Transitory behaviors in diffusively coupled nonlinear oscillators

Abstract

We study collective behaviors of diffusively coupled oscillators which exhibit out-of-phase synchrony for the case of weakly interacting two oscillators. In large populations of such oscillators interacting via one-dimensionally nearest neighbor couplings, there appear various collective behaviors depending on the coupling strength, regardless of the number of oscillators. Among others, we focus on an intermittent behavior consisting of the all-synchronized state, a weakly chaotic state and some sorts of metachronal waves. Here, a metachronal wave means a wave with orderly phase shifts of oscillations. Such phase shifts are produced by the dephasing interaction which produces the out-of-phase synchronized states in two coupled oscillators. We also show that the abovementioned intermittent behavior can be interpreted as in-out intermittency where two saddles on an invariant subspace, the all-synchronized state and one of the metachronal waves play an important role.

Introduction

Electrically coupled interneurons via gap junctions (GJ) have been widely observed in the mammalian neocortex (Galarreta and Hestrin 1999; Gibson et al. 1999). Motivated by this observation, many theoretical studies of collective behaviors of GJ-coupled interneurons have been conducted by the use of neural oscillators interacting via diffusion type couplings.

Although it had been widely believed that the GJ-coupling among interneurons promotes their synchronous firing (Galarreta and Hestrin 1999; Gibson et al. 1999), Sherman and Rinzel (1992) first observed that weak GJ-coupling can lead dephasing and phase locking by anti-phase in two coupled neural oscillators. Later, Han et al. (1995, 1997) demonstrated how diffusion type weak couplings can lead to dephasing of two oscillators by using the Morris-Lecar model. They also found alternations between synchronization and desynchronization of oscillations in a large population of coupled oscillators. Also in other systems, for instance, in electrochemical dynamical systems, Karantonis and Nakabayashi (2001) found a similar phenomenon in diffusively coupled electrochemical oscillators. The present authors also found that some kinds of neural oscillator, when diffusively coupled, exhibit transitory behaviors, i.e., spontaneously irregular transitions between various dynamical states such as synchronized states and chaotic states (Tsuda et al. 2004).

The earlier studies mentioned above (Sherman and Rinzel 1992; Han et al. 1995, 1997; Karantonis and Nakabayashi 2001) and our previous study (Tsuda et al. 2004) show different synchronies in two-oscillator system with weak diffusion coupling. While two oscillators synchronize in anti-phase in the former cases, they synchronize in out-of-phase indicating the phase difference not necessarily equal to π in the latter. Little is known about collective behaviors in large populations of coupled oscillators which have the out-of-phase synchrony in the case of the two oscillator system. In this paper, we show that there appear various transitory behaviors in such diffusively coupled oscillators and one of them can be interpreted as in-out intermittency.

Transitory behaviors with aperiodic changes between dynamic states have been observed as cortical activity, such as fluctuations of local field potentials accompanying with transient synchrony (Gray et al. 1992) and spontaneous cortical activity which encompasses a set of dynamically switching cortical states (Kenet et al. 2003). Two of the present authors have proposed a hypothesis on the mechanism of the observed chaotic field (Fujii and Tsuda 2004), that the origin of temporal fluctuations and transient synchrony observed in the neocortical activity stems from chaotic transitions among dynamic states appearing in gap junction (GJ)-coupled interneuron systems. The GJ-coupled interneurons to pyramidal cells can influence in the fluctuations of membrane potentials of the pyramidal cells, which could be experimentally observed.

This paper is organized as follows: In section "A nonlinear oscillator model, its diffusively coupled system, and dynamics of diffusively coupled two oscillator system", we describe the nonlinear oscillator and its diffusively coupled system which we treat here. We also discuss the stability of synchronous states in diffusively coupled two oscillators. In section "Dynamics of diffusively coupled N-oscillator system", we discuss behaviors appearing in diffusively coupled N-oscillator system in the case of N > 2. In subsection "Bifurcation structures of periodic states and system size dependence", we discuss bifurcation structures of periodic states of coupled N-oscillator systems. And we also show that they, when viewed as functions of rescaled coupling strength g/N², are not depend on N in the case of slightly large N. In subsection "Transitory behaviors", we discuss one of typical transitory behaviors appearing in the coupled systems. We also show that it can be interpreted as in-out intermittency. In subsection "Markov chain model", we calculate the probability distribution of laminar duration of the in-out intermittency and compare it with the one calculated by using a Markov chain model of the in-out-intermittency. In subsection "Bifurcations along with variation in the coupling strength", we describe various behaviors appearing through successive bifurcations with variations in the coupling strength. Section "Summary and discussion" is devoted to summary and discussion.

