Heteroclinic cycles between unstable attractors

Transcription

1 Heteroclinic cycles between unstable attractors Henk Broer, Konstantinos Efstathiou and Easwar Subramanian Institute of Mathematics and Computer Science, University of Groningen, P.O. 800, 9700 AV Groningen, The Netherlands Abstract. We consider networks of pulse coupled linear oscillators with non-zero delay where the coupling between the oscillators is given by the Mirollo-Strogatz function. We prove the existence of heteroclinic cycles between unstable attractors for a network of four oscillators and for an open set of parameter values. AMS classification scheme numbers: 34D45, 37L30, 92B99 Submitted to: Nonlinearity 1. Introduction In this study, we analyze heteroclinic cycles that occur in global networks of pulsecoupled oscillators. By definition, a heteroclinic cycle is a collection of orbits that connect sequences of saddle equilibria in a topological circle [1]. Robust heteroclinic cycles constitute a generic feature of certain dynamical systems with symmetry [1, 2]. They have been found to be relevant in a number of physical phenomena that include rotating convection [3], population dynamics [4], climate models [5] and coupled oscillator networks [6, 7]. The pulse coupled oscillator networks that we study in the current work are used, among other things, to model the synchronization in the flashing patterns of fireflies [8,9] and in biological neuron networks [6, 10 12]. The primary motivation, however, lies in several studies [13 18] which propose that unstable attractors connected by heteroclinic cycles could be used to model information processing in neural systems. In the model that we study, neurons are represented by linear oscillators and their membrane potential is related to the phase of the oscillator through a Mirollo-Strogatz function [8]. When the membrane potential reaches a particular threshold, the neuron fires, and the potential is reset to a lower value. As a consequence of the firing, the membrane potential (i.e. the phase) of all the other neurons (oscillators) is increased by a constant amount ε. In the original Mirollo-Strogatz model [8], this increase occurs simultaneously with the firing. Here, following later investigations (for example [9, 12]),

2 we assume that there is a time delay τ between the firing of an oscillator and the time the other oscillators receive the pulse. In networks consisting of three or more such oscillators, previous work [6,10 12,19] has established the existence of unstable attractors for an open set of parameter values. In addition, numerical studies [6, 10, 11] show that for certain values of parameters and for a sufficiently large number of oscillators the network has heteroclinic cycles between unstable attractors. The main aim of the present paper is to prove the existence of such heteroclinic cycles. In particular, we have the following theorem. Theorem 1. Global networks of four pulse coupled oscillators with delay where the coupling is given by the Mirollo-Strogatz function have heteroclinic cycles between two unstable attractors for an open set of parameters. A more detailed version of this statement is given in section 3 (theorem 2). In order to prove theorem 1 we use the metric in the infinite dimensional state space, introduced in [19], that allows to study instability in a rigorous way. The unstable attractors we study in this paper are saddle periodic orbits, or saddle fixed points of a suitably defined Poincaré map. This means that they have local stable and unstable manifolds that are both non-zero dimensional. At the same time, there exists an open set of points in the state space that converges to the attractor. The situation is sketched in figure 1. The attractor Q is a saddle point and its stable set W s (Q) contains an open set S(Q). Initial states from S(Q) collapse onto the local stable set Wloc s (Q) and converge to Q. Since Q is a saddle point there is a neighborhood U of Q such that all initial states in U \ Wloc s (Q) leave U after some time. S(Q) W s loc (Q) Q Figure 1. A schematic picture of an unstable attractor. Q is a saddle point whose stable set W s (Q) contains an open set S. Initial states from S collapse onto the local stable set Wloc s (Q) and converge to Q. Since Q is a saddle point, almost all nearby initial states move away from Q. The unstable attractors in a system can be connected by heteroclinic cycles. Consider the case that a system has N unstable attractors Q 1,..., Q N such that Q j lies in the interior of the closure of the basin of Q j+1, and Q N lies in the interior of the closure of the basin of Q 1. The dynamics in this case is not very interesting because any initial state in the neighborhood of Q 1 will end up to Q 2 and stay there forever. But if we add small noise to the system, then the system can leave Q 2 to reach Q 3 and so on. In this way the existence of heteroclinic connections together with some external noise can make the system move from one state to another. 2

3 The paper is organized as follows. In section 2 we discuss the setting of the problem by providing a description of the system and defining its state space. In section 3, we prove analytically the existence of heteroclinic cycles between unstable attractors in a network of 4 oscillators. Finally in section 4, we compare the theoretical results that we obtained with a numerical study of the system and we present our conclusions. 2. Definition of the dynamics In this section we follow closely [12] and in particular [19]. We repeat only the definitions that are necessary for the current paper. For more details we refer to [12, 19]. The system studied in this paper is a delay system [20]. The state space of such systems is an appropriate space Pτ n of functions (see definition 1) defined on the interval ( τ, 0], where τ > 0 is the delay of the system, and taking values in an n-dimensional manifold N. The state space thus is infinite dimensional. In our case, points in N represent the phases of the n coupled oscillators, which implies that N = T n, the n- dimensional torus. For a given φ Pτ n and for each t ( τ, 0], φ(t) N represents the phases of the oscillators at time t. Using the dynamics of the system, φ can be extended to a unique function φ + : ( τ, + ) N, such that φ + (t) = φ(t) for t ( τ, 0] and φ + (t) N represents the phases of the oscillators at any time t τ. Then the evolution operator Φ t : Pτ n Pτ n is defined by Φ t (φ)(s) = φ t (s) = φ + (t + s) for any t 0 and s ( τ, 0]. In other words, the evolution operator maps the initial state φ = φ 0 to the state φ t of the system at time t. The latter is the restriction of φ + in (t τ, t] shifted back to the interval ( τ, 0] Pulse coupled oscillator networks with delay We now specialize the above notions of the theory of delay equations to the current setting. Definition 1 (State space, cf. [12]). The state space Pτ n of the system of n pulse coupled oscillators with delay τ > 0 is the space of phase history functions φ : ( τ, 0] T n : s φ(s) = (φ 1 (s),..., φ n (s)), that satisfy the following conditions: (i) Each φ i is upper-semicontinuous, i.e., φ i (s + ) := lim t s + φ i (t) = φ i (s) and φ i (s ) := lim t s φ i (t) φ i (s) for all s ( τ, 0]. (ii) Each φ i is only discontinuous at a finite (or empty) set S i = {s i,1,..., s i,ki } ( τ, 0] with k i N and s i,1 > s i,2 > > s i,ki. (iii) dφ i (s)/ds = 1 for s S i. The coupling between the n oscillators is defined using the pulse response function. 3

