Frequencies locking in countable cellular systems, localization of quasi-periodic

solutions of autonomous differential systems

Laurent Gaubert

Centre Européen de Réalité Virtuelle, LISYC EA3883 UBO/ENIB, 25 rue Claude Chappe, 29280 Plouzané, France

Abstract

We address the question of frequencies locking in coupled differential systems, related to the existence of quasi-periodic solutions of differential systems. Our tool is what we call “cellular systems”, quite general as it deals with countable number of coupled systems in some general Banach spaces. Moreover, the inner dynamics of each subsystem does not have to be specified. We reach some general results about how the frequencies locking phenomenon is related to the structure of the coupling map, and therefore about the localization of quasi-periodic solutions of some differential systems that may be seen as cellular systems. This paper gives some explanations about how and why synchronized behaviors naturally occur it a wide variety of complex systems.

Key words: coupled differential systems, synchronization, frequencies locking PACS:

1 Introduction

Synchronization is an extremely important and interesting emergent property of complex systems. The first example found in literature goes back to the 17th century with Christiaan Huygens’ work [?,?]. This kind of emergent behavior can be found in artificial systems as well as in natural ones and at many scales (from cell to whole ecological systems). Biology abounds with periodic and synchronized phenomena and the work of Ilya Prigogine shows that such behaviors arise within specific conditions: a dissipative structure generally associated to a nonlinear dynamics [?]. Biological systems are open, they evolve far from thermodynamic equilibrium and are subject to numerous regulating processes, leading to highly nonlinear dynamics. Therefore periodic behaviors appear (with or without synchronization) at any scale [?]. More generally, life

Preprint submitted to Elsevier 2 March 2009

itself is governed by circadian rhythms [?]. Those phenomena are as much attractive as they are often spectacular: from cicada populations that appear spontaneously every ten or thirteen years [?] or networks of heart cells that beat together [?] to huge swarms in which fireflies, gathered in a same tree, flash simultaneously [?]. This synchronization phenomenon occupies a privileged position among emergent collective phenomena because of its various applications in neuroscience, ecology, earth Science, for instance [?,?,?], as well as in the field of coupled dynamical systems, especially through the notion of chaotic systems’ synchronization [?,?] and the study of coupled-oscillators [?]. This wide source of examples leads the field of research to be highly interdisciplinary, from pure theory to concrete applications and experimentations.

The classical concept of synchronization is related to the locking of the basic frequencies and instantaneous phases of regular oscillations. One of the most successful attempts to explore this emergent property is due to Kuramoto [?,?]. As in Kuramoto’s work, those questions are usually addressed by studying specific kinds of coupled systems (see for instance [?,?,?]). Using all the classical methods available in the field of dynamical systems, researchers study specific trajectories of those systems in order to get information on possible attracting synchronized state [?,?,?,?,?,?].

The starting point of this work was the following question : “Why synchronization is such a widely present phenomena ?” In order to give some mathematical answer to this question, the first step is to build a model of coupled systems that is biologically inspired. This is what is done in the second section, after having described some basic material, we define what we call cellular systems and cellular coupler. If one would summarize the specificities of cellular system, one could say that each cell (subsystem) of a cellular system receives information from the whole population (the coupled system) according to some constraints:

• a cell has access to linear transformations of all the others cell’s states • the way this information is gathered depends (not linearly) on the cell’s

state itself

In other words, a cell interprets its own environment via the states of the whole population and according to its own state.

It’s a bit surprising that despite this model arising very naturally, it gives a good framework to address the main question. Indeed, in the third section we expose a localization result concerning periodic trajectories of cellular systems, according to some sub-periods dependencies. It exhibits some links between the coupler’s properties and the structure of periodic trajectories.

The fourth section gives some example of general results that may be proved using the localization lemma. Moreover, it goes out of the scope of coupled systems as, synchronization is strongly related to the more abstract field of dynamical systems. If one thinks about presence of regular attractors (in opposition with strange attractor) in a differential system, one may for example classify those as:

• point attractor

• limit cycle

• limit torus

Those attractors can be related to coupled systems in an obvious way: roughly speaking, a point attractor may be seen as a solution of coupled systems for which each of the subsystems has a constant behavior. Similarly, a limit cycle may be thought as the situation where every subsystem oscillate, all frequencies among the whole system being locked. A limit torus is a similar situation which differs from the previous one by the fact that the frequencies are not locked (non commensurable periods of a quasi-periodic solution of the whole coupled system). Hence, the three previous cases may be translated into the coupled dynamical systems context:

• point attractor ↔ constants trajectories of each subsystem

• limit cycle ↔ periodic sub-trajectories with locked frequencies

• limit torus ↔ periodic sub-trajectories with unlocked frequencies

Therefore, we deduce some results about the localization of solutions of the third type, quasi-periodic solutions, using the point of view of coupled dynamical systems. The results of this third section may help to understand why the second case is the most observed in natural systems, which may be seen as coupled dynamical systems, at many levels. Indeed, the section ends with a sketch of how the cellular systems point of view may be applied to a wide class of differential systems in order to systematically address those questions with algebraic tools.

2 Basic material and notations

As our model is inspired by cellular tissues, some terms clearly come from the vocabulary used to describe those kinds of complex systems.

2.1 Model of population behavior

Here are the basic compounds and notations of our model:

A population I is a countable set, so we may consider it as a subset I ⊂ ℕ.

Moreover, as it’s only the cardinality of I that’s important, I may be chosen as an interval of integer. Elements of I are called cells.

We suppose that the systems we want to study are valued in some Banach spaces. Thus, for any i ∈ I, (E_{i}, ∥.∥_{i}) is a Banach space, and the state space of I is the vector space S = ∏_{i∈I} E_{i}.

We will sometimes identify E_{i} with

∏_{j<i}{0} × E_{i} × ∏_{j>i}{0} ⊂ S

and then consider it as a subspace of S.

We denote S_{b} the space of uniformly bounded states:

S_{b} = {x ∈ S, sup_{i∈I} ∥x_{i}∥_{i} < ∞}

This subspace will sometimes be useful as, embodied with the norm ∥x∥_{∞} = sup_{i∈I} ∥x_{i}∥_{i}, it’s a Banach space, allowing the classic PicardLindelf theorem to be valid.