A nonlinear oscillator model, its diffusively coupled system, and dynamics of diffusively coupled two oscillator system

A nonlinear oscillator model and its diffusively coupled system

The nonlinear oscillator treated here is described by (Tsuda et al. 2004)

1

When it is viewed as a neuron model, the variable V represents the membrane potential of a neuron and the other variable R is a recovery variable representing an activation state, and the parameter I represents an injected current. This model has another parameter μ which will be fixed to the value, 1.65 throughout this article. Although this model is essentially the same as the Hindmarsh-Rose model (Hindmarsh and Rose 1984) with two variables, it was derived as a minimal one-parameter family of two-dimensional neural dynamical systems showing both Hopf bifurcation and saddle-node bifurcation (Tsuda et al. 2004). We called it μ-model.

Figure 1a shows the bifurcation diagram of the model when the injected current I is a bifurcation parameter. The model exhibits saddle-node bifurcation (I = 0) and Hopf bifurcation (I ∼ 0.19), and there appears a limit cycle oscillation between these bifurcation points. Figure 1b shows nullclines for V and R together with a limit cycle for I = 0.004.

Fig. 1.

a Bifurcation diagram for a single μ-model showing V as a function of I. μ = 1.65. For stable periodic states, maximum values of V are plotted. b Nullclines for V and R together with a limit cycle. I = 0.004

The diffusively coupled system of the μ-models obeys the following equations:

2

3

where N is the number of oscillators and g is the coupling strength. Here the couplings are introduced in variables V_i. We treat the case of one-dimensional nearest neighbor coupling with a free boundary condition, which is in accord with the Neumann type of boundary condition in the continuous space limit.

A coupled two oscillator system

Before showing cases of large numbers of oscillators, in this subsection we treat the case of N = 2. In the weak interaction limit, the evolution of the phase difference δϕ = ϕ₁ − ϕ₂ between two oscillators with phases ϕ₁ and ϕ₂ is governed by (Kuramoto 1984)

4

where is the antisymmetric part of the effective coupling defined by

5

Here V(ϕ′) denotes the variable V at phase ϕ′ on the limit cycle. The phase sensitivity function Z(ϕ) gives the change of phase along the limit cycle caused by the change of state vector (V, R) in phase space.

Figure 2a shows for I = 0.004. In this figure, it is shown that the system has three types of synchronized states: the unstable in-phase and anti-phase states, and the stable out-of-phase states. The time-series of V₁ and V₂ in one of the out-of-phase synchronized states for I = 0.004 and g = 0.005 is shown in Fig. 2b.

Fig. 2.

a Antisymmetric part of effective coupling, . I = 0.004. b Time-series of V₁ and V₂ in an out-of-phase synchronized state of two interacting oscillators for g = 0.005. c Time-series of V₁ and V₂ in the anti-phase synchronized state for g = 0.02

In the case of coexistence of a limit cycle and a stable node with an intervening saddle, it was shown that the saddle causes the deformations of the phase flow and thus the diffusion type of coupling between such oscillators can produce dephasing interaction under a certain condition (Han et al. 1995). In the case of the μ-model, a limit cycle does not coexist with a saddle and a stable node (see Fig. 1). Nevertheless, such dephasing interaction can survive even after the saddle disappears via saddle-node bifurcation, because the main features of the phase flow in the vicinity of the limit cycle continue to remain unchanged (Han et al. 1995). Therefore, it is probable that the dephasing interaction in the case of the μ-model is also attributed to this mechanism.

In the case of relatively strong coupling, where the discussion based on the effective coupling cannot be applicable, in-phase and anti-phase synchronization can be stabilized, and either of them appears depending on the initial phase difference of two oscillators. The time-series of V₁ and V₂ in the anti-phase synchronized state for g = 0.02 is shown in Fig. 2c.