4 Definition 2 (Pulse response function, cf. [12]). A pulse response function is a map V : T R + R : (θ, ε) V (θ, ε), (1) that satisfies the following conditions: (i) V is smooth on (T \ {0}) R +. (ii) V (θ, ε)/ θ > 0 on (T \ {0}) (R + \ {0}). (iii) V (θ, ε)/ ε > 0 on T R +. (iv) V (θ, 0) = 0 for all θ T. (v) 0 < V (0, ε) < 1 for all ε (0, 1). (vi) H, given by (4), satisfies H m (θ) = H 1 H m 1 (θ) = m-times {}}{ H 1... H 1 (θ). (2) Notice that in the above definition V/ θ > 0, therefore V cannot be smooth everywhere on T. This is reflected in condition (i) of the definition. The pulse response function depends on the parameter ε 0, called coupling strength. As a shorthand notation we introduce V m (θ) = V (θ, mˆε), for m = 1, 2, 3,..., (3) where ˆε = ε/(n 1). Given a pulse response function V we also define and H : T R + R : (θ, ε) H(θ, ε) = θ + V (θ, ε), (4) H m (θ) = H(θ, mˆε), for m = 1, 2, 3,.... (5) Definition 3 (Dynamics, cf. [12]). A system of n pulse coupled oscillators with delay is a quadruple D = (n, V, ε, τ), where V is as in definition 2, ε 0 and τ 0. Given a system D and an initial state φ P n τ, we extend φ to a function φ + : ( τ, + ) T n using the following rules: (i) φ + (t) = φ(t) for t ( τ, 0]. (ii) dφ + i (t)/dt = 1 for t 0, if φ+ j (t τ) 0 (mod Z) for all j i. (iii) φ + i (t) = min{1, H m(φ + i (t ))} (mod Z), if there are j 1,..., j m i such that φ + j k (t τ) = 0 (mod Z) for all k = 1,..., m. The dynamics described in definition 3 can be interpreted in the following way. The phase φ i of each oscillator O i, i = 1,..., n, increases linearly. When the phase reaches the value 1 = 0 (mod Z), then the oscillator O i fires and all the other oscillators O j, j i receive a pulse after a time delay τ. In general, an oscillator O j may receive m simultaneous pulses at time t if m oscillators O i1,..., O im have fired simultaneously at time t τ. Then the phase of O j is increased to H(u j, mˆε) = H m (u j ) where u j = φ + j (t ), unless the pulse causes the oscillator to fire and then the phase becomes exactly 1. 4

5 The evolution operator Φ t for t 0 is then defined by Φ t : P n τ P n τ : φ Φ t (φ) = φ t = φ + (t τ,t] T t, (6) where T t is the shift s s + t and the positive semiorbit of φ P n τ is given by O + (φ) = {Φ t (φ) : t 0}. (7) In [19] it was proven that the evolution operator Φ t is well defined. For a given system D = (n, V, ε, τ), the accessible state space is P D = Φ τ (Pτ n ). In other words, φ P D if there is a state ψ Pτ n such that Φ τ (ψ) = φ, i.e., P D includes only those states that are dynamically accessible. From now on, we restrict our attention to P D The Mirollo-Strogatz model A pulse response function V that satisfies all the requirements of definition 2 is provided by the Mirollo-Strogatz model [8] where the pulse response function is V MS (θ, ε) = f 1 (f(θ) + ε) θ, (8) and f is a function which is concave down (f < 0) and monotonically increasing (f > 0). Moreover, f(0) = 0 and f(1) = 1. A concrete example is given by f b (θ) = 1 b ln(1 + (eb 1)θ). (9) We present a sketch of the function f b for various values of b in figure 2a. For any given positive value of ε, the pulse response function V MS (θ, ε) for f = f b as in (9) is affine: V MS (θ, ε) = m ε + K ε θ, (10) where m ε = (e bε 1)/(e b 1) and K ε = e bε 1. The graph of V MS (10) is depicted in figure 2b for different values of ε. In the numerical computations in this paper, we use the Mirollo-Strogatz model with f b as in (9) with fixed b = 3. After fixing b, the parameter space of the system is {(ε, τ) : ε > 0, τ > 0} = R 2 + where we recall that τ is the delay and ε is the coupling strength Metric We introduce a metric d on P D which we later use to define a neighborhood of states. Recall that given a phase history function φ Pτ n, we can define the extended phase history function φ +. We define a lift [21] of an extended phase history function φ + as any function L φ : ( τ, + ) R n such that: (i) L φ (s) (mod Z) = φ + (s), and (ii) for any s ( τ, + ) and for i = 1,..., n, (L φ ) i (s) (L φ ) i (s ) = φ + i (s) φ+ i (s ). 5

6 (a) 1 (b) b = f b (θ) b = 3 b = 1 V MS (θ, ε) ε = ε = θ 0 ε = θ Figure 2. (a) Graph of f b (9) as a function of θ for different values of b. (b) Graph of V MS (10) as a function of θ for f = f b, b = 3 and different values of ε. It follows from these properties that if L (1) φ and L (2) φ are two lifts of the same extended phase history function φ + then they differ by a constant integer vector, i.e., L (1) φ (s) L(2) (s) = k Zn, for all s ( τ, ). φ Definition 4 (Metric on P D ). The metric d : P D P D R is given by n τ d(φ, ψ) = min k Z n (L φ ) i (s) (L ψ ) i (s) k i ds, (11) i=1 where L φ and L ψ are arbitrary lifts of φ and ψ respectively. τ 2.4. Other representations of the dynamics It is often useful in what follows to use alternative representations of the dynamics. In this section we introduce, following [12], the past firings and the event representation The past firings representation It follows from definition 3 that the evolution of an initial state φ P D only depends on the values φ i (0) and the firing sets Σ i (φ) that are defined as follows: Definition 5. Given a phase history function φ P D, the firing sets Σ i (φ) ( τ, 0], i = 1,..., n are the sets of solutions of the equation φ i (s) = 0 for s ( τ, 0]. The total firing set is given by Σ(φ) = {(i, σ) : i = 1,..., n, σ Σ i (φ)} Therefore, if we are interested only in the future evolution of the system we can consider the following equivalence relation in P D. Definition 6. Two phase history functions φ 1, φ 2 in P D are equivalent, denoted by φ 1 φ 2, if φ 1 (0) = φ 2 (0) and Σ(φ 1 ) = Σ(φ 2 ). Let P D = P D / the quotient set of equivalence classes and by [φ] P D denote the equivalence class of φ P D. 6

7 Points [φ] P D are completely determined by the values of the phases φ i (0) and the firing sets Σ(φ) (which may be empty). We denote the elements of Σ i (φ) by σ i,1 > σ i,2 > > σ i,ki where k i is the cardinality of Σ i (φ). Note that by definition, φ i (0) σ i,1, and φ i (0) = 0 if and only if σ i,1 = 0. It is possible to give an equivalent description of the dynamics described by definition 3, using only the variables φ i (0) and σ i,j. For such a definition see [12]. Notice also that, Proposition 1. If φ 1 φ 2 then (i) Φ t (φ 1 ) Φ t (φ 2 ) for t 0, and (ii) Φ t (φ 1 ) = Φ t (φ 2 ) for t τ Poincaré map Given a network of n oscillators with dynamics defined by the pulse response function V, with pulse strength ε and with delay τ, we can simplify the study of the system D = (n, V, ε, τ) by considering intersections of the positive semiorbits O + (φ) with the set P = {φ P D : φ n (0) = 0}. (12) The set P is called a (Poincaré) surface of section [22, 23] and it inherits the metric d, see (11). The evolution operator Φ, see (6), defines a map R : P P in the following way. Consider any φ P, i.e., such that φ n (0) = 0. Since the phases of the oscillators are always increasing there is a minimum time t(φ) such that the phase of O n becomes 0 again, i.e., such that Φ t(φ) (φ) n (0) = 0. We define R(φ) = Φ t(φ) (φ). (13) The map R is called Poincaré (return) map. Furthermore, we can define the quotient map R : P/ P/ : [φ] [R(φ)], of the Poincaré (return) map R, where is the equivalence relation given by definition 6. By proposition 1 the map R is well defined The event representation Given a phase history function φ, the firing sets Σ i (φ) = {σ i,1,..., σ i,ki } describe at which moments in the interval ( τ, 0] the oscillator O i fires. Hence, they also describe at which instants in the interval (0, τ] the oscillators O l, for l i would receive a pulse from O i, making φ + l, l i, discontinuous at σ i,j + τ (0, τ], for j = 1,..., k i. Also, notice that if the phase of the oscillator O i at time 0 is φ i (0) then the oscillator will fire after time 1 φ i (0), unless it receives a pulse before it fires. Hence, the numbers σ i,j and φ i (0) where i = 1,..., n and j = 1,..., k i can completely describe the future evolution of the system. 7