Given an interval Ω ⊂ ℝ, a trajectory x of I is an element of C^{∞} (Ω, S). Such x is then described by a family of C^{∞} applications (x_{i})_{i∈I} such that ∀i ∈ I:

\begin{aligned}x_{i} : & Ω & ⟶ & E_{i} \\ & t & ⟼ & x_{i}(t) \end{aligned}

The space of trajectories on I is denoted T .

A period on I is a map τ : I → ℝ_{+}^{∗}. A trajectory x ∈ T is said to be τ-periodic if for any i ∈ I, x_{i} is τ(i)-periodic and non constant. τ(i) is then said to be a period of the cell i. The space of such trajectories is written T_{τ}.

Each cell i is supposed to behave according to an autonomous differential system given by a vector field F_{i} : E_{i} → E_{i}. Thus, given a family of functions {F_{i}}_{i∈I} we define the vector field F_{I} on S:

\begin{aligned}F_{I} : & S & ⟶ & S \\ & x & ⟼ & F_{I}(x) \end{aligned}

with, for any i ∈ I:

[F_{I}(x)]_{i} = F_{i}(x_{i})

Remark 1 The definition of periodic trajectory handle both classical concepts of periodic and quasi-periodic solutions of a differential system. From the point of view of coupled systems, it describes the situation in which each subsystem of the whole system oscillates. We stress the point that a period of a periodic trajectory needs not to be a minimal period (τ(i) isn’t necessarily a generator of the group of periods of x_{i}). Nevertheless, our definition of T_{τ} avoid any trajectory which contains some constant component (none of the x_{i} can be a constant map) as they may be seen as degenerate (localized into an hyperplane of S).

We recall that a (finite) subset {τ_{1}, …, τ_{k}} of ℝ is said to be rationally dependent if there exists some integers l_{1}, …, l_{k} non all zero and such that:

l_{1}τ_{1} + … + l_{k}τ_{k} = 0

Then there exists a unique lowest common multiple (lcm) τ_{0} for which there exits n_{1}, …, n_{k} such that:

n_{1}τ_{1} = … = n_{k}τ_{k} = τ_{0}

An infinite set of real numbers is said to be rationally dependent if any finite subset is rationally dependent.

Now, any period τ on I (or, equivalently, any periodic trajectory) defines a equivalence relation on I as:

i ∼ j ⇔ {τ(i), τ(j)} is a dependent set

Hence we may consider the (countable) partition I(τ) of I into equivalence classes (K countable):

I(τ) = {I_{k}}_{k∈K}

2.2 Cellular coupler and cellular systems

In this section we build what we call cellular systems by way of cellular coupler. Most of the works always deal with a specific way of coupling dynamical systems: one adds a quantity (that models interactions between subsystems) to the derivative of the systems. This leads to equations with the following typical shape (here, there are only two coupled systems):

x′_{1}(t) = F (x_{1}(t)) + G_{1}(x_{1}(t), x_{2}(t))
x′_{2}(t) = F (x_{2}(t)) + G_{2}(x_{1}(t), x_{2}(t))

The functions G_{1} and G_{2} are the coupling functions. The problem is then restated in terms of phase-shift variables and efforts are made to detect stable states and to prove their stability.

Our approach to coupling is different. We study exclusively a way of coupling where the exchanges are made on the current state of the system. This means that the coupling quantity applies inside the map F, which leads us to the following type of equation:

x′_{1}(t) = F (x_{1}(t) + H_{1}(x_{1}(t), x_{2}(t)))  (1)
x′_{2}(t) = F (x_{2}(t) + H_{2}(x_{1}(t), x_{2}(t)))

Remark 2 We stress the point that those two different ways of handling coupled systems are quite equivalent in most cases. Indeed, starting with the first two equations, as soon as G_{1} and G_{2} stay in the range of F (which is likely if the coupling functions are small), we can rewrite them in the second shape involving H_{1} and H_{2}.

The last kind of coupled systems is sometimes studied (for instance in [?]) but never broadly (indeed, if one wants some quantitative results about convergence of trajectories, one must work with specific equations and dynamical systems). Even in a few papers that are quite general (as the very interesting [?]) some strong assumptions are made (in [?] authors deal with symmetric periodic solutions). The kind of coupled systems we handle are a generalization of the one describe in equation (??). Its general shape is:

x′_{i}(t) = F_{i} (∑_{j∈I} c_{ij}(x_{i}(t))x_{j}(t))

Each cell i ∈ I holds it’s own differential system represented by a map F_{i} (hence, all the dynamical systems are not forcibly identical nor have the same shape, nor that they are weakly coupled (as in the classical paper of Art Winfree [?]). A cell i “interprets” it’s own environment by mean of the functions c_{ij}.

Before giving the exact definition of a cellular coupler, we recall that, as S may be seen as a module on the ring ∏_{i∈I} L(E_{i}), L(S) has to be understood as the space of linear operators on S with coefficients in the L(E_{i}, E_{j}). Any M ∈ L(S) may then be written as an infinite (if I isn’t finite) matrix:

M = [m_{ij}]_{(i,j)∈I^{2}} , m_{ij} ∈ L(E_{j}, E_{i})

In this context, here is the definition of a cellular coupler on I:

Definition 3 A cellular coupling map on I is a map c:

\begin{aligned}c : & S & ⟶ & L(S) \\ & x & ⟼ & c(x) \end{aligned}

such that the matrix [c_{ij}]_{(i,j)∈I^{2}} satisfies:

(1) ∀(i, j) ∈ I^{2}, ∀x ∈ S, c_{ij}(x) depends only on x_{i}, so that we may consider it as a map c_{ij} : E_{i} → L(E_{j}, E_{i})

(2) ∀(i, j) ∈ I^{2}, c_{ij} has continuous values: c_{ij} : E_{i} → L_{c}(E_{j}, E_{i})

a garder ?

