6 Synchronization and Oscillation for a System Comprising Two
6.4 Global convergence to multiple equilibria
In this section, we shall establish the convergence to multiple synchronous equilibria for (2.8). In addition, the existence and stability of nontrivial equilibrium and basins of attraction for stable synchronous equilibria will be derived.
Let us consider the convergence of dynamics for (2.8). First, we define F (ξ) = −ξ + cg(ξ) + 2τˆ Tc(1 + 2c)L + 1,
F (ξ) = −ξ + cg(ξ) − 2τˇ Tc(1 + 2c)L − 1.
For 0 < λ < 1, we impose the following conditions:
Condition (C1a)λ: L > 1/c > 0, τT < 1/[2L(1 + 2c)], λ(1 − 2cτTL) > LK. Condition (C2a)λ: ˇF (˜qλ) > 0, ˆF (˜pλ) < 0.
Condition (C3a)λ: g′(ξ) > (1 + λ)/c, for all ξ ∈ [ ˆmF, ˇmF].
Condition (C1a)λ is a multi-dimensional version of condition (A1a). Notably, ˜pλand
˜
qλ were defined in (3.21) where −β is replaced by c. Under conditions (C1a)λ and (C2a)λ, there exist exactly three zeros ˆlF, ˆmFand ˆrF (respectively ˇlF, ˇmFand ˇrF ) of ˆF (ξ) = 0 (resp. ˇF (ξ) = 0). Moreover, ˇlF < ˆlF < ˜pλ < ˆmF < ˇmF < ˜qλ < ˇrF < ˆrF, cf. Fig. 6.1(a).
Let us introduce three sets in R2K as follows:
Ωl:= ˜Ωl× ˜Ωl, Ωm := ˜Ωm× ˜Ωm, Ωr:= ˜Ωr× ˜Ωr,
where ˜Ωl = [−1, 1]K−1× [ˇlF, ˆlF], ˜Ωm = [−1, 1]K−1 × [ ˆmF, ˇmF], ˜Ωr = [−1, 1]K−1 × [ˇrF, ˆrF]. We can then derive the convergence to multiple synchronous equilibria for
(2.8). We say that an equilibrium (x1, · · · , xK, y1, · · · , yK) of (2.8) is synchronous if xi = yi, for all i = 1, · · · , K.
Theorem 6.4.1. Assume that conditions (S1a), (C1a)λ-(C3a)λ hold for some fixed λ ∈ (0, 1), then (2.8) achieves global convergence to synchronous equilibria; if in addition, 0 < L < 1 and λ ∈ (L, 1), then (2.8) admits exactly three synchronous equilibria. Each of regions Ωl, Ωm, Ωr contains one of these equilibria .
Proof. Recall that (2.8) can be rewritten into (6.6). Obviously, for i = 1, · · · , K − 1, the ith component of (6.6) is of the form as (3.9); hence, xi(t) converges to [wimin(∞), wmaxi (∞)], as t → ∞, as observed from the equation for xi. Restated, for i = 1, · · · , K −1, xi(t) converges to some compact interval Ii whose length di satisfies di ≤ wimax(∞)−wimin(∞). The Kth component of (6.6) has the form as (3.20) under condition (S1a) and satisfies conditions (A1a), (A2a)λ and (A3a)λ, under condition (C1a)λ, (C2a)λ and (C3a)λ. By Theorem 3.2.5, xK(t) converges to some compact interval whose length dK satisfies dK ≤ [wKmax(∞) − wKmin(∞)]/[λ(1 − 2τTcL)]. By similar arguments as the ones for Theorem 4.2.2 or 6.3.1, we can show that each xi(t) actually converges to some singleton, for i = 1, · · · , K. Similar arguments can apply to yi(t), for i = 1, · · · , K.
