Adaptation of oscillatory systems in networks — a learning signal approach

We consider a network of coupled periodic stable signals (PSS) interacting through the gradient of a coupling potential. Each PSS has its own set of parameters Ωk, characterizing the time scale of the signal and its shape. The Ωk are allowed to modify their values (i.e. to adapt) by introducing adap...

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Date:	2014
Main Author:	Rodriguez, J.
Format:	Article
Language:	English
Published:	Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України 2014
Series:	Системні дослідження та інформаційні технології
Subjects:	Прогресивні інформаційні технології, високопродуктивні комп’ютерні системи
Online Access:	http://dspace.nbuv.gov.ua/handle/123456789/85498
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spelling	irk-123456789-854982015-08-07T03:02:11Z Adaptation of oscillatory systems in networks — a learning signal approach Rodriguez, J. Прогресивні інформаційні технології, високопродуктивні комп’ютерні системи We consider a network of coupled periodic stable signals (PSS) interacting through the gradient of a coupling potential. Each PSS has its own set of parameters Ωk, characterizing the time scale of the signal and its shape. The Ωk are allowed to modify their values (i.e. to adapt) by introducing adaptive mechanisms on them. Together with the state variable interactions, the adaptive mechanisms drive all PSS towards a consensual oscillatory state where they all have a common, constant set of parameters Ωk. Once reached, the consensual oscillatory state remains invariant to the interactions. This implies that if the interactions are removed, all PSS continue to deliver the consensual signal. This situation is to be contrasted with classical synchronization problems where common dynamical patterns are attained and maintained thanks to the interactions. Hence, if the interactions are removed, all PSS converge back towards their individual behavior. The resulting value Ωk is analytically calculated. It does not depend on the network’s topology. However, the conditions for convergence do depend on the connectivity of the network and on the coupling potential. Розглянуто мережу зв’язаних періодичних стійких сигналів (PSS) взаємодіючих через градієнт потенціалу зв’язку. Кожен PSS має свій власний набір параметрів Ωk, що характеризує часову шкалу сигналу і його форму. Ωk можуть змінювати їх значення (тобто, адаптуватися) шляхом введення в них адаптивних механізмів. Разом з взаємодіями змінних стану, адаптивні механізми приводять усі PSS до узгодженого коливального стану, де вони всі мають спільну, постійну множину параметрів Ωk. Будучи досягнутим, узгоджений коливальний стан залишається інваріантним до взаємодій. Це означає, що якщо взаємодії прибирають, то усі PSS продовжують видавати узгоджений сигнал. Ця ситуація відрізняється від класичних проблем синхронізації, де загальні динамічні характеристики досягаються і підтримуються завдяки взаємодіям. Таким чином, якщо взаємодії прибирають, усі PSS сходяться назад до їх індивідуальної поведінки. Результат значення Ωk обчислюється аналітично. Він не залежить від топології мережі. Однак умови збіжності все ж залежать від зв’язності мережі і від сполученого потенціалу. Рассмотрена сеть связанных периодических устойчивых сигналов (PSS) взаимодействующих через градиент потенциала связи. Каждый PSS имеет свой собственный набор параметров Ωk, характеризующий временную шкалу сигнала и его форму Ωk могут менять их значения (то есть, адаптироваться) путем введения в них адаптивных механизмов. Вместе с взаимодействиями переменных состояния, адаптивные механизмы приводят все PSS к согласованному колебательному состоянию, где они все имеют общее, постоянное множество параметров Ωk. Будучи достигнутым, согласованное колебательное состояние остается инвариантным к взаимодействиям. Это означает, что если взаимодействия убирают, то все PSS продолжают выдавать согласованный сигнал. Эта ситуация отличается от классических проблем синхронизации, где общие динамические характеристики достигаются и поддерживаются благодаря взаимодействиям. Таким образом, если взаимодействия убирают, все PSS сходятся обратно к их индивидуальному поведению. Результат значения Ωk вычисляется аналитически. Он не зависит от топологии сети. Однако условия сходимости все же зависят от связности сети и от сопряженного потенциала. 2014 Article Adaptation of oscillatory systems in networks — a learning signal approach / J. Rodriguez // Системні дослідження та інформаційні технології. — 2014. — № 2. — С. 53-67. — Бібліогр.: 10 назв. — англ. 1681–6048 http://dspace.nbuv.gov.ua/handle/123456789/85498 518.58 en Системні дослідження та інформаційні технології Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Прогресивні інформаційні технології, високопродуктивні комп’ютерні системи Прогресивні інформаційні технології, високопродуктивні комп’ютерні системи
spellingShingle	Прогресивні інформаційні технології, високопродуктивні комп’ютерні системи Прогресивні інформаційні технології, високопродуктивні комп’ютерні системи Rodriguez, J. Adaptation of oscillatory systems in networks — a learning signal approach Системні дослідження та інформаційні технології
description	We consider a network of coupled periodic stable signals (PSS) interacting through the gradient of a coupling potential. Each PSS has its own set of parameters Ωk, characterizing the time scale of the signal and its shape. The Ωk are allowed to modify their values (i.e. to adapt) by introducing adaptive mechanisms on them. Together with the state variable interactions, the adaptive mechanisms drive all PSS towards a consensual oscillatory state where they all have a common, constant set of parameters Ωk. Once reached, the consensual oscillatory state remains invariant to the interactions. This implies that if the interactions are removed, all PSS continue to deliver the consensual signal. This situation is to be contrasted with classical synchronization problems where common dynamical patterns are attained and maintained thanks to the interactions. Hence, if the interactions are removed, all PSS converge back towards their individual behavior. The resulting value Ωk is analytically calculated. It does not depend on the network’s topology. However, the conditions for convergence do depend on the connectivity of the network and on the coupling potential.