(3) ∀i ∈ I, ∀x_{i} ∈ E_{i}, ∑_{j∈I} ∥c_{ij}(x_{i})∥_{i} < +∞

Then, c defines a cellular coupler c̃ on I in the following way:

\begin{aligned}c̃: & S & ⟶ & S \\ & x & ⟼ & c(x).x \end{aligned}

In other words (for the sake of simplicity, we only take examples with a finite population), for any x ∈ S, the matrix c(x) has the following shape:

c(x) = [ \begin{matrix}c_{11}(x_{1}) & ⋯ & c_{1k}(x_{1}) \\ ⋮ & ... & ⋮ \\ c_{k1}(x_{k}) & ⋯ & c_{kk}(x_{k}) \end{matrix} ] ∈ L(S)

And then :

c̃(x) = c(x).x = c(x) = [ c_{11}(x_{1}).x_{1} + … + c_{1k}(x_{1}).x_{k} ; ⋮ ; c_{k1}(x_{k}).x_{1} + … + c_{kk}(x_{k}).x_{k} ] ∈ S

Now we can define a cellular system:

Definition 4 Let F_{I} be a vector field on S given by a family {F_{i}}_{i∈I} of vector fields on the E_{i}. Let c̃ a cellular coupler on I. (I, F_{I}, c̃) is called a cellular system. A trajectory of this system is a trajectory x ∈ T that satisfies:

x′ = F_{I} ∘ c̃(x) = F_{I}(c(x).x)

in other words:

∀i ∈ I, ∀t ∈ Ω, x′_{i}(t) = F_{i} (∑_{j∈I} c_{ij}(x_{i}(t)).x_{j}(t))

This equation may be naturally interpreted in biological terms: the cell i behaves according to a mean of the states of all other cells x_{j}, but only its state defines how this mean is computed (the cell interprets its own environment), and this link state↔ interpreting function has no reason to be linear in x_{i}.

Remark 5 In order to avoid any confusion, we stress the differences between trajectory and solution regarding periodic behaviors. In this paper, periodic trajectory has a specific meaning related to the cells. A periodic trajectory of a cellular system is a trajectory for which each cell has a periodic behavior. From the classic point of view of differential equations, periodic and quasiperiodic solutions of the cellular system are periodic trajectories. In the case of a periodic solution, τ(I) admit a lcm, which false in the case of a quasiperiodic solution.

In the next section we start of by exposing algebraic links between a cellular coupler and a periodic trajectory, and then we turn to our localization lemma.

3 Localisation lemma

If M is a matrix indexed on I^{2}, and if J ⊂ I, we write \overline{J} = I − J and we define M^{J} as the matrix:

M = (m_{ij})_{(i,j)∈J×\overline{J}}

For x ∈ S (resp. x ∈ T ) we denote x^{J} the vector (resp. the map) [x_{i}]_{i∈J} (see figure ??).

If I(τ) = {I_{1}, ..., I_{K}} is a partition of I, we define the matrix M^{I(τ)} as (see

Fig. 1. Matrix and vectors associated to a subset J of I.

figure ??):

m_{ij}^{I(τ)} = \{ \begin{matrix}0 & if (i, j) ∈ I_{1} × I_{1} ∪ … ∪ I_{K} × I_{K} \\ m_{ij} & if not \end{matrix}

Fig. 2. Matrix associated to a partition of I.

We can then go further and begin to work on the heart of our matter. The forthcoming result that can be used in many ways and generalized as, for the sake of simplicity, we did not use the weakest assumptions under which it holds (for example, the series convergence in the proof can be insured in many other contexts).

Lemma 6 Let (I, F_{I}, c̃) be a cellular system and τ a period on I. Let U ⊂ S on which F_{I} is injective. If x ∈ T^{τ} is a periodic trajectory of cellular system that satisfies:

(1) x(Ω) ⊂ S_{b} ;

(2) c̃(x)(Ω) ⊂ U

then there exists b ∈ S_{b} such that for any t ∈ Ω:

x (t) − b ∈ ker (c(x(t))^{I(τ)})

Remark 7 Note that the first condition on x is useless if I is finite.

The previous result is not very practical as it involves the trajectory x itself, which is unknown. As there is no ambiguity, we define the kernel of c^{I(τ)} as:

ker (c^{I(τ)}) = ⋃_{x∈S} ker (c(x)^{I(τ)})

Hence we may give a weaker version of the previous lemma

Corollary 8 Under the conditions of lemma ?? there exists b ∈ S such that:

x (Ω) − b ∈ ker (c^{I(τ)})

PROOF. [of lemma ??] First of all, let’s check that c̃(x) is τ-periodic. For any i ∈ I, x′_{i} is τ(i)-periodic and non constant for x_{i} is so. Let’s U_{i} = U∩E_{i}, F_{i} has to be injective on U_{i}. Hence, as x is a trajectory of the cellular system, F_{i} (c̃(x)_{i}) must be periodic and then c̃(x)_{i} is τ(i)-periodic. Therefore, c(x) is τ-periodic.

Now, according to the partition I(τ) = {I_{k}}_{k∈K} defined by τ (see section ??), let k ∈ K and i ∈ I_{k}. For any M ∈ ℕ we define the following set:

I_{k}^{M} = I_{k} ∩ ⟦0, M⟧

The set τ (I_{k}^{M} ) is now a finite dependent set, so that we can consider its lcm τ_{k}^{M}. Now, for any j ∈ I_{k}^{M}, x_{j} and c̃(x)_{j} are τ_{j}^{M}-periodic, so that, for any integer N:

c̃(x)_{i}(t) = \frac{1}{N + 1} ∑_{l=0}^{N} c̃(x)_{i} (t + lτ_{k}^{M} )

= \frac{1}{N + 1} ∑_{l=0}^{N} ∑_{j∈I} c_{ij}(x_{i} (t + lτ_{k}^{M} ) ).x_{j} (t + lτ_{k}^{M} )

= \frac{1}{N + 1} ∑_{l=0}^{N} ∑_{j∈I} c_{ij}(x_{i}(t)).x_{j} (t + lτ_{k}^{M} )

= \frac{1}{N + 1} ∑_{l=0}^{N} ∑_{j∈I_{k}^{M}} c_{ij}(x_{i}(t)).x_{j}(t)

+ \frac{1}{N + 1} ∑_{l=0}^{N} ∑_{j∈I_{k}−I_{k}^{M}} c_{ij}(x_{i}(t)).x_{j} (t + lτ_{k}^{M} )

+ \frac{1}{N + 1} ∑_{l=0}^{N} ∑_{j∈\overline{I_{k}}} c_{ij}(x_{i}(t)).x_{j} (t + lτ_{k}^{M} )

= ∑_{j∈I_{k}^{M}} c_{ij}(x_{i}(t)).x_{j}(t)

+ \frac{1}{N + 1} ∑_{l=0}^{N} ∑_{j∈I_{k}−I_{k}^{M}} c_{ij}(x_{i}(t)).x_{j} (t + lτ_{k}^{M} )