8 The event representation is a symbolic description of the dynamics in which the state of the system is represented by a sequence of events consisting of firings and pulse receptions that would occur. Each event E in the sequence is characterized by a triplet [K(E), O(E), T (E)] where K(E) denotes the type of the event F or mp. The event F denotes a firing event and mp (m a natural number) stands for the simultaneous reception of m pulses. The event K(E) is associated with oscillator O(E) {1,..., n}. Finally, T (E) [0, 1] denotes after how much time is left for the event to occur. For example, the event denoted by [F, 2, 0.4] signifies that the oscillator O 2 will fire after time 0.4 (and this means that its current phase is = 0.6), while the event denoted by [P, 1, 0.3] signifies that O 1 is set to receive a pulse after time 0.3. We use the shorthand notation [F, (i 1,..., i k ), t] and [mp, (i 1,..., i k ), t] to indicate that the oscillators O i1,..., O ik fire or receive m pulses respectively after time t. Given a particular initial state φ P D, such that its equivalence class [φ] P D is characterized by the phases φ i (0) and firing times σ i,j for i = 1,..., n and j = 1,..., k i, consider the space A of event sequences (E 1, E 2,..., E k ) of finite (but not fixed) length and the map E : P D A : [φ] E([φ]), (14) which maps [φ] to the event sequence E([φ]) constructed in the following way. First, consider the set Y consisting of the following events: (i) [F, i, 1 φ i (0)] for i = 1,..., n, and (ii) [P, l, τ + σ i,j ] for l = 1,..., n with l i. Then, impose time-ordering on Y (i.e., order the events so that events that occur earlier appear first) and in the case that there are m > 1 identical events [P, i, t] collect them together to [mp, i, t] to obtain E([φ]). It follows that E is injective and hence the inverse map E 1 : E(P D ) A P D is well defined. Next, define the map Φ A : E(P D ) E(P D ) (15) using the following algorithm: (i) For Z E(P D ), consider the first event E 1 Z and let t = T (E 1 ). If T 1 0 then set T (E) to T (E) t for all E Z. (ii) Take the sequence Z 0 of events E Z with T (E) = 0 and define Z + = Z \ Z 0. For each event E Z 0 do the following: (a) If K(E) = F, then 1. append to Z + the event [F, O(E), 1]; 2. append to Z + the events [P, l, τ] for all l {1,..., n} with l O(E). (b) If K(E) = mp, then 1. find the (unique) event E Z + with K(E ) = F and O(E ) = O(E); 2. set T (E ) to max{t (E ) V (1 T (E ), mˆε), 0}. 8

9 (iii) Impose time-ordering on Z + and collect together identical pulse events. (iv) Set Φ A (Z) = Z +. It follows from the definition of Φ A that: Proposition 2. (i) The map Φ A : P D E(P D ) is well defined. (ii) [Φ t (φ)] = E 1 (Φ A (Z)) where Z = E([φ]) and t is determined at the first step of the algorithm. (iii) Consider an initial state φ P D and the corresponding event sequence E([φ]). If we apply Φ A, m times to E([φ]) and the time that elapses at the jth (j = 1,..., m) application is t j with t = j t j, then it is possible to reconstruct the extended phase history function φ + on the interval [0, t]. The last part of proposition 2 implies that if t τ then it is possible to obtain from the sequence {Z, Φ A (Z), Φ 2 A (Z),..., Φm A (Z)}, where Z = E([φ]), not only the equivalence class [Φ t (φ)] but also the phase history function Φ t (φ) = φ + (t τ,t] T t for any time t [τ, t]. 3. Heteroclinic cycles We begin by defining, where g 1 (τ) = H 2 (H 1 (2τ) + τ), g 2 (τ) = H 1 (α + τ + β), g 3 (τ) = 1 H 2 (W 1 + τ), g 4 (τ) = H 1 (W 2 + τ), α β = H 1 (H 1 (τ) + τ), = 1 H 1 (H 2 (2τ) + τ), W 1 = 1 + H 1 (τ) H 2 (H 1 (τ) + τ) + τ H 1 (2τ), W 2 = 1 + H 1 (τ) H 2 (H 1 (τ) + τ). Using the terminology introduced in section 2, we restate the main theorem of this paper (theorem 1) as follows. Theorem 2. Consider a system D = (n, V, ε, τ) such that (i) n = 4, (ii) V is given by a Mirollo-Strogatz model (section 2.2), (iii) g 1 (τ) < 1, (iv) g 2 (τ) < 1, (v) g 3 (τ) < τ, (vi) g 4 (τ) < 1, 9

10 Then, there exist two unstable attractors φ Q 1, φ Q 2 P with a heteroclinic cycle between them and the conditions (i-vi) define an open set in the parameter space (b, ε, τ). The fixed points φ Q 1 and φ Q 2 are defined in the following way. Definition 7. φ Q 1 P for H 2 (τ + H 1 (τ)) < 1 is defined by { φ Q 1 τ + s, for i = 1, 2, i (s) = τ + W 2 + s, for i = 3, 4, for s ( τ, 0), while for s = 0, { φ Q H1 (τ), for i = 1, 2, 1 i (0) = 0, for i = 3, 4, where W 2 = 1 + H 1 (τ) H 2 (τ + H 1 (τ)). (16a) (16b) (a) 1 (b) 1 φ Q 2 3,4 φ Q 1 3,4 W 2 φ Q1 i φ Q2 i φ Q 2 1,2 τ φ Q 1 1,2 0 τ 0 τ t τ 0 τ 0 τ t Figure 3. Graphs of the phase history functions φ Q1 and φ Q2. Definition 8. φ Q 2 P for H 2 (τ + H 1 (τ)) < 1 is defined by { φ Q W2 + s, for i = 1, 2, 2 i (s) = s, for i = 3, 4, for s ( τ, 0], where W 2 = 1 + H 1 (τ) H 2 (τ + H 1 (τ)). The parameter region in which theorem 2 is valid for b = 3 is represented by the gray region in figure 4. This represents an intersection of the plane b = 3 and the open parameter region in the space (b, ε, τ) in which theorem 2 holds Proof of existence of heteroclinic cycles In this section we prove theorem 2. We show that almost all points in an open neighborhood of φ Q 1 are mapped in finitely many iterations to φ Q 2. Finally, using a symmetry argument we show that almost all points in an open neighborhood of φ Q 2 are mapped in finitely many iterations to φ Q 1, thus establishing the existence heteroclinic cycle between φ Q 1 and φ Q (17)

