CHAOTIC ATTRACTOR IN THE KURAMOTO MODEL

2005 ◽  
Vol 15 (11) ◽  
pp. 3457-3466 ◽  
Author(s):  
YURI L. MAISTRENKO ◽  
OLEKSANDR V. POPOVYCH ◽  
PETER A. TASS
Keyword(s):  
Chaotic Attractor ◽  
Nonlinear Dynamical System ◽  
Kuramoto Model ◽  
Phase Oscillators ◽  
Bifurcation And Chaos ◽  
Nonlinear Dynamical ◽  
Transition To Chaos ◽  
Quasiperiodic Dynamics ◽  
The Kuramoto Model ◽  
Low Dimensional

The Kuramoto model of globally coupled phase oscillators is an essentially nonlinear dynamical system with a rich dynamics including synchronization and chaos. We study the Kuramoto model from the standpoint of bifurcation and chaos theory of low-dimensional dynamical systems. We find a chaotic attractor in the four-dimensional Kuramoto model and study its origin. The torus destruction scenario is one of the major mechanisms by which chaos arises. L. P. Shilnikov has made decisive contributions to its discovery. We show also that in the Kuramoto model the transition to chaos is in accordance with the torus destruction scenario. We present the general bifurcation diagram containing phase chaos, Cherry flow as well as periodic and quasiperiodic dynamics.

scholarly journals Deterministic Chaos in the Microstructure of Radio Pulses of PSR 0809+74

1992 ◽  
Vol 128 ◽  
pp. 329-331
Author(s):  
V. I. Zhuravlev ◽  
M. V. Popov
Keyword(s):  
Radio Emission ◽  
Random Sequence ◽  
Nonlinear Dynamical System ◽  
Deterministic Chaos ◽  
Relativistic Plasma ◽  
Emission Region ◽  
Polar Cap ◽  
Nonlinear Dynamical ◽  
Developed Turbulence ◽  
Low Dimensional

Abstract480 single pulses from PSR 0809+74, recorded at 102.5 MHz with a time resolution of 10μs, have been analyzed by the time delay technique in order to look for the parameters of deterministic chaos in the microstructure of radio pulses. The correlation dimension n was shown to be less than 5 in more than 20% of the analyzed pulses. This means that on such occasions the microstructure of pulsar radio emission with the time scales 10 to 100μs may be determined by the behavior of a nonlinear dynamical system with comparatively small numbers of independent parameters. For example, the observed low dimensional chaos may result from a turbulence process associated with the outflow of plasma—as in versions of the polar cap model where microstructure can be interpreted as reflecting the spatial structure of relativistic plasma outflow in the radio emission region.However, the correlation-time distribution demonstrates the tendency for microstructure to consist of a random sequence of unresolved micropulses in the majority of cases, which means in the framework of polar cap models the presence of well developed turbulence in the relativistic plasma outflow.

scholarly journals Distribution of Order Parameter for Kuramoto Model

2015 ◽  
Vol 3 (9) ◽  
pp. 52-68
Author(s):  
Hui Wu ◽  
Dongwook Kim
Keyword(s):  
Order Parameter ◽  
Initial Conditions ◽  
Oscillatory System ◽  
Kuramoto Model ◽  
Chemical System ◽  
Phase Oscillators ◽  
Numerical Result ◽  
Coupled Nonlinear Oscillators ◽  
Large Populations ◽  
The Kuramoto Model

The synchronization in large populations of interacting oscillators has been observed abundantly in nature, emergining in fields such as physical, biological and chemical system. For this reason, many scientists are seeking to understand the underlying mechansim of the generation of synchronous patterns in oscillatory system. The synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. The Kuramoto model can be used to understand the emergence of synchronization in nextworks of coupled, nonlinear oscillators. In particular, this model presents a phase transition from incoherence to synchronization. In this paper, we investigated the distribution of order parameter γ which describes the strength of synchrony of these oscillators. The larger the order parameter γ is, the more extent the oscillators are synchronized together. This order parameter γ is a critical parameter in the Kuramoto model. Kuramoto gave a initial estimate equation for the value of the order parameter by giving the value of the coupling constant. But our numerical results show that the distribution of the order parameter is slightly different from Kuramoto’s estimation. We gave an estimation for the distribution of order parameter for different values of initial conditions. We discussed how the numerical result will be distributed around Kuramoto’s analytical equation.

scholarly journals Analytical CPG model driven by single-limb velocity input generates accurate temporal locomotor dynamics