Dynamics of diffusively coupled N-oscillator system

Bifurcation structures of periodic states and system size dependence

In this subsection, we discuss bifurcation structures of periodic states of coupled N-oscillator systems, and show that they are almost independent of N when viewed as function of rescaled coupling strength g/N², in the case of relatively large N.

In the continuous space limit, the coupling term of Eq. 3, g(V_i+1 + V_i−1 − 2V_i) should be equivalent to D ∂²V/∂x², where x is a space variable. The coefficient D is a diffusion coefficient defined by g/N² as N →∞ and g →∞, keeping g/N² constant. Thus, we expect that two systems with sufficiently large numbers of oscillators N₁ and N₂ (≠ N₁) exhibit the same behaviors at g₁ and g₂, respectively, if their rescaled coupling strengths are equal each other:

6

Since this continuous limit is not mathematically straightforward, the scaling relation should be restricted to a certain range of finite system size. To demonstrate whether this scaling is realized or not, we study the bifurcation structures of periodic states of the coupled N-oscillator system for different N viewing the (rescaled) coupling strength g (g/N²) as a bifurcation parameter.

Figure 3a–c show bifurcation structures in the case of N = 8, 16 and 17, respectively. The strength of g (g/N²) is shown on the lower (upper) abscissa. To follow the branching of solutions, we use AUTO (Doedel et al. 2002), a software for continuation and bifurcation problems. It is clear from Fig. 3a–c that the bifurcation structures, when viewing g/N² as a bifurcation parameter, are almost independent of N, except in the case that g/N² is very small.

Fig. 3.

Bifurcation diagrams of periodic states of N oscillators interacting diffusively via one-dimensional nearest-neighbor coupling. The (rescaled) coupling strength g (g/N²) is viewed as a bifurcation parameter, and is shown on the lower (higher) abscissas. The axis of the ordinate is for the L₂-norm divided by . Branches located at g close to 0 which are not important for the discussion in this article are omitted. Stable and unstable branches are represented by solid and dashed lines, respectively. The branch of the all-synchronized state is marked with AS. The branches of out-of-phase synchronized state are marked with and . Branching points marked with PF₁, PF₂ and PF₃ are subcritical pitchfork bifurcation points. aN = 8. bN = 16. cN = 17. d Enlarged view of b near the branching point PF₂. Branching points marked with TR₁ and TR₂ are torus bifurcation points

The horizontal straight line marked with AS in Fig. 3 represents the branch of the all-synchronized state in which all the oscillators are in-phase synchronized and are just on the unperturbed limit cycle orbit. The spatio-temporal contour maps of the amplitude of V-variables of a state on the AS branch in the case of N = 16 is shown in Fig. 4a.

Fig. 4.

The contour plots of V_i time-series of periodic states on the branches shown in Fig. 3. The vertical and horizontal directions indicate the oscillators’ indices and time, respectively. Red, green, and blue curves are isopotential lines of V_i = 0.55, V_i = 0.3, and V_i = 0.05, respectively. Patterns are depicted during the period of two cycle. aN = 16. The states on the AS branch. bN = 16. The state at g = 1.02 (g/N² ≈ 0.003984) and rescaled L₂-norm of 0.29. cN = 16. The state at g = 0.255 and rescaled L₂-norm of 0.2913. dN = 16. The SMW⁺⁻ state at g = 0.255 and rescaled L₂-norm of 0.2937. e N = 8. The at g = 0.255 (g/N² ≈ 0.003984) and rescaled L₂-norm of 0.2937. (Color figure online)

Three curves marked with and SMW^±∓ in Fig. 3 represents out-of-phase synchronous branches on which oscillators are synchronized in out-of-phase. They arise due to the dephasing interaction that allows out-of-phase synchronous states in the case of N = 2. The states on these branches exhibit spatio-temporal patterns of metachronal waves because of their orderly phase shifts in oscillators’ indices. Examples of such metachronal wave patterns are shown in Fig. 4b–e. These metachronal wave patterns can be classified into mirror asymmetric ones or symmetric ones with respect to oscillators indices. Thus we term these out-of-phase synchronous branches “asymmetric metachronal wave” (AMW^±_1,2) branches or “symmetric metachronal wave” (SMW^±∓) branches. In fact, each of AMW^±₁, AMW^±₂ and SMW^±∓ curves in Fig. 3 does not represent one branch but a pair of branches which is degenerated in L₂-norm. For instance the SMW ^± ∓ curve corresponds to a pair of branches, SMW⁺⁻ and SMW⁻⁺. The superscripts such as “+” and “+−” of the labels are for distinction between paired branches. The essential difference between the paired branches is in the direction of phase shift with respect to oscillators’ indices.

