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Time-delay effects on synchronization features of delay-coupled slow-fast van der Pol systems are investigated in the present paper. The synchronization mechanism of “slow-manifold adjustment” is firstly described on the basis of geometric singular perturbation theory. Then, the impact of time delay on the structure of the slow manifold of synchronized system is obtained by using the method of stability switch, and thus, time-delay effects on synchronization features are stated. It is shown the time delay cannot qualitatively affect the synchronization mechanism, however, it can result in the drift of the optimal coupling strength.

Real system is often with two kinds of different dynamical variables, evolving on very different timescales [

Coupled systems will adjust their behaviors with each other, such that there are some relations between their dynamical behaviors, this phenomenon is called synchronization. Since synchronization was first reported in the study of two coupled pendulums [

Coupled slow-fast systems have different synchronization features to that of the coupled systems with uniform timescale. Coupled relaxation oscillators with heaviside coupling can get synchronization through the mechanism of “fast threshold modulation” [

Time delay should be considered in coupled systems due to finite information transmission and processing speed [

Synchronization of delay-coupled slow-fast systems has also been studied in literature [

Two identical delay-coupled slow-fast van der Pol systems are described as

{ x ˙ i = − y i + x i − 1 3 x i 3 + I 1 ( x j ( t − τ ) − x i ) y ˙ i = ε x i . (1)

where i = 1 , 2 and j = 2 , 1 , a dot represents derivative with respect to time t, 0 < ε ≪ 1 such that x i is the fast variable and y i is the slow variable, I 1 ( x j − x i ) = c ( x j − x i ) is the linearly coupled term with coupling strength c ≥ 0 , and τ = O ( 1 ) represents the connection delay.

To study time-delay effect on synchronization features of coupled systems (1), the synchronization mechanism of coupled slow-fast van der Pol systems without connection delay should be clarified firstly. Without connection delay, Equation (1) read

{ x ˙ i = − y i + x i − 1 3 x i 3 + I 1 ( x j − x i ) y ˙ i = ε x i . (2)

In the synchronization manifold of coupled systems (2), the dynamics of synchronized state ( x 1 , y 1 ) = ( x 2 , y 2 ) = ( x , y ) is described by

{ x ˙ = − y + x − 1 3 x 3 y ˙ = ε x . (3)

Equation (3) is actually the single slow-fast van der Pol system with one slow and one fast variables. Let ε → 0 in Equation (3), one has the fast subsystem

x ˙ = − y + x − 1 3 x 3 ,

where y is taken as a system parameter, thus, the fast subsystem governs the fast variable x only. Geometric singular perturbation theory defines the slow manifold of Equation (3) as the set of equilibrium points of the fast subsystem

M 1 = M 11 ∪ M 12 ∪ M 13 = { ( x , y ) | − y + x − 1 3 x 3 = 0 } .

The structure of slow manifold, including the stability and bifurcation points, can be determined through the stability analysis of the fast subsystem, as shown in

To describe the synchronization mechanism of Equations (2) analytically, simplify the coupling term as follows

I 1 ( w ) = c w ⇒ I 2 ( w ) = { c w w > δ 0 | w | ≤ δ c w w < − δ ⇒ I 3 ( w ) = { c w > δ 0 | w | ≤ δ − c w < − δ

where w = x j − x i ( i ≠ j ) and δ is a small positive constant, and this process of simplification is illustrated in

With the simplified coupling I 3 ( w ) , Equations (2) read

{ x ˙ 1 = − y 1 + x 1 − 1 3 x 1 3 + I 3 ( x 2 − x 1 ) y ˙ 1 = ε x 1 . (4)

and

{ x ˙ 2 = − y 2 + x 2 − 1 3 x 2 3 + I 3 ( x 1 − x 2 ) y ˙ 2 = ε x 2 . (5)

Though the coupling term of Equations (4) - (5) is the simplified version of that of Equations (2), the synchronization mechanism of Equations (4) - (5) is qualitatively the same as that of Equations (2), which will be illustrated in following discussion, and will also be confirmed by numerical results.

To describe the synchronization features of Equations (4) - (5), the phase and phase difference for relaxation oscillation is firstly introduced. As shown in

Denote the initial points of Equations (4) and (5) with A 10 ( x 10 , y 10 ) and B 20 ( x 20 , y 20 ) respectively, and the analysis of the synchronization features is divided into cases depending on the initial values of the two coupled system.

Case 1: When x 20 − x 10 > δ , as shown in

Case 2: When x 20 − x 10 < δ and y 20 − y 10 < c , as shown in

Case 3: When x 20 − x 10 < δ and y 20 − y 10 > c , as shown in

when point B 22 jumps to B 23 due to x 23 − x 12 < δ , and at almost the same time, point A 12 will jump to A 13 , and one has T B 20 A 10 < T A 13 B 23 < T . As a result, the phase difference is contracted when systems (4) and (5) adjust their slow manifolds. Furthermore, as shown in

The analysis of cases 1 - 3 indicates the coupled slow-fast van der Pol systems can get synchronization by adjusting their slow-manifolds with each other, and this synchronization mechanism of “slow-manifold adjustment” is more complex than the mechanism of “fast threshold modulation” proposed in [

Results 1: When c > 0 , the coupled slow-fast van der Pol systems (2) can get synchronization through the mechanism of “slow-manifold adjustment”. And there is an optimal coupling strength c ∗ , such that the coupled systems (2) can get synchronization quickly.

