Networks on the edge of chaos: Global feedback control of turbulence in oscillator networks (original) (raw)

Generating macroscopic chaos in a network of globally coupled phase oscillators

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2011

We consider an infinite network of globally-coupled phase oscillators in which the natural frequencies of the oscillators are drawn from a symmetric bimodal distribution. We demonstrate that macroscopic chaos can occur in this system when the coupling strength varies periodically in time.

Frequency assortativity can induce chaos in oscillator networks

Physical review. E, Statistical, nonlinear, and soft matter physics, 2015

We investigate the effect of preferentially connecting oscillators with similar frequency to each other in networks of coupled phase oscillators (i.e., frequency assortativity). Using the network Kuramoto model as an example, we find that frequency assortativity can induce chaos in the macroscopic dynamics. By applying a mean-field approximation in combination with the dimension reduction method of Ott and Antonsen, we show that the dynamics can be described by a low dimensional system of equations. We use the reduced system to characterize the macroscopic chaos using Lyapunov exponents, bifurcation diagrams, and time-delay embeddings. Finally, we show that the emergence of chaos stems from the formation of multiple groups of synchronized oscillators, i.e., meta-oscillators.

Complex self-organized dynamics in oscillator networks and methods of its control

2010

Models of phase oscillators are a universal tool for the study of collective dynamics in periodic systems. The interactions of otherwise regular periodic elements may produce various kinds of behavior, amongst which chaos and incoherence may not be desirable. The objective of this work is to study possible methods to control chaos and disorder in systems of interacting oscillators, and to study the different kinds of dynamical states that can be induced in the process. Systems of coupled phase oscillators are considered, which include phase shifts in the interactions between each pair of elements. This phase shift can lead to a desynchronization transition in a globally coupled system of identical oscillators. Under these conditions, we investigate the effect of external common noise acting on all elements. We observe that when such noise is weak, it gives rise to the formation of clusters in the system, whereas strong noise intensities bring the system to a synchronized state. When...

Diffusion-induced instability and chaos in random oscillator networks

Physical Review E, 2009

We demonstrate that diffusively coupled limit-cycle oscillators on random networks can exhibit various complex dynamical patterns. Reducing the system to a network analog of the complex Ginzburg-Landau equation, we argue that uniform oscillations can be linearly unstable with respect to spontaneous phase modulations due to diffusional coupling -the effect corresponding to the Benjamin-Feir instability in continuous media. Numerical investigations under this instability in random scale-free networks reveal a wealth of complex dynamical regimes, including partial amplitude death, clustering, and chaos. A dynamic mean-field theory explaining different kinds of nonlinear dynamics is constructed.

Traveling phase waves in asymmetric networks of noisy chaotic attractors

Physical review. E, 2016

We explore identical Rössler systems organized into two equally sized groups, among which differing positive and negative in- and out-coupling strengths are allowed. With this asymmetric coupling, we analyze patterns in the phase dynamics that coexist with chaotic amplitudes. We specifically investigate traveling phase waves where the oscillators settle on a new rhythm different from their own. We show that these waves are possible even without coherence in the phase angles. It is further demonstrated that the emergence of these incoherent traveling waves depends on the type of coupling, not on the individual dynamics of the Rössler systems. Together with the study of noise effects, our results suggest a promising new avenue toward the interplay of chaotic, noisy, coherent, and incoherent collective dynamics.

Disorder-induced dynamics in a pair of coupled heterogeneous phase oscillator networks

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2012

We consider a pair of coupled heterogeneous phase oscillator networks and investigate their dynamics in the continuum limit as the intrinsic frequencies of the oscillators are made more and more disparate. The Ott/Antonsen ansatz is used to reduce the system to three ordinary differential equations. We find that most of the interesting dynamics, such as chaotic behaviour, can be understood by analysing a gluing bifurcation of periodic orbits; these orbits can be thought of as "breathing chimeras" in the limit of identical oscillators. We also add Gaussian white noise to the oscillators' dynamics and derive a pair of coupled Fokker-Planck equations describing the dynamics in this case. Comparison with simulations of finite networks of oscillators are used to confirm many of the results.

Emergence of chaos in starlike networks of dissipative nonlinear oscillators by localized parametric excitations

Physical review, 2017

We study the effectiveness of locally controlling the impulse transmitted by parametric periodic excitations at inducing and suppressing chaos in starlike networks of driven damped pendula, leading to asynchronous chaotic states and equilibria, respectively. We found that the inducing (suppressor) effect of increasing (decreasing) the impulse transmitted by the parametric excitations acting on particular nodes depends strongly on their number and degree of connectivity as well as the coupling strength. Additionally, we provide a theoretical analysis explaining the basic physical mechanisms of the emergence and suppression of chaos as well as the main features of the chaos-control scenario. Our findings constitute proof of the impulse-induced control of chaos in a simple model of complex networks, thus opening the way to its application to real-world networks.

Synchronization and Peak-To-Peak Dynamics in Networks of Low-Dimensional Chaotic Oscillators

International Journal of Bifurcation and Chaos, 2006

In this paper we study the relationships between local and global properties in networks of dynamical systems by focusing on two global properties, synchronization and peak-to-peak dynamics, and on two local properties, coherence of the components of the network and coupling strength. The analysis is restricted to networks of low-dimensional chaotic oscillators, i.e. oscillators which have peak-to-peak dynamics when they work in isolation. The results are obtained through simulation, first by considering pairs of coupled Lorenz, Rössler and Chua systems, and then by studying the behavior of spatially extended tritrophic food chains described by the Rosenzweig–MacArthur model. The conclusion is that synchronization and peak-to-peak dynamics are different aspects of the same collective behavior, which is easily obtained by enhancing local coupling and coherence. The importance of these findings is briefly discussed within the context of ecological modeling.

Random symmetry breaking and freezing in chaotic networks

Physical Review E, 2012

Parameter space of a driven damped oscillator in a double well potential presents either a chaotic trajectory with sign oscillating amplitude or a non-chaotic trajectory with a fixed sign amplitude. A network of such delay coupled damped oscillators is shown to present chaotic dynamics while the amplitude sign of each damped oscillator is randomly frozen. This phenomenon of random broken global symmetry of the network simultaneously with random freezing of each degree of freedom is accompanied by the existence of exponentially many randomly frozen chaotic attractors with the size of the network. Results are exemplified by a network of modified Duffing oscillators with infinite range pseudo-inverse delayed interactions.