format	Article
author	Rodriguez, J.
author_facet	Rodriguez, J.
author_sort	Rodriguez, J.
title	Adaptation of oscillatory systems in networks — a learning signal approach
title_short	Adaptation of oscillatory systems in networks — a learning signal approach
title_full	Adaptation of oscillatory systems in networks — a learning signal approach
title_fullStr	Adaptation of oscillatory systems in networks — a learning signal approach
title_full_unstemmed	Adaptation of oscillatory systems in networks — a learning signal approach
title_sort	adaptation of oscillatory systems in networks — a learning signal approach
publisher	Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України
publishDate	2014
topic_facet	Прогресивні інформаційні технології, високопродуктивні комп’ютерні системи
url	http://dspace.nbuv.gov.ua/handle/123456789/85498
citation_txt	Adaptation of oscillatory systems in networks — a learning signal approach / J. Rodriguez // Системні дослідження та інформаційні технології. — 2014. — № 2. — С. 53-67. — Бібліогр.: 10 назв. — англ.
series	Системні дослідження та інформаційні технології
work_keys_str_mv	AT rodriguezj adaptationofoscillatorysystemsinnetworksalearningsignalapproach
first_indexed	2025-07-06T12:46:28Z
last_indexed	2025-07-06T12:46:28Z
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fulltext	 Julio Rodriguez, 2014 Системні дослідження та інформаційні технології, 2014, № 2 53 УДК 518.58 ADAPTATION OF OSCILLATORY SYSTEMS IN NETWORKS — A LEARNING SIGNAL APPROACH JULIO RODRIGUEZ We consider a network of coupled periodic stable signals (PSS) interacting through the gradient of a coupling potential. Each PSS has its own set of parameters k , characterizing the time scale of the signal and its shape. The k are allowed to modify their values (i.e. to adapt) by introducing adaptive mechanisms on them. To- gether with the state variable interactions, the adaptive mechanisms drive all PSS towards a consensual oscillatory state where they all have a common, constant set of parameters .c Once reached, the consensual oscillatory state remains invariant to the interactions. This implies that if the interactions are removed, all PSS continue to deliver the consensual signal. This situation is to be contrasted with classical syn- chronization problems where common dynamical patterns are attained and main- tained thanks to the interactions. Hence, if the interactions are removed, all PSS converge back towards their individual behavior. The resulting value c is ana- lytically calculated. It does not depend on the network’s topology. However, the conditions for convergence do depend on the connectivity of the network and on the coupling potential. 1. INTRODUCTION Producing stable oscillatory motion is of great importance for a device delivering stable periodic signals. Due to its stability mechanism, the apparatus sends out signals that are not drastically altered even if it is placed in a noisy environment. However, structural changes within the device may occur (e.g. the stability mechanism itself may be perturbed), and these create permanent discrepancies, thus lowering the quality of the output signal. To overcome this problem, a signal can be coupled to another of its like. As an example, consider two coupled signals )(1 tr and )(2 tr in the setup 2,1),,();( 21     krr r V rRr k kkk (1) with the gradient of a potential V as coupling function. Here, the set of parame- ters 10 21  due to a structural change. Synchronizing signals may enhance the overall quality in the sense that now, under suitable conditions, )()(lim , trtr Vkk   (for )2,1k with signals )(, tr Vk having the same perio- dicity .Vt However, synchronized signals )(,1 tr V and )(,2 tr V only exist at the cost of maintaining the coupling — if coupling vanishes (i.e. )0V , the two individual signals return, respectively, towards the signals produced by the vector fields Julio Rodriguez ISSN 1681–6048 System Research & Information Technologies, 2014, № 2 54 ,);( kR  .2,1k Furthermore, ,)(, tr Vk and consequently period ,Vt is subject to any change in the coupling: if V