Since only lower half parts of these out-of-phase synchronous branches, that is, the parts rescaled L₂-norms of which are less than about 0.48 are important in the present study, hereafter when we say, for example, branch, it means its lower half part. We call a state on the branch, etc., simply a state, etc.

Figure 4b–d show spatio-temporal patterns of and SMW⁺⁻ states, respectively, in the case of N = 16. The that are paired with the state exhibits the same pattern shown in Fig. 4b, except that the pattern is turned upside down. The same thing holds for the relation between the and states. The spatio-temporal pattern of SMW⁻⁺ state, that is paired with the SMW⁺⁻ state, is the same as the one shown in Fig. 4d, except that the upper and lower half parts are interchanged.

The scaling mentioned above appears also in the relation between the spatio-temporal patterns shown in Fig. 4b, e. The former figure shows the pattern of the state at g/N² ≈ 0.00398 (g = 1.02) in the case of N = 16, and the latter shows the pattern of the AMW⁺₁ state at the same g/N² (g = 0.255) in the case of N = 8. The latter pattern (N = 8) become almost the same as the former (N = 16) except that the vertical scale of the former is doubled.

It should be noted that the SMW^±∓ branches in the case of N = 16 and AMW^±₁ branches in the case of N = 8 appear at the same position when measured in g (not g/N²) (see Fig. 3a, b). It should be also noted that the lower half part of the spatio-temporal pattern of the SMW⁺⁻ state at g = 0.255 in the case of N = 16 (Fig. 4d) is the same as the one of the (Fig. 4e) state at the same g in the case of N = 8. The same thing holds for the relation between the upper half part in the case of N = 16 and the pattern of the AMW⁻₁ in the case of N = 8. In the case of even N ≥ 4, a SMW ^±∓ state at certain g for N is composed of a pair of states and for N/2 at the same g, satisfying the mirror symmetry as a whole. Even in the case that N(≥5) is odd, the same relation between a SMW ^± ∓ state for N and states for (N − 1)/2 is effectively satisfied for relatively large N.

The reason why this hierarchical structure arises is as follows. The coupled systems discussed here have an invariant subspace due to the mirror symmetry with respect to the reverse of oscillators’ indices. This mirror symmetric invariant subspace is defined by:

7

The SMW^±∓ states are inside the invariant subspace. In the case of even N, within the invariant subspace, since the interaction between N/2-th oscillator and N/2 + 1-th one disappears, a coupled N-oscillator system can be decomposed into two identical coupled N/2-oscillator systems each of which obeys the free boundary condition. Thus if there exist a pair of out-of-synchronous states, for example, in the case of N/2, there should exist, in the case of N, a pair of states each of which is composed of the paired states for N/2 satisfying the mirror symmetry. In the case of odd N, a coupled N-oscillator system within the invariant subspace can be considered to consist of two identical coupled (N − 1)/2-oscillator systems and one oscillator which locates between them. The analyses using AUTO suggest that in a SMW⁺⁻ state, the trajectory of the centermost oscillator is approximately the same as its neighbor oscillators, i.e., (N − 1)/2-th and (N + 1)/2-th oscillators for relatively large N. Then the interactions between the centermost oscillator and the subsystems consisting of (N − 1)/2 oscillators are approximately zero. Thus a coupled N-oscillator system in a SMW^±∓ state can be effectively considered to consist of two identical coupled (N − 1)/2-oscillator systems with the free boundary condition.