Finding synchronous equilibrium for (2.8) amounts to solving
−xi+ g(xi−1) = 0, i = 1, · · · , K − 1, (mod K)
−xK+ g(xK−1) + cg(xK) = 0. (6.9)
Note that under condition (S1a), all equilibria for (2.8) must be synchronous. Con-sider a fixed Ω ∈ { ˜Ωl, ˜Ωm, ˜Ωr}. For a given (η1, · · · , ηK) ∈ Ω, we define
Λi(ξ) = −ξ + g(ηi−1), i = 1, · · · , K − 1, (mod K), ΛK(ξ) = −ξ + cg(ξ) + g(ηK−1).
Note that |g(·)| < 1 and ˇF (ξ) ≤ ΛK(ξ) ≤ ˆF (ξ). Thus, there exists a unique point (η1∗, · · · , ηK∗) ∈ Ω such that η∗i is the solution of equation Λi(·) = 0 for all i = 1, · · · , K, under conditions (C1a)λ, (C2a)λ, cf. Fig. 6.1 (a). Consequently, we can define a mapping GΩ : Ω → Ω by GΩ(η1, · · · , ηK) = (η∗1, · · · , ηK∗) ∈ Ω. Thanks to conditions (C1a)λ-(C3a)λ and L < 1, by using arguments similar to Theorem 4.1.1, we can show that GΩ is a contraction mapping and there exists a unique fixed point x = (x1, · · · , xK) of GΩ, lying in Ω. Restated, x satisfies (6.9). Thus, (x1, · · · , xK, x1, · · · , xK) is the unique equilibrium point of (2.8) lying in Ω × Ω.
Remark 6.4.1. (i) Let us observe what parameters satisfy conditions (S1a) and (C1a)λ-(C3a)λ. It can be seen that L < 1, sufficiently large c, and sufficiently small τT are apt to meet these conditions. (ii) The existence of equilibrium for (2.8) should have nothing to do with delay. In respecting the conditions (S1a) and (C1a)λ-(C3a)λ involving delay, one can just take delay τT = 0 in these inequalities, if existence of equilibrium is the only issue of concern. (iii) Notably, the third inequality in condition (C1a)λ becomes more difficult to hold if L > 1 and K (the sub-network size) is large. (iv) Obviously, the equilibrium point lying in Ωmis the trivial and the others are nontrivial.
To discuss the stability of nontrivial equilibria in Theorem 6.4.1 and basins of attraction for the stable equilibria, some additional conditions are needed. First, we define
I(ξ) := −ξ + 2cg(ξ) − 1 − c, if ξ ≥ 0,
−ξ + 2cg(ξ) + 1 + c, if ξ < 0.
There exist two values pc < 0 and qc > 0 such that g′(pc) = g′(qc) = 1/(2c) if L > 1/(2c) > 0. The first additional condition is
Condition (C4a): L > 1/(2c) > 0, I(qc) > 0, I(pc) < 0.
Under condition(C4a), there exit exactly two zeros of function I, say ˇκ and ˆκ, in intervals (pc, 0) and (0, qc) respectively, cf. Fig. 6.1(b). Next, we impose
Condition (C5a)λ: g′(ˇκ) ≥ (1 − λ)/c, g′(ˆκ) ≥ (1 − λ)/c.
Let us define the following two sets in C([−τmax, 0]; R6):
Ω+:= {φ : φ ∈ C([−τmax, 0]; R6), φi(θ) ≥ ˆκ, θ ∈ [−τT, 0], for i = K, 2K}, Ω−:= {φ : φ ∈ C([−τmax, 0]; R6), φi(θ) ≤ ˇκ, θ ∈ [−τT, 0], for i = K, 2K}.
Theorem 6.4.2. Under the conditions for the existence of nontrivial equilibria in Theorem 6.4.1, if in addition, (C4a) and (C5a)λ hold, then the nontrivial equilibria are asymptotically stable. Moreover, Ω+ (resp. Ω−) is contained in the basin of attraction for the equilibrium in Ωr (resp. Ωl).