changes, the synchronized signals, as well as their periodic behavior, are perturbed. One way to tackle this problem is to construct systems that can synchronize and simultaneously “adapt” local characteristics (i.e. k ) in order to be:  closer to their likes (i.e. reduce the difference ;)11   less dependent on the coupling (i.e. find k  such that  ),( 21  Z ))(),((( 21 trtrV  is minimum over a period and )(trk  solves Equations (1)). An optimum solution for I and II is when there exists a consensual parameter set ,kc  2,1k such that .0),(  ccZ In this situation, if the coupling is removed, the devices continue to deliver the same signal. Furthermore, at this consensual state, any changes in the coupling does not affect the signals since they are now independent of it. Technically, for local parameters k to adapt, they must become time- dependent (i.e. )(tkk  and have their own dynamics. For n coupled sig- nals, having each an additional phase variable controlling their time scale, the general complex networks dynamics is ,),(),,( r V crP k kkkkk       ,),(),,( dynamicscoupling dynamicslocal   r r V crRr k kkkkk     ,,,1 nk  (2)   mechanismsadaptive ),( rAkk  with ,),( 1 n  ),,( 1 nrrr  and coupling strengths .0kc The local dy- namics belong to the class of PR systems (i.e phase-radius systems): P and R gov- ern the dynamics of the local oscillator’s phase and radius, respectively. Adapting parameters in complex systems has long been a busy field of research [1–10]. Whereas in other contributions adaptation occurs in the coupling strength [5] or directly in the connections [2], Equations (2) describe adaptation in the local sys- tems. As mentioned in [6], for local systems’ parameter adaptation, there exist two types: flow parameters controlling the frequency or time scale on an attractor, and geometric parameters determining the shape of the attracting set. Frequency or time scale controlling parameters have, in general, a high propensity for adap- tation and have been well studied in [3, 10, 1, 7]. However, not much has been accomplished for shaping local attractors, which, by nature, is a more delicate task — as stated in [8, 9]. In this paper, we present new adaptive mechanisms for modifying the local system’s attractor. Whereas in [8, 9] the adaptive mechanisms implicitly depend Adaptation of Oscillatory Systems in Networks — A Learning Signal Approach Системні дослідження та інформаційні технології, 2014, № 2 55 on the parameter set k via a functional, ours solely depend on the state vari- ables k and .kr Note that adaptive mechanisms should only depend on the state variables since, in practice, these are the only information available. In [8, 9], one needs to calculate or numerically compute an integral beforehand to know the sign of the function for the adaptive mechanism. Our approach is systematic for all parameters. This contribution is organized as follows: We present individually the com- ponents of our network’s dynamical system in Section 2. In Section 3 we discuss the resulting dynamics and present two related alternatives to our system. Nu- merical simulations are reported in Section 4, and we conclude in Section 5. 2. NETWORKS OF PERIODIC STABLE SIGNALS WITH ADAPTIVE MECHANISMS Consider a n — vertex connected and undirected network with positive adjacency entries. To each node corresponds a local dynamical system defined in Sec- tion 2.1. While the network topology (i.e. adjacency matrix) of the underlying network indicates if the thk local system is connected to the thf (and vice versa), it is the coupling dynamics discussed in Section 2.2 that describes how the neighboring local dynamics interact. Described in Section 2.3, supplementary in- teractions directly acting on the local systems’ parameters will play the role of adaptive mechanisms. Let us now individually present each three dynamical com- ponents. 2.1. Local Dynamics The local systems belong to the class of PR systems. We here focus on Periodic Stable Signals (PSS), which we define as kkkk wrP  );,( ,)())(();,( dynamicsyoscillatirdynamicsedissipativ    kkkkkkkk wFFrrR   ,,,1 nk  (3) with    q m kmkkmkkkk mvmuuF 1 ,,0, )(sin)(cos)(  . The set of parameters is .