- Published in: IEEE Conference on Decision and Control (CDC) 2018
- Authors: Jin Gyu Lee and Hyungbo Shim
- Abstract: The behavior of a network of heterogeneous Van der Pol oscillators under output diffusive coupling is studied. By the intuition on strong coupling given by the series of `blended dynamics’ research, we first show that under sufficiently strong coupling gain the network achieves practical synchronization among heterogeneous oscillators even when there is no common internal model. Then, a further analysis finds a necessary and sufficient condition for the network to achieve synchronous and oscillatory behavior. It should be noted that the condition is the existence of a stable limit cycle for an emergent Van der Pol oscillator we introduce in this paper. Since the stability of each oscillator is not asked, the individual can be any system having the same structure with the Van der Pol oscillator, for instance, double integrator, linear oscillator, and a Van der Pol oscillator with an unstable limit cycle. Finally, by applying a modified singular perturbation theory, we show that the individual agent oscillates near the stable limit cycle of the emergent Van der Pol oscillator.
A network of Van der Pol oscillators appears in many disciplines including biology. Many biological models, for instance, cardiac oscillation, is modeled as a network of Van der Pol oscillators. Biological systems, as you can see from the following video of beating heart cells, has many interesting characteristics.
- First of all, there are no exactly equivalent cells, and hence, there is heterogeneity.
- They show some collective behaviors, for instance, synchronization and oscillation, and this happens even when there are some mal-functioning cells if the majority is good.
- Usually, a biological system uses single communication, like an ion channel.
However, the internal model principle in multi-agent system says that a network of heterogeneous Van der Pol oscillators cannot achieve synchronization unless they have a common internal model. This can be also seen from the previous research on the synchronization of a network of Van der Pol oscillators, where most of them considered a homogeneous case. Some works considered heterogeneity, however, the heterogeneity is given to the frequency of the oscillator, hence still a common internal model exists.
On the contrary, when both heterogeneity and mal-functioning cells exist, the assumption of a common internal model seems hard to satisfy. One of the main topics of this paper is that even though the network of heterogeneous Van der Pol oscillators that we consider doesn’t have a common internal model, still we can observe some synchronous and oscillatory behavior.
At this point, let us see some simulation videos.
The first simulation is about the homogeneous Van der Pol oscillators, and it is quite intuitive that this network will achieve synchronization as the following video shows.
On the other hand, heterogeneous Van der Pol oscillators, as written earlier, the internal model principle prevents synchronization. However, as you can see from the following video, we can observe that the heterogeneous Van der Pol oscillators do not exactly synchronize, but they are moving in groups, and thus, approximately synchronize. Besides that, we can observe another interesting phenomenon, that they seem to oscillate near some limit cycle. For heterogeneous Van der Pol oscillators, the limit cycle is all different and it is hard to characterize what the synchronized behavior would be. This is the second main topics of this paper, which is to characterize the synchronized behavior when heterogeneous Van der Pol oscillators achieve synchronous and oscillatory behavior.
To summarize, we answer two questions, where one is about the condition required for approximate synchronization to happen and the other is about the achieved synchronized behavior.
The main tool used in the paper is blended dynamics, which is briefly introduced as follows. Consider a heterogeneous multi-agent system given by
$$\dot{x}_i = f_i(t, x_i, y_i), \quad \dot{y}_i = g_i(t, x_i, y_i) + u_i$$
where the input 
$$u_i=k \sum_{j \in \mathcal{N}_i} (y_j – y_i).$$
It was shown that (see this paper by J. G. Lee and H. Shim) with sufficiently high coupling gain 
$$\dot{\hat{x}}_i = f_i(t, \hat{x}_i, s), \quad \dot{s} = \frac{1}{N} \sum_{i=1}^N g_i(t, \hat{x}_i, s)$$
under appropriate assumptions. This property is useful since the complex behavior of the heterogeneous multi-agent system can be described by the blended dynamics which is much simpler to study.
Slides used in the presentation at the IEEE Conference on Decision and Control 2018 is given below for the reference.
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