Now, let us pay attention to the normal and tangential stabilities of the AS and SMW⁺⁻ branches with respect to the invariant subspace, that play important roles in transitory behaviors discussed in the next subsection. While the AS branch is stable at large g/N², it is unstable below g/N²∼ 0.0035. It is plausible to think that this g-dependence originates from the fact that the in-phase synchronized state in two interacting oscillators is unstable in the case of weak couplings and stable in the case of relatively strong couplings ("A nonlinear oscillator model, its diffusively coupled system, and dynamics of diffusively coupled two oscillator system"). The dependence of the stability of the AS branch and the aforementioned scaling produce a region of g where it is unstable for N and is stable for N/2. Thus, in such a region, the AS state for N is a saddle, which is unstable transversely to the mirror symmetric invariant subspace and stable within it (ones are urged to recall that a coupled N-oscillator system within the invariant subspace can be decomposed into two identical coupled oscillator systems with free boundary condition). According to analyses of Floquet multipliers by using AUTO, whereas the SMW⁺⁻ branch is unstable inside the invariant subspace, it is stable in the normal direction to the invariant subspace. Thus, below g/N²∼ 0.0035, there exist two saddles, i.e., AS and SMW⁺⁻ states in the invariant subspace. In the next subsection, we show that trajectories between these saddles are formed and transitory behaviors which consist of the saddles as dynamic components appear.

Transitory behaviors

Various transitory behaviors appear depending on the (rescaled) coupling strength. Figure 5 shows the spatio-temporal contour map of the amplitude of V-variables of one of such transitory behaviors. This is the case of N = 16, I = 0.004 and g = 0.255. The same behaviors as shown in this figure appear in the case of other N(>10) because of the scaling mentioned in the previous subsection.

Fig. 5.

The contour plot of V_i time-series in a typical transitory behavior exhibited by 16 coupled oscillators for I = 0.004 and g = 0.255. Initial conditions of (V_i, R_i) are randomly chosen from the interval (−0.2, 0.8). Characteristic patterns are marked with labels, AS, SMW⁺⁻, MSMW⁺⁻, and . Meanings of labels MSMW⁺⁻ and should be refer to the text

In Fig. 5, there appear the same patterns as shown in Fig. 4a, d. They are marked with AS and SMW⁺⁻. These states are unstable, but each of them has a stable direction. Thus they appear as dynamic components of this transitory behavior. The pattern AS, which is sustained for some time, is transformed into the pattern labeled or the one labeled . The patterns which we call modulated asymmetric metachronal waves, represent dephasing processes of oscillators. The difference between and are simply in the direction of phase shift with respect to oscillators’ indices. The dynamic states originate from quasiperiodic states bifurcating from the bifurcation point on the branches, which are marked with TR₁ in Fig. 3d. After the appearance of one of these dephasing processes, the coupled system reaches the SMW⁺⁻ state. The mirror symmetric pattern labeled MSMW⁺⁻ (modulated symmetric metachronal wave) or the one labeled WCS follows after SMW⁺⁻. The pattern MSMW⁺⁻, in which phase shifts between neighboring oscillators become less with the passage of time, corresponds to the transitions from SMW⁺⁻ to AS. The pattern WCS corresponds to the other transition processes from SMW⁺⁻ to AS. This pattern includes a slightly complex pattern unlike MSMW⁺⁻. In certain regions of the coupling strength, a state exhibiting WCS pattern exists as a stable attractor ("Bifurcations along with variation in the coupling strength") and its largest Lyapunov exponent become small but still positive. Thus we name this dynamic state weakly chaotic state.

To make clear essential structure of the transitory behavior, we take a Poincaré section by cutting the trajectory, a part of which has the spatio-temporal structure shown in Fig. 5, with a hyper-plane defined by (Tsuda et al. 2004)

8

Figure 6a shows the projection to the (V₁, V₈) plane of the Poincaré map. Here we set C at 0.2. An attractor representing the transitory behaviors is clearly visible. Parts of the attractor are marked with the same labels as the ones used in Fig. 5. The saddle states AS and SMW⁺⁻ are located at (V₁, V₈) = (0.2, 0.2) and ∼(0.15, 0.25), respectively. The transitional state MAMW⁺₂ (MAWM⁻₂) from AS to SMW⁺⁻ state appears as the left (right) curve. The straight line-like path from SMW⁺⁻ to represents the transitional state MSMW⁺⁻. The other transitional state from SMW⁺⁻ to AS, WCS appears as the twisted string.