11 0.3 g g 3 τ g 2 g ε Figure 4. The region defined by the inequalities in theorem 2 is depicted by gray. In this figure b = 3. Proposition 3. Given a system D = (n, V, ε, τ) that satisfies the conditions of theorem 2, the point φ Q 1 (16) is a fixed point of the Poincaré map R. Proof. φ Q 1 evolves in the event sequence representation as follows: ([2P, (1, 2), τ], [P, (3, 4), τ], [F, (1, 2), 1 H 1 (τ)], [F, (3, 4), 1]) 1 ([2P, (1, 2), 0], [P, (3, 4), 0], [F, (1, 2), 1 τ H 1 (τ)], [F, (3, 4), 1 τ]) Since H 2 (τ + H 1 (τ)) < 1, the evolution is, 2 ([F, (1, 2), 1 H 2 (τ + H 1 (τ))], [F, (3, 4), 1 H 1 (τ)]) 3 ([F, (1, 2), 0], [F, (3, 4), H 2 (τ + H 1 (τ)) H 1 (τ)]) 4 ([2P, (3, 4), τ], [P, (1, 2), τ], [F, (3, 4), H 2 (τ + H 1 (τ)) H 1 (τ)], [F, (1, 2), 1]) 5 ([2P, (3, 4), 0], [P, (1, 2), 0], [F, (3, 4), H 2 (τ + H 1 (τ)) H 1 (τ) τ], [F, (1, 2), 1 τ]) 6 ([F, (3, 4), 0], [F, (1, 2), 1 H 1 (τ)]) 7 ([2P, (1, 2), τ], [P, (3, 4), τ], [F, (1, 2), 1 H 1 (τ)], [F, (3, 4), 1]) Transition 6 requires H 2 (W 2 + τ) 1 which will be proven in proposition 16. Since we return to the initial state, φ Q 1 is a fixed point of R. Proposition 4. Given a system D = (n, V, ε, τ) that satisfies the conditions of theorem 2, the point φ Q 2 (17) is a fixed point of the Poincaré map R. Proof. φ Q 2 evolves in the event representation as follows: ([2P, (1, 2), τ], [P, (3, 4), τ], [F, (1, 2), H 2 (τ + H 1 (τ)) H 1 (τ)], [F, (3, 4), 1]) 1 ([2P, (1, 2), 0], [P, (3, 4), 0], [F, (1, 2), H 2 (τ + H 1 (τ)) H 1 (τ) τ], [F, (3, 4), 1 τ]) 11

12 Since H 2 (W 2 + τ) 1, 2 ([F, (1, 2), 0], [F, (3, 4), 1 H 1 (τ)]) 3 ([2P, (3, 4), τ], [P, (1, 2), τ], [F, (3, 4), 1 H 1 (τ)], [F, (1, 2), 1]) 4 ([2P, (3, 4), 0], [P, (1, 2), 0], [F, (3, 4), 1 H 1 (τ) τ], [F, (1, 2), 1 τ]) Since H 2 (τ + H 1 (τ)) = H 1 (α) < g 2 (τ) < 1 (where α = H 1 (H 1 (τ) + τ)), 5 ([F, (3, 4), 1 H 2 (τ + H 1 (τ))], [F, (1, 2), 1 H 1 (τ)]) 6 ([F, (3, 4), 0], [F, (1, 2), H 2 (τ + H 1 (τ)) H 1 (τ)]), 7 ([2P, (1, 2), τ], [P, (3, 4), 0], [F, (1, 2), H 2 (τ + H 1 (τ)) H 1 (τ)], [F, (3, 4), 1]) We note that we return to the initial state and that φ Q 2 is a fixed point of R. Lemma 5. If the assumptions stated in theorem 2 hold, then there is an open neighborhood U P of φ Q 1 such that U/ is 1-dimensional and all states [φ] (U/ ) \ {[φ Q 1 ]} converge to φ Q 2 in finitely many iterations of the Poincaré map R. Proof. According to proposition 7 there is an open neighborhood U P of φ Q 1 such that the equivalence class [φ] of each state φ U is characterized by the event sequence ([2P, (1, 2), τ], [P, (3, 4), τ], [F, 1, 1 v], [F, 2, 1 w], [F, (3, 4), 1]). (18) where v and w can be made to be arbitrarily close to H 1 (τ) and moreover if v > w then A ε w+(a ε 1)v = (2A ε 1)H 1 (τ), while if v < w then A ε v+(a ε 1)w = (2A ε 1)H 1 (τ). This shows that U/ is 1-dimensional. Since all the oscillators are identical, the system is invariant under the permutation 1 2, therefore it is enough to consider only the case v > w. We denote by [φ v,w ] the equivalence class that corresponds to the event sequence (18), where now v and w take any value in [τ, 1), and by Λ the set of equivalence classes {[φ v,w ] : v [τ, 1), w [τ, 1)}. Notice that [φ Q 2 ] = [φ W 2,W 2 ] Λ. Let Λ be the subset of [τ, 1) 2 such that if (v, w) Λ then R([φ v,w ]) Λ. Since there is an one-to-one correspondence between equivalence classes [φ v,w ] and pairs (v, w) we can define a map R Λ : Λ [τ, 1) 2 [τ, 1) 2 given by [φ RΛ(v,w) ] = R([φ v,w ]). Therefore, we can follow the evolution of an initial state [φ v,w ] on [τ, 1) 2 with coordinates v and w as long as RΛ m(v, w) Λ for m N. In the course of the proof we show that all the equivalence classes that we consider belong in Λ. The next step of the proof is to divide the space [τ, 1) 2 in different regions for which we can solve the dynamics and show that the initial state with v and w close to H 1 (τ) goes through a succession of regions until it reaches the point Q 2 = (W 2, W 2 ) that corresponds to [φ Q 2 ]. These regions are shown in figure 5. First, define the line segment l, given by v > w, H 1 (τ) < v < H 1 (2τ), w > τ and A ε w + (A ε 1)v = (2A ε 1)H 1 (τ). Define also l 1 as the subset of l for which H 2 (v + τ) H 2 (w + τ) < τ and l 2 as the subset of l for 12

13 1 Q 2 A 2 (v w) = τ W 2 w B d 2 B c 2 H 2(w + τ) = 1 τ + H 2(w + τ) = 1 H 1(τ) Q 1 l 1 l 2 B b 2 B 1 B a 2 H 1(τ + H 2(w + τ)) = 1 τ 0 0 v 1 H 1(2τ) H 2(v + τ) = 1 1 τ Figure 5. The regions used in the proof of lemma 5. The line segments l 1 and l 2 are indicated by the thick solid and dashed lines respectively. The line segment l 1 begins at Q 1 and is separated from l 2 by the thin dashed line given by H 2 (v + τ) H 2 (w + τ) = A 2 (v w) = τ which is parallel to the diagonal that joins Q 1 and Q 2. Points on l 1 are mapped to l 2, and next in B 1 or B 2 = B a 2 B b 2 B c 2 B d 2. Points in B 1 are mapped to B 2 and finally, points in B 2 are mapped in finite iterations to Q 2. which H 2 (v + τ) H 2 (w + τ) τ. Then, proposition 7 states that the part of the open neighborhood U of φ Q 1 for which v > w lies in l 1. Next, according to proposition 9 the point (v, w) l 1 is mapped in finite iterations of R Λ to l 2 and according to proposition 10, points in l 2 are mapped either in the region B 1 defined by the relations v > w, w τ, v H 1 (2τ), H 2 (v + τ) < 1 and H 2 (v + τ) H 2 (w + τ) τ or in the region B 2 defined by the relations v > w, v < 1 τ, w > τ, H 2 (v + τ) > 1, w < W 2. Denote by B2 a the subset of B 2 given by H 1 (τ + H 2 (w + τ)) < 1, by B2 b the set given by H 1 (τ +H 2 (w+τ)) > 1 and τ +H 2 (w+τ) < 1, by B2 c the set given by τ +H 2 (w+τ) > 1 and H 2 (w + τ) < 1, and finally by B2 d the set given by H 2 (w + τ) > 1. Notice that B 2 = B2 a B2 b B2 c B2. d Next, according to proposition 11 all points in B 1 are mapped in B 2. Proposition 12 states that all points in B2 a are mapped in finite iterations in B2 b B2 c B2. d Next, proposition 13 states that all points in B2 b are mapped to a single point in B2. c According to proposition 14 all points in B2 c are mapped in B2 d and finally, proposition 15 states that all points in B2 d are mapped to [φ Q 2 ]. Therefore, the initial states φ U near φ Q 1 are mapped in a finite number of iterations of R to [φ Q 2 ]. Since R([φ Q 2 ]) = φ Q 2 (from proposition 4) we conclude that φ U is mapped in a finite number of iterations of R to φ Q 2. Lemma 6. Under the assumptions of theorem 2 there is an open set W P of initial states around φ Q 2 such that W/ is 3-dimensional and all states [φ] W/, except 13