Proof. It can be justified that Ω+ and Ω− are invariant under (2.8). We merely discuss the former case. We define the function ˜I(ξ) : (−∞, ˇκ] ∪ [ˆκ, ∞) → R as
follows:
I(ξ) :=˜ −ξ + cg(ξ) + cg(ˆκ) − 1 − c, if ξ ≥ ˆκ,
−ξ + cg(ξ) + cg(ˇκ) + 1 + c, if ξ ≤ ˇκ.
We shall show that, for all t ≥ t0,
˙xK(t) > ˜I(xK(t)), ˙yK(t) > ˜I(yK(t)). (6.10) If (6.10) holds, then both xK(t), yK(t) remain in [ˆκ, ∞) for all t ≥ t0, due to I(ˆ˜κ) = I(ˆκ) = 0 and ˜I(ˇκ) = I(ˇκ) = 0. The invariance of Ω+ will then be justified.
Let us now confirm (6.10). From (2.8), it can be seen that xK(t) satisfies, for t ≥ t0,
˙xK(t) = −xK(t) + cg(xK(t)) + g(xK−1(t − τI)) + c[g(yK(t − τT)) − g(xK(t))].
The assertion holds for t = t0. Indeed, ˙xK(t0) > −xK(t0) + cg(xK(t0)) − 1 − c[1 − g(ˆκ)] = ˜I(xK(t0)), since yK(t0− τT) ≥ ˆκ and xK(t0) ≥ ˆκ . By similar arguments, we can also verify that ˙yK(t0) > ˜I(yK(t0)). Assume (6.10) holds for t ∈ [t0, ˜t) but does not at some ˜t > t0. One of the possibilities is that ˙xK(˜t) = ˜I(xK(˜t)), and
˙xK(t) > ˜I(xK(t)), ˙yK(t) > ˜I(yK(t)) for all t ∈ [t0, ˜t). In this situation, xK(t) and yK(t) remain in (ˆκ, ∞), for t ∈ [t0, ˜t). Then, ˙xK(˜t) > −xK(˜t) + cg(xK(˜t)) − 1 − c[1 − g(ˆκ)] = ˜I(xK(˜t)) and yields a contradiction. The other possibilities can also be ruled out, by similar arguments. Therefore (6.10) is valid. The remaining arguments are similar to Theorem 4.3.1. We merely sketch the idea. Note that the two nontrivial equilibria lie in Ω+, Ω− respectively. Thus, the solution starting near the nontrivial equilibrium remains in the invariant set Ω+ or Ω−. Hence, we have the slope control on the coupling terms, namely cg′(xi(t)) < 1 for i = K, 2K, and g′(xi(t)) < 1, for i 6= K, 2K, for all t ≥ t0, and so that the solution of (2.8) evolved near the nontrivial equilibrium is dominated by the degradation term −xi(t) in (2.8). Solutions starting near the equilibrium thus converge to the equilibrium.
According to Theorem 6.4.1 and the invariance, we conclude that Ω+ (resp.
Ω−) is contained in the basin of attraction for the equilibrium lying in Ωr(resp. Ωl).
Remark 6.4.2. (i) Although conditions (C4a), (C5a)λ seem complicated, it can be observed that they are apt to be satisfied for large c. (ii) In fact, it is not difficult to observe that the nontrivial equilibria are stable under the conditions on the existence of exactly three equilibria (Remark 6.4.1), and conditions (C4a), (C5a)λ. These conditions are independent of delays and are easy to satisfy if 0 < L < 1 and c is sufficiently large.
6.5 Bifurcation and oscillations
In Theorem 6.1.2, we have shown that if the transmission delay τT is small enough, (2.8) can attain global synchronization in spite of the magnitude of internal delay τI. In this section, via bifurcation analysis, we shall show that there exist nontrivial synchronous periodic solutions for (2.8) induced by internal delay τI. To simplify the presentation, we consider (2.8) with K = 3 in this section. Moreover, most of arguments in this section are similar to the ones in Section 5.2.3. Hence, we merely sketch the main process and adopt the same symbols for some settings used in Section 5.2.3.