Fig. 6.

A Poincaré map for the transitory behavior shown in Fig. 5. The projection to the (V₁, V₈) plane of the Poincaré map are shown. Correspondence between the parts of the attractor on the Poincaré section and the dynamic states appearing in Fig. 5 are shown by using the same labels used in Fig. 5. Positions of the unstable periodic states which are mentioned in the previous section are also indicated by the crosses

We estimated the largest Lyapunov exponents of this attractor, using the standard method (Shimada and Nagashima 1979; Benettin et al. 1980a, b) and obtained λ₁ = 1.537 × 10⁻³. This indicate that the attractor is chaotic.

Figure 7a, b shows the time-series of distance of trajectories of the coupled oscillators from the mirror symmetric invariant subspace, which is mentioned in the previous subsection. One can see a characteristic of an intermittent behavior, that is, the existence of the trajectory intermittently approaching and leaving the invariant subspace. Figure 7c shows time-series of V-variables of the first and the eighth oscillators on the Poincaré section. The comparison between Fig. 7b, c makes clear that exponential increase and decrease in the distance with constant rates occur when (V₁, V₈) is close to (0.2, 0.2) indicating AS, and close to around (0.15, 0.25) indicating SMW⁺⁻, respectively. This intermittent behavior can be interpreted as in-out intermittency.

Fig. 7.

a Distance of the dynamical orbit in Fig. 6 from the mirror symmetric invariant subspace as a function of time in the linear scale. b The same plot as a but in a logarithmic scale. c The time-series of V₁ and V₈ on the Poincaré section. Solid and dashed curves are for V₁ and V₈, respectively

According to Ashwin, if the intersection of an attractor that exhibits intermittency with an invariant subspace is a minimal attractor, then the intermittency is called on-off. Otherwise, it is called in-out (Ashwin et al. 1999). The intersection of the attractor treated here with the invariant subspace is not a minimal attractor but contains at least two invariant sets, AS and SMW⁺⁻, which cause the out-phase and the in-phase, respectively.

Markov chain model

The statistical properties of in-out intermittency can be described by the Markov chain model (Ashwin et al. 1999). The Markov chain model for this intermittecy is schematically shown in Fig. 8. This model consists of two semi-infinite chains of states p_i and q_i representing in-phase and out-phase, respectively. Being in a state p_i implies that the system are near the SMW⁺⁻ which is transversely attracting; q_i implies that it is near the AS state which is transversely repelling. The index i corresponds to the distance from the mirror symmetric invariant subspace; the larger the value of i, the smaller is the distance from the invariant subspace. Since both of the in-phase and out-phase are periodic, only one-way transitions are allowed in both the in-chainp_i and the out-chainq_i. The probability of the transitions p_i−1 →p_i is denoted by α_D. The probability of the transitions p_i → q_i is denoted by ε. Transitions q_i →q_i−1 occur with probability 1.

Fig. 8.

Schematic diagram showing transitions in the Markov model for the in-out intermittency. The probability of the transition p_i−1 →p_i is denoted by α_D. The probability of the transition p_i →q_i is denoted by ε

By using this model, we can investigate the asymptotic behavior of the probability distributions of the duration of laminar phases in this intermittency P(n), where n is the laminar length. Details of the method are described in Ashwin et al. (1999). We have P(n) ∼ε (1 − ε)ⁿ for large n. This distribution has exponential fall. We also calculate P(n) for the attractor shown in Fig. 6a. Figure 9 shows the calculated P(n), which has exponential fall off for large n. The behavior of our calculated P(n) for large n is compatible with the predicted asymptotic behavior by the Markov chain model.

Fig. 9.

The probability distribution of the duration of laminar phases of the attractor shown in Fig. 6

Bifurcations along with variation in the coupling strength

In the previous section, we discuss one of typical transitory behaviors. In this section, we show that the system exhibits further curious behaviors depending on g (g/N²).

Typical Poincaré maps in the range of g/N² = 0.0008 ∼ 0.0012 are shown in descending order of g/N² in Fig. 10. These figures are in the case of N = 33.

Fig. 10.