To illustrate the validity of analytical results 1, the largest condition Lyapunov exponent of the following master stability function is calculated numerically in

{ ζ ˙ 1 = − ζ 2 + ( 1 − x 2 ) ζ 1 − 2 c ζ 1 ζ ˙ 2 = ε ζ 1 . (6)

where x is one of the variable of synchronized system (3). The coupled slow-fast van der Pol systems (2) can get synchronization when the largest condition Lyapunov exponent L m is negative. And there is a optimal coupling strength c ∗ , such that the coupled systems can get synchronization quickly, as shown in

Once the synchronization mechanism of coupled slow-fast van der Pol systems (2) without connection delay is clarified, time-delay effect on synchronization features can be discussed, the key step is to clarify the time-delay influence on the structure of the slow manifold due to the coupled systems (2) get synchronization by adjusting their slow-manifolds with each other, as shown in previous section.

In the synchronization manifold of coupled systems (1), the synchronized state ( x 1 , y 1 ) = ( x 2 , y 2 ) = ( x , y ) is governed by

{ x ˙ = − y + x − 1 3 x 3 + c ( x ( t − τ ) − x ) y ˙ = ε x . (7)

Let ε → 0 in Equation (7), one has the fast subsystem

x ˙ = − y + x − 1 3 x 3 + c ( x ( t − τ ) − x ) . (8)

where y is taken as a system parameter.

Geometric singular perturbation theory defines the slow manifold of Equation (7) as the set of the equilibrium points of the fast subsystem, denoted as

M 2 = M 21 ∪ M 22 ∪ M 23 = { ( x , y ) | − y + x − 1 3 x 3 = 0 } .

Note that the set M 2 is the same as M 1 , thus, the time delay has no influence on the shape of the slow manifold, as shown in

To decide time-delay influence on the stability and bifurcation of the slow manifold M 2 , consider the local stability of the fast subsystem (8) around the

equilibrium point ( x 0 , y 0 ) ∈ M 2 , which is governed by the characteristic equation

D ( λ ) = λ − ( 1 − x 0 2 ) + c − c e − λ τ .

When | x 0 | ≤ 1 and τ = 0 , D ( λ ) = 0 has one nonnegative real eigenvalue λ = 1 − x 0 2 ≥ 0 . According to the theory of stability switch [

When | x 0 | > 1 and τ = 0 , D ( λ ) = 0 has one negative real eigenvalue λ = 1 − x 0 2 < 0 , and D ( 0 ) ≠ 0 for any τ . Thus, as τ increases, stability may switch only when a couple of eigenvalues arrive in the right half plane through crossing the imaginary axis. Let λ = ± i ω ( ω > 0 ), and separating the real and imaginary parts of D ( λ ) = 0 leads to

{ ω + c sin ( ω τ ) = 0 , ( x 0 2 − 1 ) + c ( 1 − cos ( ω τ ) ) = 0. (9)

The second equation of Equation (9) cannot hold for any τ due to x 0 2 − 1 > 0 . Thus, there is no stability switch as τ increases from zero to infinity. So, time-delay τ cannot change the stability of the parts of the slow-manifold M 21 and M 23 , and the parts of the slow manifold M 21 and M 23 are stable for any time delay τ , as shown in

The analysis above indicates time delay cannot qualitatively influence the structure of the slow manifold M 2 , and thus cannot qualitatively influence the dynamics of Equation (7), as shown in

Since coupled slow-fast van der Pol systems get synchronization through the mechanism of “slow-manifold adjustment”, one can conjecture that the connection delay τ cannot qualitatively affect the synchronization features of delay-coupled slow-fast van der Pol systems (1) because the time delay cannot qualitatively influence the structure of the slow manifold. To verify this conjecture numerically, the largest condition Lyapunov exponent, which governs the stability

of synchronization manifold of coupled systems (1), is calculated from the following master stability function

{ ζ ˙ 1 = − ζ 2 + ( 1 − x 2 ) ζ 1 − c ( ζ 1 ( t − τ ) + ζ 1 ) , ζ ˙ 2 = ε ζ 1 , (10)

where x is the variable of Equation (7).

The largest condition Lyapunov exponents as a function of c and τ are illustrated in

Summarize the analytical and numerical results, one has:

Results 2: The time delay τ cannot qualitatively affect the synchronization mechanism of “slow-manifold adjustment”, however, it can result to the drift of the optimal coupling strength c ∗ .

Coupled slow-fast van der Pol systems can describe the dynamics of various real systems in physics, electronic, and biology. And connection delay should be considered due to limit of information transmission and processing speed. It is well known time-delay that is often a negative factor to the stability of synchronization, and deteriorates the synchronizability of coupled systems with uniform timescale. The studies in this paper indicate even large time-delay cannot destroy the synchronization mechanism of “slow-manifold adjustment” of coupled slow-fast systems. The finding is helpful to understand the universal synchronization in real systems with long connection delay.

This paper was supported by NSF of China under Grants 11872197, 11202187, JXNSF and JXECF of China under Grant 20192BAB202002 and DB201407058 respectively.

The authors declare no conflicts of interest regarding the publication of this paper.

Zheng, Y.G. and Zeng, J.J. (2021) Time-Delay Effects on Synchronization of Coupled Slow-Fast Systems. Journal of Applied Mathematics and Physics, 9, 635-647. https://doi.org/10.4236/jamp.2021.94046