},,,,,,{ ,,1,1,0, qkqkkkkkk vuvuuw  Parameter kw controls the time scale of the phase, which here oscillates uniformly (i.e. .))( kkk twt   The rest of the parameters determine the shape of the stable periodic signal produced by a PSS. Stable here means that if the system endures a perturbation, it will converge back to its oscillatory motion and continue to deliver the signal with its original shape given by the compact set .}0)(\|),{( 1   kk Frr  The convergence towards k is discussed in Appendix A. It is the dissipative dynamics that is responsible for driving the orbits towards k . This term is the gradient (with respect to the variable )r of the potential .))(( 2 1 2kFr  It is seen as Julio Rodriguez ISSN 1681–6048 System Research & Information Technologies, 2014, № 2 56 an energy controller that takes in and/or gives out energy (depending on the system’s state) until it reaches its equilibrium state k . On k , the PSS’s dynamics is governed by the oscillatory dynamics and so )(trk ,0)())((  ttF kkk   which is consis- tence with Equation (3) when the dissi- pative dynamics is zero. When ,)( 0,kkk uF  the PSS is a limit cycle oscillator with constant angular velocity and a circle of radius 0,ku as an attractor. As sketched in Fig. 1, PSS may form more complicated and interesting attractors. 2.2. Coupling Dynamics The coupling dynamics is here given by the gradient of a positive semi-definite coupling potential 0),( rV  (see Section 1.1.2 in [6] for a precise definition). On ,V we have the following assumptions 0),(,,1and1  rVzyzry   with ,),,( 1 n  ),,( 1 nrrr  and .)1,,1(1  Bellow, we present two ex- amples. Example 1. Laplacian Potential Define V as    n j jkjjk n k kr rlrLrrV 1 , 1 cos )(cos 2 1 \| 2 1 ),(  with ),,( 1 n  and ,),,( 1 nrrr  and where the matrix rLcos has entries )(cos, jkjjk rl   with L being the corresponding Laplacian matrix ,( ADL  where D is the diagonal matrix with    n j jkkk ad 1 ,, ). Matrix rLcos is positive semi-definite since, by Гершгорин’s circle theorem [4], all its eigenvalues are positive (i.e. nonnegative). Explicitly, the coupling dynamics for this potential is ,)(sin),( 1 ,     n j jkjkjkk k k rrlcr V c   ,,,1 nk  ,)(cos),( 1 ,     n j jkjjkk k k rlcr r V c  ,,,1 nk  where 0kc are coupling strengths. Fig. 1. Sketch of an attractor for a PSS. The dynamics evolves at a constant angular velocity twt kk )( on the black thick curve Adaptation of Oscillatory Systems in Networks — A Learning Signal Approach Системні дослідження та інформаційні технології, 2014, № 2 57 Example 2. B Potential Define V as 0))()(( 2 1 ),( , ,,,   n jk jkjkjkjkjk rrBBarV  with edge weights ,0 ,kjkj aa  and where functions jkB , satisfy ,0)(, xB jk ,00)(,  xxB jk )()( ,, xBxB jkjk  (i.e. even function) and )0(0 , jkB . We here impose kjjk BB ,,  . For the functions ,, jkB one may take one of the cases )( 2 1 )( 2 , xxB jk       Diffusion 1)(cosh)(,  xxB jk )(cos1)(, xxB jk  Kuramoto-type ))((coshlog)(, xxB jk  . Explicitly, the coupling dynamics for this potential is ,)(),( 1 ,,     n j jkjkjkk k k Bacr V c   ,,,1 nk  ,)(),( 1 ,,     n j jkjkjkk k k rrBacr r V c  nk ,,1 with coupling strengths .0kc 2.3. Adaptive Mechanisms Here, adaptive mechanisms are additional interactions that modify the values of the local parameters. For this, the fixed and constant parameters k are now time-dependent (i.e.  },,,,,,{ ,,1,1,0, qkqkkkkkk vuvuuw  ,)())(),(,),(),(),(),(( ,,1,1,0, ttvttvtttw kqkqkkkkk    for nk ,,1 ) and each have their own dynamics depending only on state vari- ables  and ,r that is, for all ,k 0   k kA with 0 a q22 dimensional vector of .0 Time scale Adaptive Mechanisms For adaptation on ,k we apply the same idea as developed in [8, 6] and so the explicit dynamics is ),(),( r V srA k k k         ,,,1 nk  where 0 k s are susceptibility constants, technically playing the role of cou- pling strengths but with the following interpretation: the smaller the value of k s , Julio Rodriguez ISSN 1681–6048 System Research & Information