14 those in a 2-dimensional subset, converge to φ Q 1 Poincaré map R. in finitely many iterations of the Proof. According to proposition 8 there is an open neighborhood W P of φ Q 2 such that the equivalence class [φ] of each state φ W is characterized by the event sequence (making the assumption that v w) where 0 < u 1, or by ([P, (1, 2, 4), τ u], [P, (1, 2, 3), τ], [F, 1, 1 W 2 v], (19) [F, 2, 1 W 2 w], [F, 3, 1 u], [F, 4, 1]) ([F, 3, u], [P, (1, 2, 3), τ], [F, 1, 1 v], [F, 2, 1 w], [F, 4, 1]) (20) where 1 u < 0. Notice that in the case u = 0 the event sequences (19) and (20) are essentially identical and the initial state is mapped in one iteration to φ Q 2. Therefore, states in the neighborhood W can be characterized by three small parameters (u, v, w) and we show that except states with u = 0, all other states in W are mapped in finite iterations to φ Q 1. Consider the surface of section Q = {φ P D : φ 2 (0) = 0} and the maps T 1 : P Q and T 2 : Q P defined so that for φ P, T 1 (φ) is the first intersection of {Φ t (φ)} with Q and for ψ Q, T 2 (ψ) is the first intersection of {Φ t (ψ)} with P. We consider φ W and compute T 1 (φ). In particular, we consider the event sequence (19), so the evolution is 1 ([P, (1, 2, 4), 0], [P, (1, 2, 3), u], [F, 1, 1 W 2 v + u τ], [F, 2, 1 W 2 w + u τ], [F, 3, 1 τ], [F, 4, 1 + u τ]) 2 ([P, (1, 2, 3), u], [F, 1, 1 H 1 (W 2 + v u + τ)], [F, 2, 1 H 1 (W 2 + w u + τ)], [F, 3, 1 τ], [F, 4, 1 H 1 (τ u)]) 3 ([P, (1, 2, 3), 0], [F, 1, 1 H 1 (W 2 + v u + τ) u], [F, 2, 1 H 1 (W 2 + w u + τ) u], [F, 3, 1 τ u], [F, 4, 1 H 1 (τ u) u]) 4 ([F, 1, 0], [F, 2, 0], [F, 3, 1 H 1 (τ + u)], [F, 4, 1 H 1 (τ u) u]) where in the last transition we used the fact that H 1 (H 1 (W 2 + v u + τ) + u) > 1 and H 1 (H 1 (W 2 + w u + τ) + u) > 1 for (u, v, w) small enough, since H 2 (W 2 + τ) > 1. The situation for w > v is identical up to interchanging oscillators O 1 and O 2. The case for the event sequence (19) is also similar and we do not analyze it separately. Therefore, we observe that T 1 (W ) = U where U is the set of ψ Q with ψ 1 (0) = 0, ψ 2 (0) = 0, ψ 3 (0) = H 1 (τ +u), ψ 4 (0) = H 1 (τ u)+u, Σ 1 (ψ) = Σ 2 (ψ) = {0} and Σ 3 (ψ) = Σ 4 (ψ) = and u > 0 can be chosen to be arbitrarily small. Consider the map K : P D P D defined by K(φ) = (φ 3, φ 4, φ 1, φ 2 ), i.e., K corresponds to a permutation of the oscillators (notice also that K 1 = K). Define also ψ Q 1 = K(φ Q 1 ) and ψ Q 2 = K(φ Q 2 ) and notice that the neighborhood U of ψ Q 1 is mapped by K to the neighborhood K(U ) = U of φ Q 1, where U is defined in the proof of lemma 5. Moreover, following the evolution of ψ Q 2 one can show that T 2 (ψ Q 2 ) = φ Q 1. 14

15 Since all the oscillators are identical, we have that K Φ t = Φ t K. Moreover, if we denote by R P the Poincaré map on P and R Q the Poincaré map on Q we have that R P = K 1 R Q K. Then it follows that for ψ U R m Q(ψ) = K 1 (R m P(K(ψ))). According to lemma 5 for every φ = K(ψ) U (except φ Q 1 ) there is m > 0 such that R m P (φ) = φq 2. Therefore, R m Q(ψ) = K(φ Q 2 ) = ψ Q 2. Therefore, for φ W we have that T 2 (R m Q(T 1 (φ))) = T 2 (ψ Q 2 ) = φ Q 1 i.e., there is some t > 0 such that Φ t (φ) = φ Q 1 is m > 0 such that RP m (φ) = φq 1 for φ W. for φ W P. This implies that there 4. Discussion 4.1. Numerical simulations and comparisons to the theoretical results In section 2, we established, that since the system studied in this paper is a delay system, the state space is the set of functions Pτ n that represent the values of the phases of the oscillators in the time interval ( τ, 0]. Nevertheless, it is often the case (see for example [11, 12]), that in numerical simulations of such systems, only the phases θ i, for i = 1,..., n 1, of the oscillators at time t = 0 are given as initial data and then the times σ i,j when the oscillators have fired in the interval ( τ, 0] are determined through a set of rules and θ n = 0, since we consider states on the surface of section. These rules essentially determine a map G : T n 1 P/ and they define an n 1 dimensional subset S = G(T n 1 ) of the infinite dimensional space P/. If θ, θ T n 1 then it appears natural to define the distance between the points G(θ), G(θ ) S by the Euclidean distance between θ and θ. We call this metric, the T n 1 metric. It would be interesting to know whether studying a pulse coupled oscillator network using the T n 1 metric gives the same results as studying the network using the metric d, given by (11). For this reason, in this section we numerically study a pulse coupled 4-oscillator network with delay by giving a map G : T 3 P/ and using the corresponding T 3 metric and we compare these numerical results to the results obtained in section 3. In particular, we consider a 4-oscillator network with the dynamics as described in section 2.1 and coupling given by a Mirollo-Strogatz pulse response function V MS (8) for b = 3. Given θ = (θ 1, θ 2, θ 3 ) T 3, we consider in the event representation the state given (up to time ordering) by G(θ) = ([P, (1, 2, 3), τ], [F, 1, 1 θ 1 ], [F, 2, 1 θ 2 ], [F, 3, 1 θ 3 ], [F, 4, 1]) 15