2018 ◽  
Author(s):  
Sergiy Yakovenko ◽  
Anton Sobinov ◽  
Valeriya Gritsenko
Keyword(s):  
Full Range ◽  
Nonlinear Dynamical System ◽  
Underlying Mechanism ◽  
Neural Pathways ◽  
Nonlinear Dynamical ◽  
Model Driven ◽  
Body Velocity ◽  
Leaky Integration ◽  
Low Dimensional ◽  
Walking Speeds

The ability of vertebrates to generate rhythm within their spinal neural networks is essential for walking, running, and other rhythmic behaviors. The central pattern generator (CPG) network responsible for these behaviors is well-characterized with experimental and theoretical studies, and it can be formulated as a nonlinear dynamical system. The underlying mechanism responsible for locomotor behavior can be expressed as the process of leaky integration with resetting states generating appropriate phases for changing body velocity. The low-dimensional input to the CPG model generates the bilateral pattern of swing and stance modulation for each limb and is consistent with the desired limb speed as the input command. To test the minimal configuration of required parameters for this model, we reduced the system of equations representing CPG for a single limb and provided the analytical solution with two complementary methods. The analytical and empirical cycle durations were similar (R2=0.99) for the full range of walking speeds. The structure of solution is consistent with the use of limb speed as the input domain for the CPG network. Moreover, the reciprocal interaction between two leaky integration processes representing a CPG for two limbs was sufficient to capture fundamental experimental dynamics associated with the control of heading direction. This analysis provides further support for the embedded velocity or limb speed representation within spinal neural pathways involved in rhythm generation.

scholarly journals Neuronal Assembly Dynamics in Supervised and Unsupervised Learning Scenarios

2013 ◽  
Vol 25 (11) ◽  
pp. 2934-2975 ◽  
Author(s):  
Renan C. Moioli ◽  
Phil Husbands
Keyword(s):  
Temporal Structure ◽  
Evolutionary Robotics ◽  
Kuramoto Model ◽  
Dynamical Analysis ◽  
Phase Oscillators ◽  
Cognitive Behaviors ◽  
The Neural Network ◽  
Conflicting Objectives ◽  
The Kuramoto Model ◽  
Neuronal Assembly

The dynamic formation of groups of neurons—neuronal assemblies—is believed to mediate cognitive phenomena at many levels, but their detailed operation and mechanisms of interaction are still to be uncovered. One hypothesis suggests that synchronized oscillations underpin their formation and functioning, with a focus on the temporal structure of neuronal signals. In this context, we investigate neuronal assembly dynamics in two complementary scenarios: the first, a supervised spike pattern classification task, in which noisy variations of a collection of spikes have to be correctly labeled; the second, an unsupervised, minimally cognitive evolutionary robotics tasks, in which an evolved agent has to cope with multiple, possibly conflicting, objectives. In both cases, the more traditional dynamical analysis of the system's variables is paired with information-theoretic techniques in order to get a broader picture of the ongoing interactions with and within the network. The neural network model is inspired by the Kuramoto model of coupled phase oscillators and allows one to fine-tune the network synchronization dynamics and assembly configuration. The experiments explore the computational power, redundancy, and generalization capability of neuronal circuits, demonstrating that performance depends nonlinearly on the number of assemblies and neurons in the network and showing that the framework can be exploited to generate minimally cognitive behaviors, with dynamic assembly formation accounting for varying degrees of stimuli modulation of the sensorimotor interactions.

CONTROLLING CHAOS IN A STATE-DEPENDENT NONLINEAR SYSTEM

2002 ◽  
Vol 12 (05) ◽  
pp. 1111-1119 ◽  
Author(s):  
TAKUJI KOUSAKA ◽  
TETSUSHI UETA ◽  
HIROSHI KAWAKAMI
Keyword(s):  
Dynamical System ◽  
Nonlinear System ◽  
Periodic Orbits ◽  
Chaotic Attractor ◽  
Discrete System ◽  
Nonlinear Dynamical System ◽  
Nonlinear Dynamical ◽  
Controlling Chaos ◽  
State Dependent ◽  
General Method

In this paper, we propose a general method for controlling chaos in a nonlinear dynamical system containing a state-dependent switch. The pole assignment for the corresponding discrete system derived from such a nonsmooth system via Poincaré mapping works effectively. As an illustrative example, we consider controlling the chaos in the Rayleigh-type oscillator with a state-dependent switch, which is changed by the hysteresis comparator. The unstable one- and two-periodic orbits in the chaotic attractor are stabilized in both numerical and experimental simulations.