+ \frac{1}{N + 1} ∑_{l=0}^{N} ∑_{j∈\overline{I_{k}}} c_{ij}(x_{i}(t)).x_{j} (t + lτ_{k}^{M} )

As x is uniformly bounded, using the 3rd property of a coupling map (def. ??), we may invert the summation orders in the previous equation and compute the limits when M → +∞ and N → +∞ in any order. Thus we have:

c̃(x)_{i}(t) = ∑_{j∈I_{k}^{M}} c_{ij}(x_{i}(t)).x_{j}(t)

+ ∑_{j∈I_{k}−I_{k}^{M}} c_{ij}(x_{i}(t)) [\frac{1}{N + 1} ∑_{l=0}^{N} x_{j} (t + lτ_{k}^{M} )]

+ ∑_{j∈\overline{I_{k}}} c_{ij}(x_{i}(t)) [\frac{1}{N + 1} ∑_{l=0}^{N} x_{j} (t + lτ_{k}^{M} )]

for the same reasons, it’s easy to show that: and

lim_{M→+∞} lim_{N→+∞} ∑_{j∈I_{k}−I_{k}^{M}} c_{ij}(x_{i}(t)) [\frac{1}{N + 1} ∑_{l=0}^{N} x_{j} (t + lτ_{k}^{M} )] = 0

Now, as for all j ∈ \overline{I_{k}}, τ_{k}^{M} and τ(j) are non commensurable, if we denote tau′_{j} the generator of x_{j} group of period, as τ(j) = n_{j}τ′_{j} for a a certain integer n_{j}, τ_{k}^{M}

and τ′_{j} as well are non commensurable. Therefore, the sequence (\frac{t + lτ_{k}^{M}}{τ′_{j}} )_{l∈ℕ}

is equidistributed mod 1, and we may apply some classic ergodic theorem (see for instance [?,?]) and write:

lim_{N→+∞} \frac{1}{N + 1} ∑_{l=0}^{N} x_{j} (t + lτ_{k}^{M} ) = \frac{1}{τ′_{j}} ∫_{0}^{τ(j)} x_{j}(s)ds = \frac{n_{j}}{τ(j)} ∫_{0}^{τ(j)} x_{j}(s)ds

We can now define the state b as:

b = [b_{j}]_{j∈I}, b_{j} = \frac{n_{j}}{τ(j)} ∫_{0}^{τ(j)} x_{j}(s)ds

so that:

lim_{N→+∞} ∑_{j∈\overline{I_{k}}} c_{ij}(x_{i}(t)) [\frac{1}{N + 1} ∑_{l=0}^{N} x_{j} (t + lτ_{k}^{M} )] = ∑_{j∈\overline{I_{k}}} c_{ij}(x_{i}(t))b_{j}

hence, we have shown that:

c̃(x)_{i}(t) = ∑_{j∈I_{k}} c_{ij}(x_{i}(t)).x_{j}(t) + ∑_{j∈\overline{I_{k}}} c_{ij}(x_{i}(t)).b_{j}

But, obviously, from the beginning we had:

c̃(x)_{i}(t) = ∑_{j∈I_{k}} c_{ij}(x_{i}(t)).x_{j}(t) + ∑_{j∈\overline{I_{k}}} c_{ij}(x_{i}(t)).x_{j}(t)

So that:

∑_{j∈\overline{I_{k}}} c_{ij}(x_{i}(t)).x_{j}(t) = ∑_{j∈\overline{I_{k}}} c_{ij}(x_{i}(t)).b_{j}

The previous work can be done for any i which belongs to I_{k}, thus we can summarize in the following way (see previously defined notations):

c(x(t))^{I_{k}}. (x(t) − b)\overline{I_{k}} = 0

Again, the previous conclusion still holds for any k ∈ K, hence we may conclude using our notations:

c(x(t))^{I(τ)}. (x(t) − b) = 0

In the next section we give some examples of results based upon this lemma. We will mainly show how lemma ?? may be applied to turn synchronization issues (and existence of quasi-periodic solutions to a differential system) into algebraic problems. One of the main argument is that one wants to avoid periodic trajectories for which one cell is inert (a constant map), as it may be discarded from the population (in the case of an infinite population, this may lead to recurrence reasoning).

4 Applications

4.1 Weakly injective coupler

In this example we just write down an elementary property of c̃ which ensure that a periodic trajectory must have an inert cell.

4.2 Localization results with bounded states and finite population

Definition 9 Let c̃ be a cellular coupler on I. c̃ is said to be weakly injective if for any non trivial partition I(τ) of I there exist i ∈ I such that:

∀x ∈ S, ker (c(x)^{I(τ)}) ∩ E_{i} = {0}

Now we can state a simple result:

Proposition 10 Under the conditions of lemma ??, if c̃ is weakly injective and if x is a τ-periodic trajectory of the cellular system, then τ(I) is a dependent set.

PROOF. Let suppose that I(τ) is not trivial, applying lemma ?? we know that:

c(x(t))^{I(τ)}. (x(t) − b) = 0

As c̃ is weakly injective, there exists i ∈ I such that:

∀t ∈ Ω, x(t)_{i} = b_{i}

which contradicts the definition of a periodic trajectory.

This result may be restated in terms of quasi-periodic solution of the cellular system:

Proposition 11 Under the conditions of lemma ??, if c̃ is weakly injective and if τ is bounded, the cellular system has no quasi-periodic solution.

The next example deals with some topological properties of a coupler (how it connects cells together).

4.3 Chained cellular system

We first study the case of differential systems for which the spaces E_{i} have same finite dimension and are coupled with k-nearest neighbors (the finite dimension condition isn’t necessary, but it makes the exposure simpler). This case is formally described by a cellular system (I, F_{I}, c̃) where I is countable, all dim(E_{i}) = n and c̃ satisfies:

∀i, j ∈ I, |j − i| > k ⇒ c_{ij} = 0

This is what we call a chained cellular system. Adding the following condition on the coupler, we may reach a general result:

Definition 12 A cellular coupler c̃ is said to have full rank if for any i, j ∈ I and x ∈ S the map c_{ij}(x) has full rank

Proposition 13 Let (I, F_{I}, c̃) be a chained cellular system coupled with knearest neighbors (all E_{i} having same finite dimension). Let F_{I} be injective on U ⊂ S and x a τ-periodic trajectory that stays in U. If c̃ has maximal rank and if there exists I ∈ I(τ) which contains 2k consecutive cells, i.e. there exists i ∈ I such that:

⟦i, i + 2k − 1⟧ ⊂ I

Then I(τ) = {I} (equivalently, τ(I) is a dependent set).