First, let us consider a circle block matrix circ(A0, A1, · · · , An−1) where Aj, j = 0, 1, · · · , n − 1, is a k × k matrix. Let vj = e2πjn i, and define a function of matrices G(x) = A0+ xA1+ · · · + xn−1An−1. In [77], it is shown that
det(λIkn− circ(A0, A1, · · · , An−1)) = Πnj=1det(λIk− G(vj)), (6.11) where Ik is the k × k identity matrix. The linearization of (2.8) at the trivial equilibrium (0, 0, 0, 0, 0, 0) is given by
Then the characteristic equation for (6.12) (cf. [34], [72]) is
∆(λ) := det
Thanks to (6.11), the characteristic equation can be factored as
∆+(λ)∆−(λ) = 0,
∆±(λ) := (1 + λ)2(1 + λ ∓ de−λτT) − L3e−3λτI.
We substitute λ = iw with w > 0 into ∆±(λ) = 0 and collect the real and imaginary parts to yield
L3sin(3τIw) = (w3− 3w) ± (w2− 1)d sin(τTw) ± 2d cos(τTw)w,
L3cos(3τIw) = (1 − 3w2) ∓ (1 − w2)d cos(τTw) ∓ 2d sin(τTw)w. (6.13) Summing up the square of equations (6.13) gives
Q±(w) = L6, (6.14)
where Q±(w) := Q1(w) ± Q2(w) and Q1(w) := w6 + (3 + d2)w4+ (3 + 2d2)w2 + d2+ 1 and Q2(w) := d[2 sin(τTw)w5− 2 cos(τTw)w4+ 4 sin(τTw)w3− 4 cos(τTw)w2+ 2 sin(τTw)w − 2 cos(τTw)]. Therefore the positive solution of (6.14) corresponds to the purely imaginary roots of ∆±(w) = 0.
Now let us introduce some settings and the condition imposed for purely imag-inary roots of ∆±(w) = 0 as follows:
Define P (w) := (6 − 2|d|τT)w5− |d|(10 + 2τT)w4+ (12 + 4d2− |d||8 − 4τT|)w3−
|d|(12 + 4τT)w2+ (6 + 4d2− |d||8 − 2τT|)w − |d|(2 + 2τT). As P is a polynomial, we set
˜
w := the largest zero of P (w). (6.15) Condition (B1a)±: 6 − 2|d|τT > 0, min{Q∓(w) : w ∈ [0, ˜w]} > L6, and max{Q±(w) : w ∈ [0, ˜w]} < L6,
By similar arguments to Lemma 5.2.6, we can derive the following lemma.
Lemma 6.5.1. Under condition (B1a)+ (resp. (B1a)−), there exist exactly one pair of purely imaginary roots, say ±iω+∗ (resp. ±iω−∗), for characteristic equation
∆(λ) = 0. In particular, ±iω+∗ (resp. ±iω−∗) are the roots of ∆+(λ) = 0 (resp.
∆−(λ) = 0).
Remark 6.5.1. (i) Note that Q+(0) = (1 − d)2 and Q−(0) = (1 + d)2. Therefore (1 − d)2 < L6 < (1 + d)2 (resp. (1 + d)2 < L6 < (1 − d)2) (6.16) is a necessary condition for (B1a)+ (resp. (B1a)−) to hold. Moreover, d = cL 6= 0 is necessary, as seen from the inequality (6.16); in particular, c = d/L > 0 (resp.
c = d/L < 0) is necessary for condition (B1a)+ (resp. (B1a)−). (ii) The following weaker condition
L6 > (1 − d)2 ( resp. L6 > (1 + d)2) (6.17)
can also provide the existence of zero to Q+(w) = 0 (resp. Q−(w) = 0), but the uniqueness of positive zero can not be guaranteed. However, the situation of multiple zeros can be ruled out with assistance of numerical computation. Basically, from (6.17), it can be observed that larger L is advantageous to the occurrence of Hopf bifurcation, hence oscillation. Herein, η+k (resp. ηk−) is positive and the critical value of bifurcation parameter with respect to τI, at which ∆(λ) = 0 has exactly one pair of purely imaginary roots
±iω+∗ (resp. ±iω−∗). To apply the Hopf bifurcation theory, it remains to verify the transversality condition:
Proposition 6.5.2. Assume that conditions (B1a)+ and (B2a)+ (respectively, (B1a)− and (B2a)−) hold for some fixed k ∈ N. The Hopf bifurcation occurs at τI = ηk+(resp. ηk−), and a periodic orbit is bifurcated from the zero solution of (2.8).