Typical Poincaré sections in the region of g/N² = 0.0008–0.0012. N = 33. ig/N² = 0.00119 (g = 1.29591). Two fixed points representing periodic states and a weakly chaotic attractor. iig/N² = 0.001175 (g = 1.279575). Two fixed points. iiig/N² = 0.001045 (g = 1.138005). Two tori. ivg/N² = 0.00103 (g = 1.12167). An eight-shaped attractor. vg/N² = 0.000975 (g = 1.061775). The in-out intermittency discussed in the previous section. vig/N² = 0.00093 (g = 1.012770). Intermittent transitions between the attractor ruin and the strong chaos. viig/N² = 0.0009 (g = 0.9801). Strong chaos

In Fig. 10i, two fixed points and a string-like shaped attractor appear on the section. The fixed points represent stable periodic states on the branches shown in Fig. 3d. The branches are stable in the range of g/N² = 0.00105 ∼ 0.0013. The largest Lyapunov exponent of string-like shaped attractor is small but still positive. Thus it is very weakly chaotic.

The weakly chaotic attractor disappears and only two fixed points survive in Fig. 10ii. The disappearance of this chaotic state results from a crisis occurring via the contact of chaotic trajectories with a stable manifold of the SMW⁺⁻ state. Then this crisis is a boundary crisis (Grebogi et al. 1982). In Fig. 10iii, the states loose stability and then two tori appear instead.

In Fig. 10iv, the attractor in an eight-shaped fashion is shown. It is made by merging of the two tori in Fig. 10iii. This attractor is the same as the one discussed in "Transitory behaviors" except the absence of the weakly chaotic state. It is not a quasi-periodic attractor but a chaotic one, in which either of the two dynamical paths from the AS to the SMW⁺⁻ is selected, not regularly but irregularly. This can also be interpreted as a kind of in-out intermittency. The distribution of the duration of laminar phases for this chaotic attractor obeys an exponential law for large n, as with the one for the attractor discussed in the previous subsection. Decrease in the coupling strength to g/N²∼0.00094 makes this chaotic attractor change into a quasi-periodic, although its Poincaré map looks almost the same as the one shown in Fig. 10iv. Nevertheless, the selection of the dynamical paths from AS to SMW⁺⁻ is regular in it.

In Fig. 10v, the weakly chaotic attractor which do not appear in Fig. 10ii–iv reappears as a dynamic component of the attractor discussed in the previous subsection. Even in the region characterized by the Poincaré sections shown in Fig. 10ii–iv, this chaotic state exists as a transient state. Once a trajectory visits the transient state, it finally goes to either of the fixed points, tori, or eight-shaped attractor depending on the coupling strength. Here, nevertheless, a collision between an stable manifold of SMW⁺⁻ and the eight-shaped attractor occurs, and consequently a gateway to the weakly chaotic state in the vicinity of SMW⁺⁻ is yielded. Thus, a trajectory can visit the weakly chaotic state over and over. This crisis is interior crisis (Grebogi et al. 1982), since it causes growth of the attractor size.

On the Poincaré section in Fig. 10vi, randomly scattered points appear in addition to an ordered structure similar to the one appearing in Fig. 10v. These randomly scattered points correspond to strong chaos. Spatio-temporal patterns corresponding to this Poincaré section consist of the patterns as shown in Fig. 5, turbulent patterns corresponding to the strong chaos, and spontaneous transitions among them (Tsuda et al. 2004). Such transitions can be interpreted as chaotic itinerancy (Tsuda et al. 2004; Ikeda et al. 1989; Kaneko 1990; Tsuda 1991, 1992; Kaneko and Tsuda 2003). In Fig. 10vii the ordered structure disappears and there remains strong chaos.

To specify the regions of g/N² characterized by such Poincaré maps as is shown in Fig. 10i–vii, we introduce useful quantities: The maximum and minimum Euclidean distances between the AS state and current trajectory are measured on the Poincaré section, and . Figure 11a, b show plots of and as a function of g/N² for N = 16, 20, 32, and 33. From these two figures, we clearly observe seven ranges of g/N². In these seven regions labeled i–vii, types of the Poincaré map shown in Fig. 10i–vii are obtained. Realization of the scaling mentioned in the "Bifurcation structures of periodic states and system size dependence" appears also in this figure: plots for different N are almost independent of N. The scaling is sufficient for N ≥ 32. Furthermore, it is seen that even for N = 16 the bifurcation structure is the same as the one for N ≥ 32, though the scaling is not sufficient.