PROOF. Let suppose that I ≠ I. There must exist ⟦i, i + 2k⟧ ⊂ I, such that i − 1 ∉ I. Then, line i + k − 1 of the matrix c(x(t))^{I(τ)} contains only one non zero element c_{i+k−1,i−1}. As this linear map is injective for any t ∈ Ω, we know that:

ker(c(x(t))^{I(τ)}) ⋂ E_{i−1} = {0}

Applying lemma ?? we know that there exists b_{i−1} ∈ E_{i−1} such that for any t ∈ Ω:

x_{i−1}(t) − b_{i−1} ∈ ker (c(x(t))^{I(τ)}) ⋂ E_{i−1}

i.e. x_{i−1}(t) = b_{i−1} is a constant map, which contradicts the definition of a periodic trajectory. So we can conclude that I = I.

If we assume that τ is bounded, this result may be restated as: “as soon as k consecutive cells are synchronized (locked frequencies), then all the population is syncrohnized”.

Moreover, we may drop the assumptions made on the E_{i} and reach an interesting connecting result concerning the case when k = 1.

Proposition 14 Let (I, F_{I}, c̃) be a chained cellular system coupled with 1nearest neighbor. Let F_{I} be injective on U ⊂ S and x a τ-periodic trajectory that stays in U. If c̃ has maximal rank and if there exists two sets I_{1} and I_{2} in I(τ) such that:

⟦i, i + 1⟧ ⊂ I_{1} ⟦i + 2, i + 3⟧ ⊂ I_{2}

Then I_{1} = I_{2}.

PROOF. Let suppose that the cells i + 1 have non commensurable periods with those of the cells i + 2 (i.e. I_{1} ≠ I_{2}). Following the previous proof, we know that the lines i + 1 and i + 2 of the matrix c(x(t))^{I(τ)} contains only one non zero element, respectively c_{i+1,i+2} and c_{i+2,i+1}. But, we recall that for any t ∈ Ω:

c_{i+1,i+2}(x_{i+1}(t)) : E_{i+2} → E_{i+1} , c_{i+2,i+1}(x_{i+2}(t)) : E_{i+1} → E_{i+2}

As the coupler has maximal rank, one of the previous map must be injective for all t ∈ Ω (here we use the Axiom of choice to ensure that there must be a linear injection between the two Banach spaces). Using the same argument we may conclude that either x_{i+1} is a constant map, either it’s x_{i+2}, leading to a contradiction.

Moreover, one could restate those results in terms of quasi periodic solutions of differential systems, but it may sound less intuitive. We will do it in the next sections.

For the next example, we add some regularity conditions on the cellular system which lead to a interesting description of S.

4.4 Localization results with bounded states

As (S_{b}, ∥.∥_{∞}) is a Banach space, the classic PicardLindelf theorem is valid and we can give a version adapted to cellular systems (we stress the point that in this section, any vector field F_{I} has to be a vector field on S_{b}, as well for any cellular coupler c̃, which brings some constraint on the families (F_{i})_{i∈I} and (c_{ij})_{(i,j)∈I^{2}}).

Proposition 15 If F_{I} : S_{b} → S_{b} and c̃ are locally lipshcitz, which is the case if for any x ∈ S_{b} there exists a neighborhood V = ∏_{i∈I} V_{i}, a positive number k

and a sequence (k_{j})_{j∈I} of positive numbers such that:

(1) ∀y, z ∈ V, ∀i ∈ I, ∥F_{i}(y_{i}) − F_{i}(z_{i})∥_{i} ≤ k∥y_{i} − z_{i}∥_{i}

(2) ∀y, z ∈ V, ∀i ∈ I, ∥c_{ij}(y_{i}) − c_{ij}(z_{i})∥_{(E_{j},E_{i})} ≤ k_{j}∥y_{i} − z_{i}∥_{i}

(3) ∑_{j∈I} k_{j} < +∞

then, given any initial condition (t^{0}, x^{0}) in ℝ×S_{b}, the cellular coupling admits a unique maximal solution x that satisfies x(t^{0}) = x^{0}.

Before stating our localization result, we need to define the sets that any non synchronized periodic trajectory of the cellular system must avoid (or, with the classical point of view, any quasi periodic solution).

Definition 16 Let c̃ be a cellular coupler on I. The set of regular points for c̃ is defined as:

R(c̃) = {x ∈ S, ∀I(τ) partition of I, c(x)^{I(τ)} is injective }

We say that c̃ is regular if R(c̃) = S.

Now we can state a localization result:

Proposition 17 Under the conditions of lemma ?? and proposition ??, if there exists a infinite compact subset V ⊂ Ω such that:

∀t ∈ V, x(t) ∈ R(c̃)

then τ(I) is a dependent set.

One can rewrite this result in terms of differential systems:

Proposition 18 Under the conditions of lemma ?? and proposition ??, and if τ is bounded, a quasi-periodic trajectory must “avoid” R(c̃) (it can’t cross this set on an infinite compact subset of Ω).

PROOF. [of proposition ??] Let suppose that I(τ) is not trivial, applying lemma ?? we know that:

c(x(t))^{I(τ)}. (x(t) − b) = 0

the assumptions made on c̃ ensure that:

∀t ∈ V, x(t) = b

As V has an accumulation point, we may conclude that there exists t_{0} ∈ V such that:

x′(t_{0}) = 0

Proposition ?? may be applied, hence we know that t ↦ x(t) is a constant map, which contradicts the definition of a periodic trajectory.

The next example gives a more precise result in the case where the maps c_{ij} don’t depend on the state of the system (homogeneous coupler).

4.5 Exact frequencies locking with homogeneous cellular coupler

If x ∈ T_{τ}, for any i ∈ I the map x_{i} equals its Fourier’s series. We write:

e_{τ(i)}^{k}(t) = exp (\frac{2iπkt}{τ(i)})

and we define :

x̂_{i}(k) = \frac{1}{τ(p)} ∫_{0}^{τ(p)} x_{i}(t)e_{τ(i)}^{k}(t)dt

so that we have :

x = ∑_{k∈ℤ} x̂(k)e^{k}

i.e. ∀i ∈ I:

x_{i}(t) = ∑_{k∈ℤ} x̂_{i}(k)e_{τ(i)}^{k}(t)

with normal convergence (note that x̂_{i}(k) is E_{i}-valued).