The proof of Proposition 6.5.2 is similar to the one of Theorem 5.2.8; hence omitted.
Let us recall Theorems 6.1.2 and 6.1.3 in which (2.8) attains synchronization under condition (S1a) or (S2a) or (S3a). As mentioned in Remark 6.5.1, (1−d)2 < L6 and c > 0 are necessary for condition (B1a)+. It can be observed that (1 − d)2 <
L6 and c > 0 is compatible only with condition (S1a), but not (S2a) and (S3a).
From this view point, positive coupling strength c (with other conditions) leads to synchronous oscillation. On the other hand, similarly, we observe that negative coupling strength c leads to anti-phase oscillation. Motivated by these observations, Theorems 6.1.2, 6.1.3, 6.2.1 and Proposition 6.5.2, we draw the following conclusion.
Theorem 6.5.3. Assume that conditions (S1a) (resp. (AP1)), (B1a)+, and (B2a)+
(resp. (B1a)−, and (B2a)−) hold for some fixed k ∈ N. Then there exists a syn-chronous (respectively, anti-phase) periodic solution bifurcated from the trivial equi-librium, at internal delay τI = ηk+ (resp. τI = ηk−) for the coupled K-loops (2.8) (resp. (2.8)0).
6.6 Description of dynamical scenarios
As mentioned earlier, coupled network system can exhibit a variety of interesting behaviors. We plan to depict the dynamical scenario for the coupled K-loops (2.8) under the influence of coupling strength, the gain of the activation function, the internal delay, and the transmission delay. Let us first mention some properties of the single K-loop (2.7). Notably, the dynamics of (2.7) without internal delay τI
has been studied extensively in [7]. By similar approach as Lemma 2 in [31] and Lemma 4.1 in [72]), it can be shown that the trivial equilibrium of (2.7) is stable for all τI ≥ 0 if L ≤ 1. On the other hand, there exist periodic solutions bifurcated from the zero solution, at suitable τI for (2.7) if L > 1. Therefore, the dynamical scenarios for cases L ∈ (0, 1) and L ∈ (1, ∞) are rather different. Below we shall discuss the two cases: L ∈ (0, 1) and L ∈ (1, ∞), for the coupled loops separately.
Notice that the trivial equilibrium for the coupled loops (2.8) can become unstable as there exist periodic solutions bifurcated from the equilibrium, cf. Theorem 6.5.3.
This already shows an effect from the coupling between these two loops.
Notably, what we have derived for global synchronization, global convergence to the origin are theories with sufficient conditions. When we say that (2.8) does
not admit certain dynamics (such as synchronization), we may need computer sim-ulations to support the arguments. In addition, caution must be used if saying that a system can not be synchronized merely through numerical simulation. It is not assured how long a simulation should be run to exclude the possibility of synchro-nization. On the other hand, anti-phase is an evidence of desynchrony that one can assure from analysis or numerical simulation. Our approach can establish anti-phase oscillations bifurcated from the zero solution at certain values of internal delay, and can also be extended to analyze bifurcation with respect to transmission delay.
Effect of coupling strength. Let us first consider the case that L ∈ (0, 1), c ≥ 0. It can be seen from Theorem 6.3.2 that if c, the coupling strength between two loops, is sufficiently small so that
c ≤ 1/L − 1, (6.19)
then the coupled loops (2.8) attain global convergence to the trivial equilibrium (hence global synchronization) in spite of delay magnitude of τI and τT. In addition, if c is larger so that (6.19) fails to hold, the coupled loops (2.8) can attain global synchronization if τT is small enough. Such an observation follows from that for arbitrarily large c > 0, condition (S1a) for Theorem 6.1.2 holds if τT is small enough.
