Bifurcation analysis on the globally coupled Kuramoto oscillators with distributed time delays☆
Introduction
The Kuramoto model was established to investigate the phenomenon of collective synchronization [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], in which each oscillator was modeled as an equation only about the phase on the unit circle. Studying such a model may have direct utilization to reveal the complicated phenomena in many subjects, such as biology, neural science, lasers, engineering, superconducting Josephson junctions and so on [11], [12], [13], [14], [15]. Earlier mathematical study about collective synchronization was exhibited as Fourier integrals by Wiener [16], and as a mean-field approximation by Winfree [17]. Kuramoto had gone further with Winfree’s idea in [6], and confirmed that one population of weakly nearly identical coupled oscillators could be depicted as the universal phase model where the instinct frequencies come from a certain ensemble with probability density function , such as a normal distribution. The coupling scheme is chosen in the form of a sine function.
In most situations, signal transmission usually takes a non-negligible period of time. Meanwhile, the receiver also needs some reaction time to process the signal. Thus a time lag is natural and necessary in many coupled systems [18], [19], [20]. Moreover, a more realistic consideration is to deal with delay as following some probability distribution after considering the inhomogeneity of oscillators. In [21], the authors found that spread in the distribution function of delays can greatly alter the system dynamics, and the method therein is due to the Ott–Antonsen manifold reduction method [22]. The results are given with respect to Gamma-distribution time lags. In numerical simulation, both supercritical and subcritical Hopf bifurcations on the mean field (in coupled system, this corresponds to the situation that coherent states, i.e. partially synchronized states, are bifurcated from incoherent states) are observed. However, universal and theoretical analysis about the bifurcation properties in such a system has not been found yet.
In this paper, we will give some theoretical explanations about the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. For the sake of usage, we consider four kinds of distributions that delay comes from, including degenerated Dirac distribution, uniform distribution, Gamma distribution and the real world case—normal distribution. Motivated by such a consideration, one can find the mean field is governed by a delay differential equation with distributed delays. Stability and bifurcation analysis in such equations is quite difficult [23]. In this paper, we will use some theoretical bifurcation diagrams to show the rich bifurcation behaviors in this system. The methods we used are the characteristic equation analysis and the center manifold reduction method near the critical points.
Consider the model where is the phase of the th oscillator and is its natural frequency coming from an ensemble with distribution density . is the coupling strength which is assumed to be a constant. The interaction delays are random values, statistically independent from , obeying the distribution density . From the point of view of statistical physics, the motion of oscillators at phase (taking the limit ) follows where the drift term with the complex-valued mean field defined by and The newly defined distribution density characterizes the state of the oscillators’ system at time in frequency and phase . Following the Feynman–Kac formula in the absence of diffusion terms, we have the system (3) is equivalent to the Fokker–Planck equation The solution depicts the behavior of system (2). For example, for some , when , the marginal distribution of is uniform and the order-parameter . Hence system (2) exhibits an incoherent state [i.e., asynchronization]. At the other extreme, if , that is , then system (2) exhibits a completely synchronized state at for some . Otherwise, characterizes the partially synchronized states.
In this paper, the Ott–Antonsen manifold reduction method is used to obtain a system on the mean field [22]. Thus a delay differential equation is obtained, and by using the center manifold reduction method, we study the stability and Hopf bifurcation for delay coupled system (2) with Dirac, uniform, Gamma and normal distributions, respectively.
The paper is organized as follows: Section 2 is used to repeat the Ott–Antonsen method for reducing the system into a delay differential system on the mean field. Section 3 is a universal analysis of the reduced equation from the viewpoint of the bifurcation approach. In Sections 4 The degenerate case: Dirac-distribution, 5 A nontrivial case: uniformly distributed interaction delay, 6 Gamma distribution case, 7 Real world case: normal distribution we consider different distributions ranging from Dirac distribution, uniform distribution, and Gamma distribution to normal distribution. Finally, a conclusion section completes this paper where some discussions are also given.
Section snippets
Ott–Antonsen manifold reduction
The Ott–Antonsen reduction of a system with distributed delay was first derived by Lee et al. [21]. For the readers’ convenience, we state the main results here. Rewriting system (6) as and restricting this partial differential equation on the Ott–Antonsen manifold, that is with c.c. the complex conjugate of the formal terms, we have that the incoherent state () and completely synchronized state (
Stability and Hopf bifurcation analysis of the incoherent states
In this section, we assume that a Hopf bifurcation occurs in Eq. (13). In fact, this means that a partially synchronized state bifurcates from the incoherent state. We aim to obtain the properties of Hopf bifurcation such as the stability of periodic solutions and the direction of bifurcation. The method we use here is due to the center manifold reduction method established in [25], [26].
First, let us make the following assumptions to ensure the occurrence of Hopf bifurcation at .
- (A)
The
The degenerate case: Dirac-distribution
Consider with , then the time delay is constant. Thus , which means the characteristic equation is the regular case.
Regarding and as bifurcation parameters, we have , when or .
When , the roots’ distribution of (39) is complicated. Consider the Hopf bifurcation case, that is with is a root, then we have Obviously, Two kinds of transversality conditions are, respectively,
A nontrivial case: uniformly distributed interaction delay
In this section, we assume
The characteristic equation now reads
Substituting into (45) and separating the real and imaginary parts yield
Eliminating in the two equations we have
Denoting and , we can obtain the intersections between the two functions as drawn in Fig. 6. Denoting the intersections , then by Eq.
Gamma distribution case
In this section we consider the time delay comes from an ensemble with Gamma distribution with probability density function where is the shape parameter and is the scale parameter. For the sake of analysis, we choose an integer, thus the Gamma distribution is degenerated to the Erlang distribution. When , the mean value is , and the variance is . The Laplace transformation is . The probability density function of the Gamma distribution
Real world case: normal distribution
In the real world case, it makes more sense in our model (2) that we choose delay to come from a normal distribution . Surely in this case the delay lies in , but we can fix such that negative delay occurs with a very, very small probability less than 0.0000003. Actually, in our simulations negative delay never occurred. Noticing that the Laplace transform of the normal distribution is , the characteristic equation is given by
Conclusion
In this paper, we give some exact results for globally coupled phase oscillators with delay following Dirac, uniform, Gamma and normal distributions. Partially synchronized states are bifurcated from the incoherent states while on the mean field a Hopf bifurcation occurs. By using the center manifold reduction, we give a method to determine the direction of the bifurcations, then the detailed information of the partially synchronized states is obtained near the critical values. The existence of
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