = (vj − vi ) (v˙j − v˙i ) (33)
S˙ij = (vj − vi ) × 1 c (vj − vi ) + (wi − wj ) − (vj3 − vi3 ) + KN (vi − vj ) 3 (34) 2 ˙ Sij = c (vj − vi ) + c (vj − vi ) (wi − wj ) c 2 2 (35) − (vj − vi ) vj2 + vi2 + vi vj − KN (vj − vi ) 3 Similarly, let the positive definite Lyapunov function Pij be given as,
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linearly coupled Fitzhugh Nagumo neural oscillators using theorem 4.2 corroborates with that computed using theorem 6 (example 4.4) based on partial contraction analysis in Wang et.al (2004) for a similar model of coupled oscillators (See Appendix A). Remark 2: This work can further be extended to obtain sufficiency conditions on coupling gain for synchronization of non-uniformly linearly coupled benchmark oscillator systems described above. For the case of non-uniformly linearly coupled Van der Pol oscillators the sufficient coupling gain should be such that Kij > µ/N and for non-uniformly linearly coupled Fitzhugh Nagumo neural oscillators Kij > c/N ∀i, j = (1, ...., N ). 5. SIMULATION RESULTS
1 2 (wj − wi ) (36) 2 The derivative of this function along the trajectories of (21) is, Pij =
P˙ij = (wj − wi ) (w˙j − w˙ i ) P˙ij = (wj − wi ) d [b (vj − vi ) + (wi − wj )] 2 P˙ij = d b (vj − vi ) (wj − wi ) − (wj − wi ) Let,
2 P˙ij = db (vj − vi ) (wj − wi ) − d (wj − wi )
(37) (38) (39) (40)
c (41) db 2 ⇒ L˙ij = (c − KN ) (vj − vi ) 2 vj2 (vi + vj ) c c vi2 2 2 + + − (wj − wi ) − (vj − vi ) 3 2 2 2 b (42) Since all the terms in (42) except the term containing (cKN) are purely negative, L˙ij would be negative definite if, c (43) K≥ N Accordingly, the linearly coupled Fitzhugh Nagumo neural oscillators synchronize asymptotically as t → ∞. For all the coupled oscillators the components of Lij i.e. Sij and Pij can be computed and accordingly synchronization of linearly coupled FHN oscillator can be shown. Lij = Sij + Pij
4.3 Remarks Remark 1: The Lyapunov approach based condition on sufficient coupling gain for synchronization of linearly coupled Van der Pol oscillator and linearly coupled Fitzhugh Nagumo neural oscillators using theorem 4.1 and 4.2 have been derived. It is important to point out here that the sufficient coupling gain calculated for synchronization of 323
Fig. 3. Synchronization of three linearly coupled Van der Pol oscillators for µ=1, N=3 and coupling gain K=0.33 introduced at t=50 secs. We have simulated linearly coupled Van der Pol oscillators as defined in (6)and (7) using ode45 in Matlab. It was observed that, linearly coupled Van der Pol oscillators synchronizes asymptotically for K ≥ µ/N . Here, we have taken µ = 1 and accordingly coupling gain K = 0.33 and K = 0.25 introduced at t=50 second for N=3 and N=4 cases respectively, (Fig.3. and Fig.4.). Also, we have simulated linearly coupled Fitzhugh Nagumo oscillators as defined in (8) and (9) using ode45 in Matlab. It was observed that, for K ≥ c/N the linearly coupled Fitzhugh Nagumo oscillators synchronizes asymptotically. Here, a = 2, b = 1, c = 1, d = 0.1 and I = 2 accordingly coupling gain K = 0.33 and K = 0.25 introduced at t= 100 seconds for N=3 and N=4 cases respectively, (Fig.5. and Fig.6.).The initial conditions were [(-1,-4),(3,-2),(6,2)] and [(-1,-4),(3,-2),(6,2),(-3,1)] for both the oscillator models. Fig. 7. and Fig.8 shows simulations of non-uniformly linearly coupled Van der Pol and Fitzhugh Nagumo neural oscillators respectively for N=4, K12 = 0.3, K23 = 0.4, K34 = 0.5 and K41 = 0.6, K42 = 0.7 and K31 = 0.8 with
IFAC ACODS 2016 324 Shyam Krishan Joshi et al. / IFAC-PapersOnLine 49-1 (2016) 320–325 February 1-5, 2016. NIT Tiruchirappalli, India
Fig. 4. Synchronization of Four linearly coupled Van der Pol oscillators for µ=1, N=4 and coupling gain K=0.25 introduced at t=50 secs.
Fig. 6. Synchronization of Four linearly coupled Fitzhugh Nagumo neural oscillators for c=1, N=4 and coupling gain K=0.25 introduced at t=100 secs.
Fig. 5. Synchronization of three linearly coupled Fitzhugh Nagumo neural oscillators for c=1, N=3 and coupling gain K=0.33 introduced at t=100 secs. initial conditions and parameters remaining the same as described above.
Fig. 7. Synchronization of Four non-uniformly linearly coupled Van der Pol oscillators for µ = 1, N=4 and coupling gains K12 = 0.3, K23 = 0.4, K34 = 0.5,K41 = 0.6, K42 = 0.7 and K31 = 0.8 introduced at t=100 secs.