16 Θ Q Q Θ Θ Figure 6. Basin of attraction, projected to S, of the unstable attractors Q 1 and Q 2 for a network of 4 oscillators with b = 3, ε = 0.1, τ = 0.2. Both fixed point attractors lie on the intersection of the plane θ 3 = 0 (bottom plane) with the plane θ 1 = θ 2. Initial states that converge to the attractors Q 1 and Q 2 are shown in dark and light gray respectively. Observe that initial states near the attractor Q 1 belong to the basin of Q 2 and vice versa, which demonstrates that there exists a heteroclinic cycle between Q 1 and Q 2. if θ i τ for i = 1, 2, 3. If θ 1 < τ then O 1 must have fired in the interval [ θ 1, 0). We make the extra assumption that in this case O 1 has fired exactly at time θ 1. Therefore, we add to G(θ) the events [P, (2, 3, 4), τ θ 1 ]. Similarly, if θ 2 < τ we add to G(θ) the events [P, (1, 3, 4), τ θ 2 ] and if θ 3 < τ we add the events [P, (1, 2, 4), τ θ 3 ]. In each case we time-order G(θ). This construction defines the mapping G : T 3 P/. By definition, G is a bijection on S = G(T 3 ). Notice that, the Poincaré map R, does not define a map on T 3 by G 1 R G, because the image of R G contains points that do not belong to S. The fixed states [φ Q i ] (i = 1, 2) of the Poincaré map, belong to S, therefore there 16

17 exist Q 1, Q 2 T 3 such that G(Q i ) = [φ Q i ] for i = 1, 2. We study numerically which states G(θ) converge to [φ Q 1 ] or [φ Q 2 ] for parameter values ε = 0.1 and τ = 0.2. The result is depicted in figure 6 where the intersection of the basins of the unstable attractors with planes θ 3 = const is shown. The basin of attraction of Q 1 is represented by dark gray and that of Q 2 by light gray. From figure 6 we conclude that there is an open, in T 3, ball B 1 around Q 1 that belongs in the basin of Q 2 (except for points on the plane θ 1 = θ 2 ). Moreover, there is an open, in T 3, ball B 2 of points around Q 2 contained (except points on the plane θ 3 = 0) in the basin of Q 1. This is harder to see in figure 6 but a magnification of the region near Q 2 reveals that the situation is as just described. Therefore, from the numerical results we conclude that if we restrict our attention to S with the T 3 metric, the fixed points Q 1 and Q 2 are unstable attractors since their basins have interior points and there is a heteroclinic cycle between them. This is exactly the result that we obtained for φ Q 1 and φ Q 2 in section 3. It is an important question whether one can infer, in all cases, from such numerical results using the T n 1 metric, the existence of unstable attractors and heteroclinic cycles on the Poincaré surface of section P with the metric d, given by (11). Notice first, that the T n 1 metric is not equivalent to d. For example, we have obtained in lemma 5 that the open neighborhood U of φ Q 1, with respect to the metric d, is characterized by 1 dynamically significant parameter, i.e., U/ is 1 dimensional. On the other hand, the open neighborhood of Q 1 T 3 with respect to the T 3 metric is 3-dimensional and moreover G(U/ ) is an 1-dimensional closed subset of T 3. In order to show that from the numerical results on the existence of unstable attractors and heteroclinic cycles we can infer similar results for P, it would be enough to show that all open sets in T n 1 correspond to open sets in P. Whether this is true remains an open question Conclusions In this paper we proved the existence of heteroclinic cycles between unstable attractors in a global network consisting of four oscillators. Such heteroclinic cycles occur for an open set of parameter values in the class of systems that we considered. For this purpose we used the mathematical framework introduced in [12] and extended in [19], that permits us to study analytically the evolution of the system and define the neighborhood of a state in the (infinite dimensional) state space. A natural question is whether similar cycles occur in networks with more than four oscillators. In figure 7 we illustrate the presence of heteroclinic cycles for a five oscillator network. Moreover, numerical simulations in [10] suggest that such cycles exist for networks with n = 100 oscillators. Another question is how the existence of heteroclinic cycles is affected if we consider instead of the Mirollo-Strogatz model other pulse response functions. The importance of heteroclinic connections like those considered in the current work, is that they provide flexibility to the system because it is possible to switch between unstable attractors. Furthermore, they can also be used to perform 17

18 φ i (t) time (t) Figure 7. Heteroclinic connections for five oscillators: The phases φ i (t) of the oscillators are shown against the time t at which the nth oscillator fires. Random noise is generated every 40 time units. The reconfiguration of the oscillators after the noise corresponds to switching between unstable attractors. The parameter values are b = 3, ε = 0.3 and τ = 0.1. computational tasks, such as, to design a multibase counter [7] and sequence learning [24]. We are not aware of any work that answers the question whether heteroclinic connections persist for non-global networks or for non-identically coupled networks. It would be worthwhile to further explore the dynamics of pulse coupled oscillators and the existence of heteroclinic cycles between unstable attractors for varied non-global networks, viz., regular, random [25], small-world [26] and fractal [27, 28] networks. Acknowledgments The authors would like to thank Marc Timme, Gert Vegter and Sijbo Holtman for useful discussions. KE acknowledges support from the NWO cluster NDNS +. [1] M. Krupa, Robust heteroclinic cycles, Journal of Nonlinear Science 7 (1997) [2] J. Guckenheimer, P. Holmes, Structurally stable heteroclinic cycles, Math. Proc. Cambridge Phil. Soc. 103 (1988) [3] F. M. Busse, K. E. Heikes, Convection in a rotating layer: A simple case of turbulence, Science 208 (1980) [4] R. M. May, W. Leonard, Nonlinear aspects of competition between three species, SIAM Journal of Applied Mathematics 29 (1975) [5] I. Stewart, Mathematics: Regime change in meterology, Nature 422 (2003) [6] M. Timme, F. Wolf, T. Geisel, Unstable attractors induce perpetual synchronization and desynchronization, Chaos 13(1) (2002) [7] P. Ashwin, J. Borresen, Discrete computation using a perturbed heteroclinic network, Physics Letters A 347 (2005)