Fig. 11.

The quantities (a) and (b) as a function of g/N². Transient trajectories are omitted. The plots do not depend on the initial values of (V_i, R_i). In the regions marked with labels i–vii, types of the Poincaré sections i–vii shown in Fig. 10 appear, respectively

Summary and discussion

We studied transitory behaviors in the population of diffusively interacting nonlinear oscillators which includes following synchrony in the case of two-oscillator system: the two oscillators synchronize in out-of-phase in the case of weak couplings, whereas, in the case of relatively strong couplings, they synchronize in in-phase or in anti-phase, depending on the initial condition. Large populations of such oscillators interacting via one-dimensional nearest neighbor couplings exhibit a variety of dynamical behaviors depending on the rescaled coupling strength. Among others, of special interest is the transitory behavior which consists of some sorts of metachronal waves, the all-synchronized state, and weakly chaotic state. We showed that it can be interpreted as in-out intermittency where the trajectory leaves and approaches the invariant subspace which stems from the mirror symmetry. Two saddles exist on the invariant subspace, which represent the all-synchronized state and the symmetric metachronal wave. Because of their stabilities in the normal direction to the invariant subspace, the former and the latter give rise to the motion of leaving and approaching the invariant subspace, respectively.

Since connections between the two saddles are formed in the transitory behavior discussed here, it seems to be similar to the heteroclinic cycle. Nevertheless the staying periods in the neighborhood of the saddles in the behavior do not become infinite. Thus the behavior is a phenomenon different from the heteroclinic cycle.

Also in the case of some types of oscillator systems such as the Morris-Lecar model, which exhibit anti-phase synchrony in the weak coupling limit, their coupled systems can exhibit transitory behaviors, i.e. alternations between synchronization and desynchronization of oscillations (Han et al. 1995, 1997; Karantonis and Nakabayashi 2001; Han and Postnov 2003). Let us compare the transitory behaviors in the present study with the ones in the case of anti-phase synchronous oscillators. While transitory behaviors in Han et al. (1995, 1997), Karantonis and Nakabayashi (2001) can appear even in the case of N = 2 (Han and Postnov 2003), the transitory behaviors in the present model cannot appear. Because, one of the two saddles in the mirror symmetric invariant subspace, i.e. the SMW⁺⁻ state, that is prerequisite for the transitory behavior in the present model, cannot exist in the case of N = 2. Thus transitory behaviors in the present study are essentially different from the ones in the case of anti-synchronous oscillators.

Are there any other oscillator systems exhibiting behaviors similar to the ones in the present study? The two-dimensional Hindmarsh-Rose neuron model (Hindmarsh and Rose 1984) exhibits out-of-phase synchrony in the weak coupling case like the μ-model (see Fig. 12a). Their diffusively coupled system can exhibit similar transitory behaviors. A Poincaré map of one of such behaviors is shown in Fig. 12b, in which an attractor similar to the one in Fig. 10iv appears. This attractor also consists of the all-synchronized state and metachronal waves. It should be noted that all out-of-phase oscillator systems do not always exhibit transitory behaviors mentioned above when diffusively coupled. The conditions for the appearance of the transitory behaviors described here are in the future study.

Fig. 12.

a Antisymmetric part of effective coupling for the two-dimensional Hindmarsh-Rose neuron (Hindmarsh and Rose 1984). Solid, dashed, and dotted curves are for I = 0.20, I = 0.5, and I = 0.9, respectively. b A Poincaré map for a diffusively coupled system of the Hindmarsh-Rose neurons. N = 16, I = 0.20, g = 9.8. The same hyperplane as in the case of the μ-model is used as a Poincaré section

In the present paper, the case of one-dimensional nearest neighbor coupling is discussed. We further investigated N × N oscillators interacting via the two-dimensional nearest neighbor coupling. We also found that such coupled systems exhibit various transitory behaviors (not shown here) in the regions of g(g/N²) and I where the all-synchronized state becomes a saddle. We will report these issues, elsewhere.