Technologies, 2014, № 2 58 the easier it is for the PSS to modify the value of its ,k and vice versa — the larger the value of , k s the harder it is for the PSS to modify the value of its .k Amplitude Adaptive Mechanisms Inspired by attractor-shaping mechanisms studied in [8, 6] we propose, for the PSS’s mkk ,0, , and mkv , , the following new adaptive mechanisms ,)(),( 1 , ,1 , 0,0, 0       n j jjkj sj j n j jjkk rls F rlsrA kk      ,,,1 nk  ,)(cos)(),( 1 , ,1 , ,,       n j jjjkj mj j n j jjkk mrls F rlsrA mkmk m      ,,,1 nk        n j jjjkvj mj j n j jjkv v k mrls v F rlsrA mkmk m 1 , ,1 , ,)(sin)(),( ,,  ,,,1 qm  where jkl , are the entries of L and strictly positive , 0,k s , ,mk s and mkvs , are susceptibility constants. 3. NETWORK’S DYNAMICAL SYSTEM WITH TIME SCALE AND AMPLITUDE ADAPTATION Combining the individual components discussed in Section 2 yields the global dynamical system ,),( r V c k kkk       ,,,1 nk  ,),()())(( dynamicscoupling dynamicslocal     r r V cFFrr k kkkkkkkk     ,,,1 nk  , ),()( mechanisms adaptive scale time     r V s k k k       ,,,1 nk  , 1 ,0, 0,    n j jjkk rls k ,,,1 nk  ),(cos 1 ,, , j n j jjkmk mrls mk      ,,,1 qm  ,)(sin mechanisms adaptive amplitude 1 ,, ,     j n j jjkvmk mrlsv mk    .,,1 qm  (4) Equations (4) describe the dynamics of n PSS (i.e. local dynamics) coupled by the gradient of a coupling potential V (i.e. coupling dynamics) with frequency Adaptation of Oscillatory Systems in Networks — A Learning Signal Approach Системні дослідження та інформаційні технології, 2014, № 2 59 adaptation (i.e. time scale adaptive mechanisms) and attractor shaping (i.e. ampli- tude adaptive mechanisms). For Equations (4), we have 1q constants of motion, the existence of a consensual oscillatory state and the convergence towards it. 12 q Constants of Motion The functions ,)( 1 0, 0 0, 0    n k k k s J     ,)( 1 , ,    n k mk m mk m s J     ,)( 1 , ,    n k v mk mv mk m s v vJ qm ,,1 (5) with ,),,( 0,0,10 n  ),,( ,,1 mnmm   , and ),,( ,,1 mnmm vvv  are constants of motion. Indeed, if ,)(0 t ,)(tm )(tvm for qm ,,1 are orbits of Equa- tions (4), then     n k n j jjk rl dt tJd 1 1 , 0 , ))](([ 0      n k j n j jjk m mrl dt tJd m 1 1 , ,)(cos, ))](([       n k j n j jjk mv mrl dt tvJd m 1 1 , ,)(sin, ))](([  qm ,,1 by Lemma D.2 in [6]. If we further suppose that ),( 1 r Vn k k            for all ),( r (and this is true for both types of coupling potentials in Example 1), then Equa- tions (4) admit another constant of motion, namely    n k k k s J 1 )(     with .1 ),,( n  Existence of a Consensual Oscillatory State Equations (4) admit a consensual oscillatory state. Indeed, for given common constants ,,,,,,,( ,,1,1,0, qcqcccccc vv   ),),(,())(),(),(( ccckkk tFtttrt   nk ,,1 (7) is a consensual orbit of Equations (4), with here cF taking the value .c Indeed, since points given by Equations (7) are extrema of the ,V then the coupling dy- namics and the adaptive time scale mechanisms are zero. Hence, )(tk is a con- stant taking value c for all ,k and ))(,())(),(( tFttrt cckk   solves each local dynamics and cancels all amplitude adaptive mechanisms for all .k Therefore ))(),(,),(),((),(( ,,1,1,0, tvttvtt qkqkkkk   are constants taking, respectively, common values ,,,,,,( ,,1,1,0, qcqcccc vv   for all .k Julio Rodriguez ISSN 1681–6048 System Research & Information Technologies, 2014, № 2 60 Convergence Towards a Consensual Oscillatory State If perturbations are introduced in Equations (7), we say that System (4) converges towards a consensual oscillatory state if we have the following limit ktFtttrt ccckkk t   0)}),(),(()(),(),({lim  (8) with constant .c This limit raises two problems: determining the limit values c and finding the conditions for convergence. Limit Values. If the constant of motion in Equation (6) exists and if Limit (8) holds, then, thanks to all the other constants of motion in Equations (5), we have , 1 )1())(lim())((lim))0(( 1    n k cc tt k s JtJtJJ    , 1 )1())(lim())((lim))0(( 1 0,0,000 0, 0000    n k cc tt k s JtJtJJ    , 1 )1())(lim())((lim))0(( 1 ,, ,    n k mcmcm t m t m mk mmmm s JtJtJJ       