19 [8] R. Mirollo, S. H. Strogatz, Synchronization of pulse-coupled biological oscillators, SIAM J. Appl. Math 50(6) (1990) [9] W. Wu, T. Chen, Desynchronization of pulse-coupled oscillators with delayed excitatory coupling, Nonlinearity 20 (2007) [10] U. Ernst, K. Pawelzik, T. Geisel, Synchronization induced by temporal delays in pulse-coupled oscillators, Physical Review Letters 74(9) (1995) [11] M. Timme, F. Wolf, T. Geisel, Prevalence of unstable attractors in networks of pulse-coupled oscillators, Physical Review Letters 89(15) (2002) [12] P. Ashwin, M. Timme, Unstable attractors: existence and robustness in networks of oscillators with delayed pulse coupling, Nonlinearity 18 (2005) [13] M. Rabinovich, A. Volkovskii, P. Lecanda, R. Huerta, H. D. I. Abarbanel, G. Laurent, Dynamical encoding by networks of competing neuron groups: Winnerless competition, Physical Review Letters 87 (2001) [14] R. Huerta, T. Nowotny, M. Garca-Sanchez, H. Abarbanel, M. I. Rabinovich, Learning classification in the olfactory system of insects, Neural Computation 16 (2004) [15] R. Huerta, M. Rabinovich, Reproducible sequence generation in random neural ensembles, Physical Review Letters 93 (2004) [16] D. Hansel, G. Mato, C. Meunier, Clustering and slow switching in globally coupled phase oscillators, Physical Review E 48 (1993) [17] H. Kori, Y. Kuramoto, Slow switching in globally coupled oscillators: robustness and occurrence through delayed coupling, Physical Review E 63 (2001) [18] P. Ashwin, M. Timme, When instablity makes sense, Nature 436 (2005) [19] H. W. Broer, K. Efstathiou, V. E. Subramanian, On unstable attractors in pulse-coupled networks with delay, submitted to Nonlinearity. [20] O. Diekmann, S. A. van Gills, S. M. Verduyn Lunel, H.-O. Walther, Delay Equations: Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, [21] A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, UK, [22] H. W. Broer, F. Takens, Dynamical Systems and Chaos, Epsilon Uitgaven, 2008 (to appear). [23] S. H. Strogatz, Nonlinear Dynamics and Chaos, Addison Wesley, [24] M. Rabinovich, P. Varona, A. I. Selverston, H. D. I. Abarbanel, Dynamical principles in neuroscience, Reviews of Modern Physics 78 (2006) [25] A. Zumdieck, M. Timme, T. Geisel, F. Wolf, Long chaotic transients in complex networks, Physical Review Letters 93 (2004) [26] D. Watts, S. Strogatz, Collective dynamics of small-world networks, Nature 393 (1998) [27] S. Raghavachari, J. Glazier, Spatially coherent states in fractally coupled map lattices, Phys. Rev. Lett. 74 (1995) [28] S. Raghavachari, Synchrony and chaos in coupled oscillators and neural networks, Ph.D. thesis, University of Notre Dame (1999). Appendix A. Propositions used in the proofs of lemmas 5 and 6 Proposition 7. There is ρ 1 > 0 and C 2 > 0 such that if d(φ Q 1, φ) = ɛ < ρ for some φ P then there is some x with 0 < x < C 2 ɛ such that either (i) φ 1 (0) = H 1 (τ + x), φ 2 (0) = H 1 (τ x) + x, φ 3 (0) = φ 4 (0) = 0, or (ii) φ 1 (0) = H 1 (τ x) + x, φ 2 (0) = H 1 (τ + x), φ 3 (0) = φ 4 (0) = 0, or (iii) φ 1 (0) = H 1 (τ), φ 2 (0) = H 1 (τ), φ 3 (0) = φ 4 (0) = 0. In all cases, Σ i (φ) = for i = 1, 2 and Σ 3 (φ) = Σ 4 (φ) = {0}. 19

20 Proof. In this proof we use results from [19]. Since the phases φ Q 1 i, i = 1,..., 4 do not have any discontinuities in the intervals ( τ, 0) and (0, τ) there is ρ 1 > 0 and constants C 1, C 2 > 0 such that if d(φ, φ Q 1 ) = ɛ < ρ 1 then φ j (s) φ Q 1 j (s) < C 1 ɛ for j = 1,..., 4 and for s ( τ + C 2 ɛ, C 2 ɛ) (C 2 ɛ, τ C 2 ɛ). This also implies that no oscillator fires in the interval ( τ + C 2 ɛ, C 2 ɛ), because then the phase of the other oscillators would have a discontinuity in the interval (C 2 ɛ, τ C 2 ɛ). Given that the size of discontinuity has to be larger than V 1 (0) > 0 we conclude that by making ρ 1 (and consequently ɛ) small enough, the condition φ j (s) φ Q 1 j (s) < C 1 ɛ would not hold, therefore we have a contradiction. Moreover, for similar reasons and because φ 4 (0) = 0 we conclude that φ 4 (s) = s for all s [0, τ C 2 ɛ). This in turn implies that the oscillators O 1, O 2 and O 3 do not fire in ( τ, C 2 ɛ). We have established that at time C 2 ɛ, the phase φ 1 ( C 2 ɛ) of the oscillator O 1 is O(ɛ) close to τ, while φ 1 (C 2 ɛ) is O(ɛ) close to H 1 (τ). From this we deduce that O 1 must receive exactly one pulse in the interval ( C 2 ɛ, C 2 ɛ). The same is also true for O 2. On the other hand, φ 3 ( C 2 ɛ) is O(ɛ) close to τ + W 2 and φ 3 (C 2 ɛ) is O(ɛ) close to 0. This means that the oscillator O 3 must receive enough pulses to fire in the interval ( C 2 ɛ, C 2 ɛ). Given that H 1 (W 2 + τ) < 1 and that H 2 (W 2 + τ) > 1 and also that ɛ can be chosen arbitrarily small we conclude that O 3 must receive exactly two pulses in the interval ( C 2 ɛ, C 2 ɛ). Similar arguments show that O 4 must also receive exactly two pulses in the interval ( C 2 ɛ, 0]. The only possibility for this combination of pulses to happen is if the oscillators O 1 and O 2 fire at moments t 1 = τ x and t 2 = τ such that x < C 2 ɛ. We should consider the cases that O 1 fires at t 1 and O 2 at t 2, that O 2 fires at t 1 and O 1 at t 2, or finally that O 2 and O 1 fire simultaneously at t 1 = t 2 = 0. Consider first the case that O 1 fires at t 1 = τ x and that O 2 fires at t 2 = τ. Then, at time x the oscillators O 2, O 3 and O 4 receive a pulse from O 1 and at time 0 the oscillators O 1, O 3 and O 4 receive a pulse from O 2. Since, O 3 and O 4 receive the second pulse at time 0, we have φ 3 (0) = φ 4 (0) = 0. Moreover, when O 2 receives the pulse from O 1 its phase is φ 2 ( x ) = τ x, so φ 2 (0) = H 1 (τ x) + x. Finally, when O 1 receives the pulse from O 2 its phase is φ 1 (0 ) = τ + x so its phase becomes φ 1 (0) = H 1 (τ + x). In the case that O 2 fires before O 1 we can use similar arguments to show that φ 1 (0) = H 1 (τ x) + x and φ 2 (0) = H 1 (τ + x). Finally, if O 1 and O 2 fire simultaneously at τ, then φ 1 (0) = φ 2 (0) = 0. Proposition 8. There is ρ 1 > 0 and C 1 > 0 such that if d(φ Q 2, φ) = ɛ < ρ for some φ P then φ i (0) W 2 < C 1 ɛ and Σ i (φ) = for i = 1, 2, φ 4 (0) = 0 with Σ 4 (φ) = {0} and either (i) 0 φ 3 (0) < C 2 ɛ and Σ 3 (φ) = { φ 3 (0)}, or (ii) C 2 ɛ < φ 3 (0) < 0 and Σ 3 (φ) =. 20