On the other hand, Theorem 6.5.3 concludes the birth of nontrivial synchronous periodic solutions, under certain criteria. It can be seen that the dominant condition (B1a)+can not hold if c = 0 or is too small, cf. Remark 6.5.1. On the other hand, for some suitable magnitude of c (not too large) such that (B1a)+ holds, the nontrivial synchronous periodic solutions of coupled loops can be induced by internal delays.
Now, it is natural to ask what will occur if magnitude of c is quite large. Theorem 6.4.1 has shown that system (2.8) with sufficiently large c achieves global convergence to nontrivial synchronous equilibria (hence globally synchronized) in spite of internal lag (τI) if transmission lag (τT) is small enough. This result has justified the the numerical finding in [7].
Accordingly, we summarize a dynamical scenario for (2.8) as c increases from
zero to ∞:
global convergence to zero
→ global synchronization (τT small)
→ global synchronization with synchronous oscillation (induced by internal delays) (τT small)
→ global convergence to nontrivial synchronous stable equilibria (τT small).
Notice that the effect of positive coupling strength on synchronization and negative coupling strength on anti-phase motion are counterparts to each other for the coupled loops. Indeed, if L ∈ (0, 1), as c varies from 0 to −∞, (2.8)0goes through global convergence to zero → birth of anti-phase oscillation (induced by internal delays) → global convergence to nontrivial stable antisynchronous equilibria.
Roughly speaking, it is easier for the coupled loops (2.8) to attain global syn-chronization for the case L ∈ (0, 1) than L ∈ (1, ∞).
In the case that L ∈ (1, ∞), the single loop (2.7) has at least three equilibria, and thus the decoupled system (2.8) with c = 0 can not be synchronized. In fact, we can observe that wether if the coupled loops (2.8) can attain synchronization strongly relies on the interaction between two coupled loops. Observe condition (S1a) in Theorem 6.1.2, it can be seen that coupling strength c is necessary (can not be zero), and τT must be small enough. It has been illustrated in Example 6.7.2 that (2.8) fails to be synchronized if coupling strength is too small, and can be synchronized if coupling strength is suitably large. However, we found that large magnitude of c does not always favor condition (S1a); it also depends on the nonlinearity of activation function g, and the magnitude of transmission delay.
Similar observation was reported in [50] and [48]. In our numerical computation in Example 6.7.2, for some situation as transmission delay is too large, large coupling strength c still can not synchronize the oscillators.
Effect of delays. As previous arguments, in some parameter regions of L and c, it is necessary that transmission delay is small enough for (2.8) to be synchronized.
Example 6.7.2 has illustrated that the coupled loops (2.8) can be synchronized as transmission lag τT is small enough, and (2.8) can not be synchronized if τT is too large. Notice that from our numerical evidence, if (2.8) becomes asynchronous in-duced by large transmission delay, stronger coupling strength c does not promote
the system to regain synchronization. In fact, the coupled loops can become asyn-chronous in the form of anti-phase oscillation. This can be confirmed by performing the bifurcation analysis as in Section 6.5, but with τT as bifurcation parameter. Then at a large coupling strength, there exists anti-phase periodic solution for certain τT. Under our formulation, whether if the coupled loops (2.8) can be synchronized does not depend on internal delays, cf. Theorems 6.1.2, 6.1.3. But transmission delay plays a role in synchronization. It is then natural to ask how the internal delay affect the dynamics in (2.8). Following our result in Theorem 6.5.3, it can be seen that oscillation is generated by internal delay of certain magnitudes.
6.7 Numerical examples
We provide two numerical examples to illustrate the present theory.