Theorem 19 Under the conditions of lemma ??, let c̃ be homogeneous and regular. If τ is a bounded period on I and x a τ-periodic trajectory of the cellular system then τ is constant on I.

Remark 20 As this result is true as soon as τ is a period of x, it may be applied to the minimal periods of each x_{i}, then its conclusion is that all cells have exactly the same minimal period.

PROOF. As c̃ is homogeneous, we may identify it with c. Moreover, applying lemma ?? we know that τ(I) is a dependent set (unless at least one of the x_{i} would be a constant map). We now have to prove that τ is constant on I.

Let’s write a partition of I according to τ’s values on I (we must recall that τ is supposed bounded):

{I_{1}, I_{2}, …, I_{K}}

such that

∀1 ≤ k ≤ K, τ(I_{k}) = τ_{k}

and τ_{l} ≠ τ_{k} if l ≠ k.

We now suppose that K > 1.

As τ(I) is a finite dependent set, there exists n_{1}, …, n_{K} integers and τ_{0} (the lcm) such that:

τ_{0} = n_{1}τ_{1} = n_{2}τ_{2} = … = n_{K}τ_{K}

The trajectory x is τ_{0}-periodic. We may therefore write its Fourier’s series:

x(t) = ∑_{l∈ℤ} x̂(l)e_{τ_{0}}^{l}(t)

and as well for c.x :

(c.x)(t) = ∑_{l∈ℤ} ĉ.x(l)e_{τ_{0}}^{l}(t)

uniqueness of Fourier coefficients forces them to satisfy:

ĉ.x(l) = c x̂(l)

So that, for any i ∈ I:

ĉ.x_{i}(l) = ∑_{j=1}^{k} c_{ij}x̂_{j}(l)

Now, let i ∈ I_{k}, the properties of Fourier decomposition ensure that x̂_{i}(l) and c.x̂_{i}(l) are zero as soon as n_{k} does not divide l (as (c.x)_{i} and x_{i} are τ_{k}-periodic and τ_{0} = n_{k}τ_{k}). So, if l ∈ ℤ, let’s define I(l) as:

I(l) = {k ∈ {1, …, K}, n_{k} |̸ l}

For any integer l, if k ∈ I(l) and i ∈ I_{k}, then x̂_{i}(l) = c.x̂_{i}(l) = 0, so that (with similar convergence arguments that in the proof of lemma ??):

c.x̂_{i}(l) = ∑_{j=1}^{k} c_{ij}x̂_{j}(l)

0 = ∑_{j∈I(l)}^{k} c_{ij}x̂_{j}(l) + ∑_{j∉I(l)}^{k} c_{ij}x̂_{j}(l)

0 = ∑_{j∉I(l)}^{k} c_{ij}x̂_{j}(l)

This last property, (observable on figure ??), can be writen as:

∀l ∈ ℤ c^{I(l)} x̂(l)\overline{I(l)} = 0

Fig. 3. Constraints on the Fourier’s coefficients x̂(l).

This property holds for any integer l, and is empty when l is a multiple of all the n_{i}. So that, if I(l) is the partition of I defined as:

I(l) = \{I(l), \overline{I(l)}\}

we can re-write it as:

∀l ∈ ℤ c^{I(l)} x̂(l) = 0

Let’s now consider I_{1} ≠ I_{2} (this is possible as K > 1). As those two classes are distinct, there exists l such that n_{1} does not divide l and n_{2} divides l. As c is regular, c^{I(l)} is thereby injective. We deduce that:

x̂(l)\overline{I(l)} = 0

This proves that for any l divisible by n_{2} and not by n_{1}, x̂(l)\overline{I(l)} is zero. Thus, for any coefficient of x̂(l)\overline{I(l)} to be non zero, n_{1} must divide l, and consequently (as none of the x_{i} is a constant map) for all i ∈ I_{2}, x_{i}(t) is n_{1}τ_{0} periodic. This is incompatible with the partition of I. Thus, K = 1 and thereby τ is a constant map (in other words, I is synchronized).

4.6 Perspectives of application to classical differential systems

In this last section, we show how the cellular systems point of view may be applied to classic differential systems and how dealing with different Banach spaces E_{i} may be useful. This discussion will be enlightened with a really simple example (finite population).

Let E be a Banach space and F a vector field on E. We want to see how this differential equation may be seen as a cellular system. For instance, one could consider a simple conservative system on E = ℝ^{4} with an Hamilton’s equation given by (see [?])

\begin{aligned}x′_{1} & = & y_{1} \\ y′_{1} & = & αx_{1} − βx_{1}^{3} + εx_{2} \\ x′_{2} & = & y_{2} \\ y′_{2} & = & −γx_{2} + εx_{1} \end{aligned}

The first step is to identify the different cells of I. The first step is to factorize each term in the equations according to the different variables. For example, the second equation may be seen as:

y′_{1} = (α − βx_{1}^{2})x_{1} + εx_{2}

So that the term (α − βx_{1}^{2}) has to be a part of the coupler we are building. Moreover, as it’s the equation giving y′_{1}, and as the way a cell computes how it interprets the population’s state depends only on its own state, x_{1} and y_{1} have to belong to the same cell. In this simple example it’s the only case where to variables have to be gathered in the same cell. In the end, this leads to the following structure of cellular system:

I = {1, 2, 3}

with the Banach spaces:

E_{1} = ℝ^{2}, E_{2} = E_{3} = ℝ

As it should often be the case, the associated vector fields are just identity maps on E_{i}, and the coupler is then:

[ \begin{matrix}c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \\ c_{31} & c_{32} & c_{33} \end{matrix} ]

with

\begin{aligned}c_{11} : & E_{1} & ⟶ & L(E_{1}) \\ & (x_{1}, y_{1}) & ⟼ & [ \begin{matrix}0 & 1 \\ α − βx_{1}^{2} & 0 \end{matrix} ] \end{aligned} \begin{aligned}c_{12} : & E_{2} & ⟶ & L(E_{2}, E_{1}) \\ & x_{2} & ⟼ & [ \begin{matrix}0 \\ ε \end{matrix} ] \end{aligned} \begin{aligned}c_{13} : & E_{3} & ⟶ & L(E_{3}, E_{1}) \\ & y_{2} & ⟼ & [ \begin{matrix}0 \\ 0 \end{matrix} ] \end{aligned}

\begin{aligned}c_{21} : & E_{1} & ⟶ & L(E_{1}, E_{2}) \\ & (x_{1}, y_{1}) & ⟼ & [ 0 0 ] \end{aligned} \begin{aligned}c_{22} : & E_{2} & ⟶ & L(E_{2}) \\ & x_{2} & ⟼ & [ 0 ] \end{aligned} \begin{aligned}c_{23} : & E_{3} & ⟶ & L(E_{3}, E_{2}) \\ & y_{2} & ⟼ & [ 1 ] \end{aligned}

\begin{aligned}c_{31} : & E_{1} & ⟶ & L(E_{1}, E_{3}) \\ & (x_{1}, y_{1}) & ⟼ & [ ε 0 ] \end{aligned} \begin{aligned}c_{32} : & E_{2} & ⟶ & L(E_{2}, E_{3}) \\ & x_{2} & ⟼ & [ −γ ] \end{aligned} \begin{aligned}c_{33} : & E_{3} & ⟶ & L(E_{3}) \\ & y_{2} & ⟼ & [ 0 ] \end{aligned}

Now, before applying some of the previous techniques, we may compute the different decomposition of c upon different non trivial partitions of I. Those partitions are:

P_{1} = {{1}, {2}, {3}}, P_{2} = {{1, 2}, {3}}, P_{3} = {{1, 3}, {2}}, P_{4} = {{1}, {2, 3}}

which give:

c^{P_{1}} = [ \begin{matrix}0 & c_{12} & c_{13} \\ c_{21} & 0 & c_{23} \\ c_{31} & c_{32} & 0 \end{matrix} ] c^{P_{2}} = [ \begin{matrix}0 & 0 & c_{13} \\ 0 & 0 & c_{23} \\ c_{31} & c_{32} & 0 \end{matrix} ] c^{P_{3}} = [ \begin{matrix}0 & c_{12} & 0 \\ c_{21} & 0 & c_{23} \\ 0 & c_{32} & 0 \end{matrix} ] c^{P_{4}} = [ \begin{matrix}0 & c_{12} & c_{13} \\ c_{21} & 0 & 0 \\ c_{31} & 0 & 0 \end{matrix} ]

Now, in order to simplify, we replace the c_{ij} that are identically zero by 0, we obtain the following different matrices:

c^{P_{1}} = [ \begin{matrix}0 & c_{12} & 0 \\ 0 & 0 & c_{23} \\ c_{31} & c_{32} & 0 \end{matrix} ] c^{P_{2}} = [ \begin{matrix}0 & 0 & 0 \\ 0 & 0 & c_{23} \\ c_{31} & c_{32} & 0 \end{matrix} ] c^{P_{3}} = [ \begin{matrix}0 & c_{12} & 0 \\ 0 & 0 & c_{23} \\ 0 & c_{32} & 0 \end{matrix} ] c^{P_{4}} = [ \begin{matrix}0 & c_{12} & 0 \\ 0 & 0 & 0 \\ c_{31} & 0 & 0 \end{matrix} ]

In the end, writing the coupler as an application from S to L(S), one finds those four matrices:

[ \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & ε & 0 \\ 0 & 0 & 0 & 1 \\ ε & 0 & −γ & 0 \end{matrix} ] [ \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ ε & 0 & −γ & 0 \end{matrix} ] [ \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & ε & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & −γ & 0 \end{matrix} ] [ \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & ε & 0 \\ 0 & 0 & 0 & 0 \\ ε & 0 & 0 & 0 \end{matrix} ]

At this point, we just have to check that the coupler is weakly injective:

ker (c^{P_{1}}) ∩ E_{2} = ker (c^{P_{4}}) ∩ E_{2} = {0}

ker (c^{P_{2}}) ∩ E_{3} = ker (c^{P_{3}}) ∩ E_{3} = {0}

So, we may apply the proposition ?? and without any analytic calculus, state that this differential system may not admit any quasi-periodic solution. In other words, in case there exists periodic trajectories (which is well known to be true) they must be synchronized.

Moreover, those conclusions may hold in a more general case were the c_{ij} are less simple, and we can easily produce a result without any effort:

Proposition 21 (Generalized coupled pendulum) Let’s consider a differential system which is driven by the following equations:

\begin{aligned}x′_{1} & = & a_{1}(x_{1}, y_{1})x_{1} + a_{2}(x_{1}, y_{1})y_{1} + a_{3}(x_{1}, y_{1})x_{2} + a_{4}(x_{1}, y_{1})y_{2} \\ y′_{1} & = & a_{5}(x_{1}, y_{1})x_{1} + a_{6}(x_{1}, y_{1})y_{1} \\ x′_{2} & = & a_{7}(x_{2})x_{2} + u(x_{2})y_{2} \\ y′_{2} & = & ε(y_{2})x_{1} + a_{8}(y_{2})y_{1} − γ(y_{2})x_{2} + a_{9}(y_{2})y_{2} \end{aligned}

If the maps u and ε never vanish, then the systems has no continuous quasiperiodic solution.

Indeed, in this really simple example, the continuity of a solution is the only property needed to ensure the validity of lemma ??. Moreover, this result doesn’t have to be deep in itself, neither has it to be the most general one we could have deduced from the previous discussion. It’s just a sketch of how one can handle some structure properties of a differential system, applying ??, without going into deep and specific calculus.

5 Conclusion and discution

In this work we have built a general framework of cellular systems in order to handle a wide variety of coupled systems, and therefore a wide class of complex systems. We focused on an emergent property of those dynamical systems: the frequencies locking phenomenon. Usually one observes solutions of particular coupled systems and shows that within suitable conditions synchronization must occur. Those results are qualitatively dependent on the systems of interest and do not stand in the general cases. We tried to change our point of view and to bring out completing results. As we choose not to address the problem of the convergence to a periodic solution, we don’t prove that synchronization ultimately happens. Instead, we consider the problem at its end: if one supposes that some coupled systems “truly” oscillates, then they must be synchronized, regardless to the individual dynamical systems as soon as the maps which define each of them are injective nearby the trajectories. The loss in time evolution information is compensated by very general results, almost independent from the individual differential systems to be coupled. For example, we proved results concerning countable coupled systems, each of them needing not to be finite dimensional. In most papers (see for instance [?]) this population of coupled systems is implicitly defined and has only two cells (sometimes a finite number N, and more rarely an infinity). Moreover, on the contrary of what most studies about synchronization issues state, we do not assume anything concerning the cells dynamics (especially, we don’t assume that they are oscillators). We only assume that they exhibit periodic behaviors under the coupling effects (the first assumption implies the second, but the opposite is clearly false).