6. CONCLUSION
that the Lyapunov approach gives sufficient condition for synchronization of both the above types of oscillators.
The non-linear analysis tool of Lyapunov approach provide a good mathematical formalism to understand synchronization of coupled oscillator dynamics. In this paper, we have carried out synchronization studies on second order oscillator models. Here, we apply this method on linearly coupled Van der Pol and Fitzhugh Nagumo oscillators. We have computed sufficient coupling gain using Lyapunov approach. It is important to point out here 324
Efforts have been made to obtain numerical simulation results when both types of oscillators viz. Van der Pol and Fitzhugh Nagumo, are non-uniformly linearly coupled for N=4 case. Prospective directions of research may include applications of non-linear approaches on coupled oscillators with more than two states. Thus, finding the sufficiency conditions for their synchronization. Further the effect of noise,
IFAC ACODS 2016 February 1-5, 2016. NIT Tiruchirappalli, India Shyam Krishan Joshi et al. / IFAC-PapersOnLine 49-1 (2016) 320–325
325
ramoto oscillators. SIAM Journal on Control and Optimization, volume 50, 1616–1642. Wang, W and Slotine, J.J.(2004) On partial contraction analysis for coupled nonlinear oscillators. Biological Cybernetics, volume 1, 92–96. Izhikevich, E.M.(2004) Simple model of spiking neurons. IEEE Transaction on Neural Networks, volume 14, 1569-1572. Strogatz, S.H.(1994) Nonlinear dynamics of chaos with applications to physics, biology, chemistry and engineering. Addison Wesley Publishing company, edition 1. Appendix A. ILLUSTRATION OF SYNCHRONIZATION CONDITION USING PARTIAL CONTRACTION ANALYSIS.
Fig. 8. Synchronization of Four non-uniformly linearly coupled Fitzhugh Nagumo neural oscillators for c=1, N=4 and coupling gains K12 = 0.3, K23 = 0.4, K34 = 0.5,K41 = 0.6, K42 = 0.7 and K31 = 0.8 introduced at t=100 secs. time delays and perturbations in synchronous regimes of the coupled oscillators can be explored. Developing mathematical formalisms for multi state synchronization of coupled dynamics of oscillators would be instrumental in explaining the coupled behaviour of large systems having coupled oscillators. Also, such studies helps to understand the rhythm adjustment phenomena in multi state dynamical systems. These results should help in formulating basis to understand the mutual interactions among coupled oscillators which finally lead to synchronization. Together, it will aid to develop stability studies of large dynamical systems wherein oscillators are coupled. Specifically, these results can be used to analyse and ensure synchronization among coupled non-linear oscillators.
The dynamical model which we are considering is the special case of the model worked out in Wang et.al (2004). And, the sufficiency condition of synchronization on coupling gain which we are obtaining using Lyapunov analysis i.e. K ≥ c/N as found in theorem 4.2 corroborates with one obtained through Partial Contraction analysis in example 4.4 in Wang et.al (2004). The statement of example 4.4 is explained as below. The authors have considered a diffusion coupled network with n-identical Fitzhugh Nagumo neurons given by, N 1 kij (vj − vi ) (A.1) v˙i = c[vi − wi − vi3 ] + I + 3 j∈Ni
1 w˙ i = − [vi − a + bwi ] i = (1, ...., n) (A.2) c Where a,b and c are strictly positive constants. The whole network synchronizes exponentially if, b λ2 (A.3) Tknij > c (ij)∈N
Where,
Tknij
REFERENCES Dorfler, F and Bullo, F.(2014) Synchronization in complex network of phase oscillators: A Survey. Automatica, volume 50, 1539–1564. Strogatz, S.H.(2000) From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators. Physica D, volume 2, 1–20. Kuramoto, Y.(1975) Self-entrainment of a population of coupled non-linear oscillators. In international symposium on mathematical problems in theoretical physics. Lecture notes in physics, volume 39, 420–422. Bharath, R and Slotine, J.J.(2013) Nonlinear observer design and synchronization analysis for classical models of neural oscillators arXiv:1310.0479v1. Chopra, N and Spong, M.(2009) On exponential synchronization of Kuramoto oscillators. Transactions on Automatic Control, volume 54, 353–357. Dorfler, F and Bullo, F.(2011) Synchronization and transient stability in power networks and non-uniform Ku325
.. .. . . . · · · K · · · −K · · · .. . . .. = . . . · · · −K · · · K · · · .. .. . . . . . n×n ..
(A.4)
All the elements in Tkni j are zero except those displayed above at four intersections of ith and j th columns. One has Tkni j =0 if K=0. Also, N = ∪ni=1 Ni denote set of active links in the network. The authors have proved using Partial Contraction analysis that the coupling gain sufficient to give synchronization among linearly coupled FHN model should be such that K ≥ c/N . Our findings which are based on Lyapunov analysis and also show analogous results. This strengthens and substantiates the outcomes which we have arrived at for synchronization of linearly coupled FHN neural oscillators.