Example 6.7.1. Consider the coupled 3-loops (2.8) with g(ξ) = tanh(0.999ξ), τI = 11.2, τT = 0.001, and c = 400/999. Then (2.8) satisfies condition (S1a). By Theorem 6.1.2, (2.8) attains global synchronization. It can be verified that condition (B1a)+ and (B2a)+, k = 1, 2, holds by direct computation. By Theorem 6.5.3, there exists a synchronous periodic solution bifurcated from the zero solution of (2.8).
Fig. 6.2 illustrates that the solution of (2.8) tends to a synchronous periodic orbit as t → ∞. Fig. 6.3 provides the the dynamics for each component of the solution in Fig. 6.2; in the panels, six different colors represent the evolutions of six components.
Example 6.7.2. Consider the coupled 3-loops (2.8) with L = 1.02. As c = 0.5 τI = 2 and τT = 0.01, it can be checked that (2.8) satisfies condition (S1a).
In such a situation, Fig. 6.4 illustrates that the solution of (2.8), evolved from (1.6 + 0.1 sin t, −1.6 + 0.1t, 1.6 + 0.1 sin t · cos t, −1.6 + 0.1 sin t, −1.6 + 0.1t, −1.6 + 0.1 sin t · cos t) is synchronized. If we consider (2.8) with smaller c = 0.001 instead, which does not satisfy condition (S1a), Fig. 6.5 illustrates that the solution of (2.8) converges to some asynchronous equilibrium point. If we consider (2.8) with the same parameters but with larger τT = 100 instead, which does not satisfy condition (S1a), Fig. 6.6 illustrates that the solution of (2.8) appears to be anti-phase, hence not synchronized. If we increase the coupling strength to c = 20 and still hold τT = 100, the system still exhibits anti-phase, hence not synchronized, as shown in Fig. 6.7.
−1.5 −1 −0.5 0 0.5 1 1.5
Figure 6.3: The dynamics for the corresponding component of solution in Fig. 6.2.
0 20 40 60 80 100
Figure 6.4: Evolutions of components (xi(t), yi(t)) for the solution of (2.8) with g(ξ) = tanh(1.02ξ), τI = 2, τT = 0.01, and c = 0.5, starting from φ(t) = (1.6 + 0.1 sin t, −1.6 + 0.1t, 1.6 + 0.1 sin t · cos t, −1.6 + 0.1 sin t, −1.6 + 0.1t, −1.6 + 0.1 sin t · cos t). It appears to be synchronized.
0 50 100 150 200 250 300
Figure 6.5: Evolutions of components (xi(t), yi(t)) for the solution of (2.8) with g(ξ) = tanh(1.02ξ), τI = 2, τT = 0.01, and c = 0.001, starting from φ(t) = (1.6 + 0.1 sin t, −1.6 + 0.1t, 1.6 + 0.1 sin t · cos t, −1.6 + 0.1 sin t, −1.6 + 0.1t, −1.6 + 0.1 sin t · cos t). The solution converges to asynchronous steady state.
0 500 1000 1500 2000
Figure 6.6: Evolutions of components (xi(t), yi(t)) for the solution of (2.8) with g(ξ) = tanh(1.02ξ), τI = 2, τT = 100, and c = 0.5, starting from φ(t) = (1.6 + 0.1 sin t, −1.6 + 0.1t, 1.6 + 0.1 sin t · cos t, −1.6 + 0.1 sin t, −1.6 + 0.1t, −1.6 + 0.1 sin t · cos t). The system tends to an anti-phase motion.
0 500 1000 1500 2000
Figure 6.7: Evolutions of components (xi(t), yi(t)) for the solution of (2.8) with g(ξ) = tanh(1.02ξ), τI = 2, τT = 100, and c = 20, starting from φ(t) = (1.6 + 0.1 sin t, −1.6 + 0.1t, 1.6 + 0.1 sin t · cos t, −1.6 + 0.1 sin t, −1.6 + 0.1t, −1.6 + 0.1 sin t · cos t). The system retains an anti-phase motion, even with larger coupling strength c.
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