scholarly journals Dynamics in the Sakaguchi-Kuramoto model with bimodal frequency distribution

PLoS ONE ◽  
2020 ◽  
Vol 15 (12) ◽  
pp. e0243196
Author(s):  
Shuangjian Guo ◽  
Yuan Xie ◽  
Qionglin Dai ◽  
Haihong Li ◽  
Junzhong Yang
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Natural Frequency ◽  
Frequency Distribution ◽  
Bimodal Distribution ◽  
Kuramoto Model ◽  
Phase Lag ◽  
Phase Oscillators ◽  
Different Types ◽  
Low Dimensional

In this work, we study the Sakaguchi-Kuramoto model with natural frequency following a bimodal distribution. By using Ott-Antonsen ansatz, we reduce the globally coupled phase oscillators to low dimensional coupled ordinary differential equations. For symmetrical bimodal frequency distribution, we analyze the stabilities of the incoherent state and different partial synchronous states. Different types of bifurcations are identified and the effect of the phase lag on the dynamics is investigated. For asymmetrical bimodal frequency distribution, we observe the revival of the incoherent state, and then the conditions for the revival are specified.

scholarly journals Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators

Nonlinearity ◽  
2012 ◽  
Vol 25 (5) ◽  
pp. 1247-1273 ◽  
Author(s):  
Giambattista Giacomin ◽  
Khashayar Pakdaman ◽  
Xavier Pellegrin
Keyword(s):  
Global Attractor ◽  
Kuramoto Model ◽  
Phase Oscillators ◽  
Asymptotic Dynamics ◽  
The Kuramoto Model

Low-Dimensional Discrete Kuramoto Model: Hierarchy of Multifrequency Quasiperiodicity Regimes

2014 ◽  
Vol 24 (07) ◽  
pp. 1430022 ◽  
Author(s):  
Alexander P. Kuznetsov ◽  
Yuliya V. Sedova
Keyword(s):  
Coupling Parameter ◽  
Kuramoto Model ◽  
Parameter Plane ◽  
Phase Oscillators ◽  
Repulsive Interactions ◽  
Discrete Phase ◽  
Model Hierarchy ◽  
Frequency Coupling ◽  
Low Dimensional ◽  
Two Parameter

The dynamics of a low-dimensional ensemble consisting of a network of five discrete phase oscillators is considered. A two-parameter synchronization picture, which appears instead of the Arnol'd tongues with an increase of the system dimension, is discussed. An appearance of the Arnol'd resonance web is detected on the "frequency–coupling" parameter plane. The cases of attractive and repulsive interactions are discussed.

scholarly journals Chaos in driven Alfvén systems: unstable periodic orbits and chaotic saddles

2007 ◽  
Vol 14 (1) ◽  
pp. 17-29 ◽  
Author(s):  
A. C.-L. Chian ◽  
W. M. Santana ◽  
E. L. Rempel ◽  
F. A. Borotto ◽  
T. Hada ◽  
...  
Keyword(s):  
Periodic Orbits ◽  
Chaotic Attractor ◽  
Unstable Periodic Orbits ◽  
Gap Filling ◽  
Intermittent Turbulence ◽  
Dynamical Processes ◽  
Nonlinear Dynamical ◽  
Derivative Nonlinear Schrödinger Equation ◽  
Low Dimensional ◽  
Chaotic Saddle

Abstract. The chaotic dynamics of Alfvén waves in space plasmas governed by the derivative nonlinear Schrödinger equation, in the low-dimensional limit described by stationary spatial solutions, is studied. A bifurcation diagram is constructed, by varying the driver amplitude, to identify a number of nonlinear dynamical processes including saddle-node bifurcation, boundary crisis, and interior crisis. The roles played by unstable periodic orbits and chaotic saddles in these transitions are analyzed, and the conversion from a chaotic saddle to a chaotic attractor in these dynamical processes is demonstrated. In particular, the phenomenon of gap-filling in the chaotic transition from weak chaos to strong chaos via an interior crisis is investigated. A coupling unstable periodic orbit created by an explosion, within the gaps of the chaotic saddles embedded in a chaotic attractor following an interior crisis, is found numerically. The gap-filling unstable periodic orbits are responsible for coupling the banded chaotic saddle (BCS) to the surrounding chaotic saddle (SCS), leading to crisis-induced intermittency. The physical relevance of chaos for Alfvén intermittent turbulence observed in the solar wind is discussed.

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