We believe that this way of reaching general results about cellular systems gives some explanations about why the frequencies locking phenomenon emerges naturally in a large variety of coupled dynamical systems. Our results show that the following alternative is natural in many cases: either the whole population is synchronized, or its cells can’t all have periodic behaviors.

Another interesting perspective is to apply this strategy to differential systems, as we outlined in the end of the fourth section. We think that it could prove useful to understand the stability or instability of limit torus when one adds some perturbations to a differential system. For example, if one already knows that there’s no periodic solution, even with perturbations, and if on is able to prove, using our strategy, that a quasi-periodic solutions disappears under the effects of those perturbation, some conclusions about strange attractor emergence may be reachable.

Moreover, we have achieved some similar work on a natural generalization of this strategy to non countable population (in order to model natural systems, it’s often necessary to handle continuous populations). We truly think that all those results are only a part of what can be done using cellular systems and that this work enlarges the possibilities of studying synchronization issues. But the scope of those kind of cellular systems may be beyond synchronization questions, as it’s quite general and allows some theoretical studies. It could be a promising theoretical tool to model complex systems by the way of coupled differential systems.

References

[1] A. C. J. Luo, Predictions of quasi-periodic and chaotic motions in nonlinear Hamiltonian systems, Chaos, Solitons & Fractals 28 (3) (2006) 627–649.

[2] M. Bennett, M. F. Schatz, H. Rockwood, K. Wiesenfeld, Huygens’s clocks, Proc. R. Soc. Lond. 458 (2002) 563–579.

[3] J. Buck, Synchronous rhythmic flashing of fireflies. ii, Quarterly Review of Biology 63 (3) (1988) 265–289.

[4] L. Bunimovich, Dynamical Systems, Ergodic Theory and Applications, Encyclopaedia of Mathematical Sciences, 2nd ed., Springer, 2000.

[5] G. B. Ermentrout, W. C. Troy, Phaselocking in a reaction-diffusion system with a linear frequency gradient., SIAM-J.-Appl.-Math. 46 (3) (1986) 359–367.

[6] L. Gaubert, Auto-organisation et émergence dans les systèmes couplés, individuation de données issues de systèmes biologiques couplés., Ph.D. thesis, Université de Bretagne Occidentale (2007).

[7] J.-P. Goedgebuer, P. Levy, L. Larger, Laser cryptography by optical chaos, in: J.-P. Goedgebuer, N. Rozanov, S. T. andA.S. Akhmanov, V. Panchenko (eds.), Optical Information, Data Processing and Storage, and Laser Communication Technologies, vol. 5135, 2003.

[8] P. Goel, B. Ermentrout, Synchrony, stability, and firing patterns in pulsecoupled oscillators, PHYSICA D 163 (3-4) (2002) 191–216.

[9] D. Gonze, J. Halloy, A. Goldbeter, Stochastic models for cicardian oscillations: Emergence of a biological rythm, International Journal of Quantum Chemistry 98 (2004) 228–238.

[10] F. C. Hoppensteadt, J. B. Keller, Synchronization of periodical cicada emergences, Science 194 (4262) (1976) 335–337.

[11] C. Huygens, Christiani Hugenii Zulichemii, Const. F. Horologium oscillatorium sive De motu pendulorum ad horologia aptato demonstrationes geometricae, Parisiis : Apud F. Muguet, Paris, France, 1673.

[12] E. M. Izhikevich, F. C. Hoppensteadt, Slowly coupled oscillators: Phase dynamics and synchronization, SIAM J. APPL. MATH. 63 (6) (2003) 1935– 1953.

[13] N. Kopell, G. B. Ermentrout, Handbook of Dynamical Systems II: Toward Applications, chap. Mechanisms of phase-locking and frequency control in pairs of coupled neural oscillators, Elsevier, Amsterdam, 2002, pp. 3–54.

[14] Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in: International Symposium on Mathematical Problems in Theoretical Physics, vol. 39, Springer Berlin / Heidelberg, 1975, pp. 420–422.

[15] Y. Kuramoto, Chemical oscillations, waves, and turbulence, Springer-Verlag, New York, 1984.

[16] S. C. Manrubia, A. S. Mikhailov, D. H. Zanette, Emergence of Dynamical Order. Synchronization Phenomena in Complex Systems, World Scientific, Singapore, 2004.

[17] D. C. Michaels, E. P. Matyas, J. Jalife, Mechanisms of sinoatrial pacemaker synchronization: A new hypothesis, Circ. Res. 61 (1987) 704–714.

[18] L. M. Pecora, T. L. Carroll, G. A. Johnson, D. J. Mar, J. F. Heagy, Fundamentals of synchronization in chaotic systems, concepts, and applications, Chaos 7 (4) (1997) 520–543.

[19] A. Pikovsky, M. Rosenblumand, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, 2001.

[20] I. Prigogine, Introduction to Thermodynamics of Irreversible Processes, Wiley, New York, 1967.

[21] I. Prigogine, Theoretical Physics and Biology, M Marois, North-Holland: Amesterdam, 1969.

[22] L. Ren, B. G. Ermentrout, Phase locking in chains of multiple-coupled oscillators, PHYSICA D 143 (1-4) (2000) 56–73.

[23] Y. G. Sinai, Introduction to ergodic theory, Princeton University Press, 1976.

[24] I. Stewart, M. Golubitsky, Patterns of oscillation in coupled cell systems, in: P. P.Holmes, A.Weinstein (eds.), Geometry, Dynamics and Mechanics: 60th Birthday Volume for J.E. Marsden, Springer-Verlag, New York, 2002, pp. 243– 286.

[25] S. Strogatz, Sync: The Emergeing Science of Spontaneous Order, New York: Hyperion, 2003.

[26] A. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, Journal of Theoretical Biology 16 (1) (1967) 15–42.

[27] A. Winfree, The geometry of biological time, Springer- Verlag, New York, 1990.

[28] C. Wu, L. O. Chua, A unified framework for synchronization and control of dynamical systems, Tech. Rep. UCB/ERL M94/28, EECS Department, University of California, Berkeley (1994).