n k v mcmcvm t vmv t mv mk mmmm s vvJtvJtvJvJ 1 ,, , 1 )1())(lim())((lim))0(( for .,,1 qm  Hence, the consensual values of c are analytically expressed as , 1 )0( , 1 )0( 1 1 0, 0, 1 1 0, 0,          n k n k j cn k n k k c k k k k s s s s         ,,,1 qm  , 1 )0( , 1 )0( 1 1 , , 1 1 , , , , , ,          n k v n k mj mcn k n k mj mc mk mk mk mk s s v s s       .,,1 qm  (9) Convergence Conditions. To prove the convergence in Limit (8), one can linearize Equations (4) around a consensual oscillatory state. In general, the re- sulting )24()24( qnqn  Jacobian depends explicitly on time (since evaluated on a consensual oscillatory state) and therefore Floquet exponents have to be computed. Note that for certain coupling potentials V and assumptions on the coupling strengths and susceptibility constants, the Jacobian can be diagonalized in order to reduce the computation of Floquet exponents to n systems, each of size .)24()24( qq  We emphasize that numerous numerical simulations show that Limit (8) holds — and this for different topologies, coupling potential and values of coupling Adaptation of Oscillatory Systems in Networks — A Learning Signal Approach Системні дослідження та інформаційні технології, 2014, № 2 61 strengths and susceptibility constants. For these numerical experiments, the cou- pling strengths were set around one and susceptibility constants around .1.0 REMARK: ADAPTATION Here, adaptation is to be interpreted as an asymptotic stability problem, which is directly related to the study of Limit (8). Indeed, for initially different PSS, if Limit (8) holds, then the adaptive mechanisms, with the help of the coupling dy- namics, drive all local systems towards a consensual oscillatory state as defined in Equations (7). Once this state is reached, the coupling dynamics, as well as the adaptive mechanisms, may be removed — and all PSS will still continue to de- liver the same signal with the same time scale (i.e. local system are no longer de- pendent on their environment to produce common dynamical patterns). This is because the values k have been permanently modified (i.e. .))(lim ck t t   If the adaptive mechanisms are not switched on initially, dynamical patterns may occur (due to the coupling dynamics) — but these are maintained because of the network interactions. If the interactions are removed, all PSS converge back to- wards their own shape, which is determined by k and their own time scale, given by .kw 3.1. Miscellaneous Remark: Time Scale or Amplitude Adaptation Only We present here two alternatives of System (4). One alternative concerns ampli- tude kr adaptation only (Section 3.1.1), whereas the other deals with time scale k adaptation only (Section 3.1.2). 3.1.1. Amplitude Adaptation Only Consider Equations (4) with no phases k (and hence no time scale adaptive mechanisms), and for each local PSS, let ttk )( for all .k The system becomes ,)()())(( dynamicscoupling dynamicslocal      r r V ctFtFrr k kkkkk    , 1 ,0, 0,    n j jjkk rls k ,,,1 nk  ),()(cos ,, r r V mts k mk mk     ,,,1 nk  ,)()(sin mechanismsadaptiveamplitude , ,     r r V mtsv k vmk mk    .,,1 qm  (10) with .)(sin)(cos)( 1 ,,0,    q m mkmkkk mtvmttF  Note that for Equations (10) we still have 1q constants of motion (as given in Equations (5)), the existence Julio Rodriguez ISSN 1681–6048 System Research & Information Technologies, 2014, № 2 62 of a consensual oscillatory state )),(())(),(( cckk tFttr  for ,,,1 nk  and the convergence towards it (i.e. 0})),(())(),(({lim   cckk t tFttr for all )k as observed by numerous numerical simulations. A priori, Equations (10) describe a system where adaptation only occurs on the shape of the local attractors. However, by adequately setting the value of one or several constants of motion in Equations (5), one can cancel the asymptotic values of the corresponding coefficients. Thus, by changing the shape of the sig- nal, one can change its frequency. 