21 Proof. Since the phases φ Q 2 i, i = 1,..., 4 do not have any discontinuities in the intervals ( τ, τ) there is ρ 1 > 0 and constants C 1, C 2 > 0 such that if d(φ, φ Q 2 ) = ɛ < ρ 1 then φ j (s) φ Q 2 j (s) < C 1 ɛ for j = 1,..., 4 and for s ( τ + C 2 ɛ, τ C 2 ɛ). This also implies that no oscillator receives a pulse in the interval ( τ + C 2 ɛ, τ C 2 ɛ) which means that no oscillator fires in the interval [ τ, C 2 ɛ). For the oscillators O 1 and O 2 we can conclude that they do not fire also in [ C 2 ɛ, 0] since their phases are O(ɛ) close to φ 1 (0) = φ 2 (0) = W 2 in this time interval. Therefore, φ j (0) W 2 < C 1 ɛ for j = 1, 2 and Σ j (φ) =. Moreover, φ 4 (0) = 0 since φ P and therefore Σ 4 (φ) = {0}. Finally, for the oscillator O 3 we have that φ 3 (0) is O(ɛ) close to 0 (mod 1), and therefore either it fires at time u with C 2 ɛ < u 0 and φ 3 (0) = u > 0, Σ 3 (φ) = {u} or it fires at time u with C 2 ɛ > u > 0 and φ 3 (0) = 1 u, Σ 3 (φ) =. Proposition 9. If φ l 1 then R(φ) l and there is a finite number m of iterations such that R m (φ) l 2. Proof. The initial event sequence in l 1 with H 2 (v + τ) H 2 (w + τ) < τ evolves as, ([2P, (1, 2), τ], [P, (3, 4), τ], [F, 1, 1 v], [F, 2, 1 w], [F, (3, 4), 1]) 1 ([2P, (1, 2), 0], [P, (3, 4), 0], [F, 1, 1 v τ], [F, 2, 1 w τ], [F, (3, 4), 1 τ]) 2 ([F, 1, 1 H 2 (v + τ)], [F, 2, 1 H 2 (w + τ)], [F, (3, 4), 1 H 1 (τ)]) 3 ([F, 1, 0], [F, 2, H 2 (v + τ) H 2 (w + τ)], [F, (3, 4), H 2 (v + τ) H 1 (τ)]) Let ν = H 2 (v + τ) H 2 (w + τ). Since ν < τ < H 2 (w + τ) H 1 (τ) < H 2 (v + τ) H 1 (τ) (which follows from equation (B.2) in proposition 16), the evolution continues as, 4 ([F, 2, ν], [P, (2, 3, 4), τ], [F, (3, 4), H 2 (v + τ) H 1 (τ)], [F, 1, 1]) 5 ([F, 2, 0], [P, (2, 3, 4), τ ν], [F, (3, 4), H 2 (v + τ) H 1 (τ) ν], [F, 1, 1 ν]) 6 ([P, (2, 3, 4), τ ν], [P, (1, 3, 4), τ], [F, (3, 4), H 2 (v + τ) H 1 (τ) ν], [F, 1, 1 ν], [F, 2, 1]) 7 ([P, (2, 3, 4), 0], [P, (1, 3, 4), ν], [F, (3, 4), H 2 (v + τ) H 1 (τ) τ], [F, 1, 1 τ], [F, 2, 1 τ + ν]) Since H 1 (1 H 2 (v + τ) + H 1 (τ) + τ)) < H 1 (W 2 + τ) < 1 and ν < 1 H 1 (1 H 2 (v + τ) + H 1 (τ) + τ) (equation (B.3) in proposition 16), we get, 8 ([P, (1, 3, 4), ν], [F, (3, 4), 1 H 1 (1 H 2 (v + τ) + H 1 (τ) + τ)], [F, 1, 1 τ], [F, 2, 1 H 1 (τ ν)]) 9 ([P, (1, 3, 4), 0], [F, (3, 4), 1 H 1 (1 H 2 (v + τ) + H 1 (τ) + τ) ν], [F, 1, 1 τ ν], [F, 2, 1 H 1 (τ ν) ν]) For ν < τ, one can show that H 1 (H 1 (1 H 2 (v + τ) + H 1 (τ) + τ) + ν) 1 (equation (B.4) in proposition 16). Therefore, 10 ([F, (3, 4), 0], [F, 1, 1 H 1 (τ + ν)], [F, 2, 1 H 1 (τ ν) ν]) 11 ([2P, (1, 2), τ], [P, (3, 4), τ], [F, 1, 1 H 1 (τ + ν)], [F, 2, 1 H 1 (τ ν) ν], [F, (3, 4), 1]) 21

22 Let v = H 1 (τ + ν) and w = H 1 (τ ν) + ν be the phases of the oscillators O 1 and O 2. Then, since 0 < ν < τ we have that H 1 (τ) < v < H 1 (2τ) and w = H 1 (τ) + (1 A ε )ν. Since A ε > 1 we obtain that w > H 1 (τ) + (1 A ε )τ = m ε + τ > τ. Furthermore, Finally, A ε w + (A ε 1)v = A ε (H 1 (τ) + (1 A ε )ν) + (A ε 1)(H 1 (τ) + A ε ν) = (2A ε 1)H 1 (τ). ν = H 2 (v + τ) H 2 (w + τ) = A 2 ε(v w ) = A 2 ε(h 1 (τ) + A ε ν H 1 (τ) + (A ε 1)ν) = A 2 ε(2a ε 1)ν. This implies that in finite iterations the initial state is mapped to a state characterized by v, w such that H 2 (v + τ) H 2 (w + τ) τ. Proposition 10. If φ l 2 then R(φ) B 1 B 2. Proof. Observe that for v < H 1 (2τ), we have that H 2 (v + τ) < g 1 (τ) < 1. Let and and note that µ = H 1 (1 + τ + H 2 (w + τ) H 2 (v + τ)) κ = H 1 (1 + τ + H 1 (τ) H 2 (v + τ)) µ κ = H 1 (H 2 (w + τ) H 1 (τ)) = H 2 (H 1 (w + τ) τ). We skip the first three transitions in the evolution of [φ v,w ] which are the same as in proposition 9. Then we have, 4 ([P, (2, 3, 4), τ], [F, 2, H 2 (v + τ) H 2 (w + τ)], [F, (3, 4), H 2 (v + τ) H 1 (τ)], [F, 1, 1]) 5 ([P, (2, 3, 4), 0], [F, 2, H 2 (v + τ) H 2 (w + τ) τ], [F, (3, 4), H 2 (v + τ) H 1 (τ) τ], [F, 1, 1 τ]) We distinguish now two cases based on the value of µ. First, for µ < 1, the evolution is, 6 ([F, 2, 1 µ], [F, (3, 4), 1 κ], [F, 1, 1 τ]) 7 ([F, 2, 0], [F, (3, 4), µ κ], [F, 1, µ τ]) Since µ κ > τ (equation (B.5) in proposition 16) 8 ([P, (1, 3, 4), τ], [F, (3, 4), µ κ], [F, 1, µ τ], [F, 2, 1]) 9 ([P, (1, 3, 4), 0], [F, (3, 4), µ κ τ], [F, 1, µ 2τ], [F, 2, 1 τ]) 10 ([F, (3, 4), 1 H 1 (1 + κ + τ µ)], [F, 1, 1 H 1 (2τ + 1 µ)], [F, 2, 1 τ]) 11 ([F, (3, 4), 0], [F, 1, H 1 (1 + κ + τ µ) H 1 (2τ + 1 µ)], [F, 2, H 1 (1 + κ + τ µ) τ]) 12 ([2P, (1, 2), τ], [P, (3, 4), τ], [F, 1, H 1 (1 + κ + τ µ) H 1 (2τ + 1 µ)], [F, 2, H 1 (1 + κ + τ µ) τ], [F, (3, 4), 0]) 22