3.1.2. Time scale Adaptation Only We here remark that PSS (i.e. belonging to the class of PR System) can be slightly modified in order to be seen as Ortho-Gradient (OG) systems. For a pre- cise definition and examples, see Section 1.1.1 in [6]. Briefly, OG systems are characterized by dissipative dynamics that are orthogonal to their canonical — or here, oscillatory-dynamics. Let us consider the following network of PSS that are also OG systems, and where there is only time scale adaptation ,)()())(( dynamicscoupling      k kkkkkk V cFFr    ,))(()( dynamicslocal    kkkkkk FrFr   ,,,1 nk  (11) .)( mechanismsadaptivescaletime       k k V s k    As shown in Lemma 1.1 in [6], each local dynamics in Equations (11), taken individually, possesses its own attractor given by . Networks of OG systems with adapting angular velocities have been studied. For the particular type of cou- pling dynamics and time scale adaptive mechanisms (i.e. only on variables k ), one can directly apply Proposition 2.2 in [6] to show that System (11) converges towards a consensual oscillatory state with consensual value c as in Equations (9). For this convergence, one needs to suppose that )(\|1 V for all  and to make a technical hypothesis on .V 4. NUMERICAL SIMULATIONS We report two sets of numerical simulations, one with time scale and amplitude adaptation (refer to Section 4.1.) and one with amplitude adaptation only (refer to Section 4.2.). 4.1. Time Scale and Amplitude Adaptation We consider 39 PSS as in Equations (4) with network topology as in Fig. 2,a). Here, each PSS is given by    3 1 ,,0, )(sin)(cos)( m mkmkkk mvmF  for Adaptation of Oscillatory Systems in Networks — A Learning Signal Approach Системні дослідження та інформаційні технології, 2014, № 2 63 .39,,1k The coupling strengths and susceptibility constants are ,1kc 1.0 ,,0,  mkmkkk vssss  for 39,,1k and .3,2,1m A Laplacian po- tential, as in Example 1, is used for the coupling dynamics. The initial conditions ))0(),0(),0(),0(),0(),0(),0(),0(),0(( 3,3,2,2,1,1,0, kkkkkkkkk vvv  are randomly uniformly drawn from  3][5,5][2,2][1,1][,] hhhhhhhh [5,5][7,7][3, hhhhhh  with .225.0h These initial con- ditions determine ,)0(kF and finally, the initial conditions )0(kr are randomly uniformly drawn from .[)0(,)0(] hFhF kk  The resulting dynamics for the variables ,kr k and 2,kv is shown in Fig. 3. Note that the variables kr converge quickly towards a common signal, whereas Fig. 2. Two 39-vertex Network Topologies, “Manhattan” 2,a and Metro of Kyiv 2,b a b Fig. 3. Time evolution of kr (Fig. 3,a), ωk (Fig. 3,b) and 2,kv (Fig. 3,c) for 39 PSS, interacting through the network in (Fig. 2,a) rk ωk a b State variables rk State variables ωk vk,2 c State variables vk,2 Julio Rodriguez ISSN 1681–6048 System Research & Information Technologies, 2014, № 2 64 the variables k and 2,kv take more time to converge towards their asymptotic values. This is due to a relatively strong coupling strength compared to the sus- ceptibility constants. For this setup, we have always observed convergence to- wards a consensual oscillatory state. With the same setup, but with a network as in Fig. 2,b, convergence was not observed for all numerical experiments — as we report in Fig. 4,a–c. However, for the exact same initial conditions as in Fig. 4, if all adaptive mechanisms are switched off (i.e. all susceptibility constants are zero), the network is still able to synchronize as shown in Fig. 4,d. 4.2. Amplitude Adaptation Only Two PSS with amplitude adaptation only as in Equations (10) are considered, with here    3 1 ,,0, )(sin)(cos)( m mkmkkk mtvmttF  for .2,1k The coup- ling strengths and susceptibility constants are ,2kc  mkmkk vsss ,,0,  5.0 for .2,1k and .3,2,1m The coupling potential is .)( 2 1 )( 2 21 rrrV  State variables rk State variables ωk rk ωk a b State variables vk,2 State variables rk vk,2 c d Fig. 4. Time evolution of kr (Fig. 4,a), k (Fig. 4,b) and 2,kv (Fig. 4,d) for 39 PSS, interact- ing through the network in (Fig. 2,b). Time evolution of kr for 39 PSS with all their adaptive mechanisms switched off (i.e. all susceptibility constants are zero), interacting through the network in (Fig. 2,d) Adaptation of Oscillatory Systems in Networks — A Learning Signal Approach Системні дослідження та інформаційні технології, 2014, № 2 65 The initial conditions ))0(),0()0(),0(),0(),0(( 3,23,23,13,10,20,1 vv  are ran- domly uniformly drawn from ,8.4][2.5,8.4][2.3,8.2][2.1,8.0][2.1,8.0]  ,[8.6,2.7][2.5  and the others given by ))0(),0(),0(),0(( 2,12,11,11,1 vv  )1,1,2,2(  and .)1,1,2,2())0(),0(),0(),0(( 2,22,21,21,2 vv  These initial conditions determine ,)0(kF and finally, the initial conditions )0(kr are randomly uniformly drawn from [2.0)0(,2.0)0(] 11  kFkF for .2,1k The resulting dynamics for variables kr and 1,k is shown in Fig. 5. For ]15,0[t , the coupling dynamics and the adaptive mechanisms are switched off and so each PSS generates its individual signal. Because of the choice of the ini- tial conditions ))0(),0(( ,, mkmk v ,2,1, mk the asymptotic values are )0,0())0(),0()][( ,, mcmc v for ,2,1m and so both amplitudes )(1 tr and )(2 tr converge towards )3(sin)3(cos)( 3,3, tvttF ccc   (i.e. Fourier series with mode )3(cos t and )3(sin t only). As a consequence, )(tFc has a higher frequency than any of the two signals before interactions are switched on. This is observed in Fig. 5,a where the two signals have a larger period in the interval ]15,0[ than when they are close to .)(tFc 5. CONCLUSION PSS form a suitable class of systems to investigate the interaction of multi-signal dynamics. Whereas adapting the time scale is a fairly straightforward procedure, shaping the attractor is more complicated. Nevertheless, our dynamical systems show that this can be implemented in a robust manner. The adaptive mechanisms depend solely on the state variables, and no pre-calculations or information on the curvature of the attractor is needed. Fig. 5. Time evolution of kr (Fig. 5,a) and 1,k (Fig. 5,b) for two PSS. Coupling dy- namics and adaptive mechanisms are switched on a 15t (black solid line). State variables rk rk a b State variables µk,1 µk,1 Julio Rodriguez ISSN 1681–6048 System Research & Information Technologies, 2014, № 2 66 The asymptotic values from the resulting dynamics are analytically calcula- ble. The network’s topology and the nature of the coupling potential itself directly influence the conditions for attaining a consensual oscillatory state. Determining basins of attraction with respect to connectivity and coupling functions remains an open question. Apart from investigating the resulting dynamics for directed network connections with time-dependent edges, prospective works would also include merging two adapting PSS communities — one belonging to the class of systems given by Equations (4), and the other described by Systems (10). APPENDIX A: CONVERGENCE TOWARDS COMPACT SET  The convergence towards the compact set }0)(\|),{( 1   Frrk  fol- lows from Lyapunov’s second method with Lyapunov function 2))(( 2 1 ),(  FrrL  . By construction, we have that  rr \|),{( 1   .}0)(  L Computing the time derivative ),(\|),( rrL   rFrFFr  ))(()())((  ))())((())(()())(( wFFrFrwFFr   = .))(( 2Fr  Hence, ),(\|),( rrL  for all  \)(),( 1 r . ACKNOWLEDGMENTS The author thanks Prof. Alexander Makarenko for the interesting conference “NONLINEAR ANALYSIS AND APPLICATIONS” (2nd Conference in mem- ory of corresponding member of the National Academy of Science of Ukraine, Valery Sergeevich Melnik, Ukraine, Kyiv, 4–6 April, 2012) for which he was the main organizer. This work was mainly developed before and at the conference. The author acknowledges the support from the DFG-IRTG 1132 (Deutsche For- schungsgemeinschaft — International Research Training Group) under the project entitled “Internationales Graduiertenkolleg — Stochastics and Real World Models”. REFERENCES 1. Acebrón J., Spigler R. Adaptive frequency model for phase-frequency synchroniza- tion in large populations of globally coupled nonlinear oscillators // Physical Re- view Letters. — 1998. — № 81. — P. 2229–2232. 2. De Lellis P., di Bernardo M., Gorochowski T.E., Russo G. 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Received 23.07.2013 From the Editorial Board: the article corresponds completely to submitted manu- script.

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