Influence of parameters inhomogeneity on the existence of chimera states in a ring of nonlocally coupled maps

2021 ◽  
Vol 29 (6) ◽  
pp. 943-952
Author(s):  
Vasiliy Nechaev ◽  
◽  
Elena Rybalova ◽  
Galina Strelkova ◽  
◽  
...  
Keyword(s):  
Coupling Strength ◽  
Noise Intensity ◽  
Initial Conditions ◽  
Random Variable ◽  
Inhomogeneous Distribution ◽  
Control Parameters ◽  
Chimera States ◽  
Intensity Parameters ◽  
Different Characteristics ◽  
Ring Elements

The aim of the research is to study the influence of inhomogeneity in a control parameter of all partial elements in a ring of nonlocally coupled chaotic maps on the possibility of observing chimera states in the system and to compare the changes in regions of chimera realization using different methods of introducing the inhomogeneity. Methods. In this paper, snapshots of the system dynamics are constructed for various values of the parameters, as well as spatial distributions of cross-correlation coefficient values, which enable us to determine the regime observed in the system for these parameters. To improve the accuracy of the obtained results, the numerical studies are carried out for fifty different realizations of initial conditions of the ring elements. Results. It is shown that a fixed inhomogeneous distribution of the control parameters with increasing noise intensity leads to an increase in the range of the coupling strength where chimera states are observed. With this, the boundary lying in the region of strong coupling changes more significantly as compared with the case of weak coupling strength. The opposite effect is provided when the control parameters are permanently affected by noise. In this case increasing the noise intensity leads to a decrease in the interval of existence of chimera states. Additionally, the nature of the random variable distribution (normal or uniform one) does not strongly influence the observed changes in the ring dynamics. The regions of existence of chimera states are constructed in the plane of «coupling strength – noise intensity» parameters. Conclusion. We have studied how the region of existence of chimeras changes when the coupling strength between the ring elements is varied and when different characteristics of the inhomogeneous distribution of the control parameters are used. It has been shown that in order to increase the region of observing chimera states, the control parameters of the elements must be distributed inhomogeneously over the entire ensemble. To reduce this region, a constant noise effect on the control parameters should be used.

CELLULAR NEURAL NETWORKS CAN MIMIC SMALL-WORLD NETWORKS

1999 ◽  
Vol 09 (10) ◽  
pp. 2105-2126 ◽  
Author(s):  
TAO YANG ◽  
LEON O. CHUA
Keyword(s):  
Neural Networks ◽  
Coupling Strength ◽  
Initial Conditions ◽  
Computing Time ◽  
Small World ◽  
Cellular Neural Networks ◽  
Clustering Coefficient ◽  
High Survival ◽  
Game Of Life ◽  
Hole Detection

Small-world phenomenon can occur in coupled dynamical systems which are highly clustered at a local level and yet strongly coupled at the global level. We show that cellular neural networks (CNN's) can exhibit "small-world phenomenon". We generalize the "characteristic path length" from previous works on "small-world phenomenon" into a "characteristic coupling strength" for measuring the average coupling strength of the outputs of CNN's. We also provide a simplified algorithm for calculating the "characteristic coupling strength" with a reasonable amount of computing time. We define a "clustering coefficient" and show how it can be calculated by a horizontal "hole detection" CNN, followed by a vertical "hole detection" CNN. Evolutions of the game-of-life CNN with different initial conditions are used to illustrate the emergence of a "small-world phenomenon". Our results show that the well-known game-of-life CNN is not a small-world network. However, generalized CNN life games whose individuals have strong mobility and high survival rate can exhibit small-world phenomenon in a robust way. Our simulations confirm the conjecture that a population with a strong mobility is more likely to qualify as a small world. CNN games whose individuals have weak mobility can also exhibit a small-world phenomenon under a proper choice of initial conditions. However, the resulting small worlds depend strongly on the initial conditions, and are therefore not robust.

The multitype continuous-time Markov branching process in a periodic environment

1980 ◽  
Vol 12 (1) ◽  
pp. 81-93 ◽  
Author(s):  
B. Klein ◽  
P. D. M. MacDonald
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Cell Kinetics ◽  
Initial Conditions ◽  
Random Variable ◽  
Diurnal Rhythms ◽  
Biological Applications ◽  
Regular Case ◽  
Markov Branching Process ◽  
Periodic Environment

The multitype continuous-time Markov branching process has many biological applications where the environmental factors vary in a periodic manner. Circadian or diurnal rhythms in cell kinetics are an important example. It is shown that in the supercritical positively regular case the proportions of individuals of various types converge in probability to a non-random periodic vector, independent of the initial conditions, while the absolute numbers of individuals of various types converge in probability to that vector multiplied by a random variable whose distribution depends on the initial conditions. It is noted that the proofs are straightforward extensions of the well-known results for a constant environment.

scholarly journals Chimera states and the interplay between initial conditions and non-local coupling

2017 ◽  
Vol 27 (3) ◽  
pp. 033110 ◽  
Author(s):  
Peter Kalle ◽  
Jakub Sawicki ◽  
Anna Zakharova ◽  
Eckehard Schöll
Keyword(s):  
Initial Conditions ◽  
Chimera States ◽  
Non Local ◽  
Local Coupling

scholarly journals Phase Transition in Dynamical Systems: Defining Classes of Universality for Two-Dimensional Hamiltonian Mappings via Critical Exponents

2009 ◽  
Vol 2009 ◽  
pp. 1-22 ◽  
Author(s):  
Edson D. Leonel
Keyword(s):  
Phase Transition ◽  
Critical Exponents ◽  
Scaling Laws ◽  
Scaling Function ◽  
Initial Conditions ◽  
Invariant Tori ◽  
Dynamical Variable ◽  
Two Dimensional ◽  
Control Parameters ◽  
Average Value

A phase transition from integrability to nonintegrability in two-dimensional Hamiltonian mappings is described and characterized in terms of scaling arguments. The mappings considered produce a mixed structure in the phase space in the sense that, depending on the combination of the control parameters and initial conditions, KAM islands which are surrounded by chaotic seas that are limited by invariant tori are observed. Some dynamical properties for the largest component of the chaotic sea are obtained and described in terms of the control parameters. The average value and the deviation of the average value for chaotic components of a dynamical variable are described in terms of scaling laws, therefore critical exponents characterizing a scaling function that describes a phase transition are obtained and then classes of universality are characterized. The three models considered are: The Fermi-Ulam accelerator model, a periodically corrugate waveguide, and variant of the standard nontwist map.

scholarly journals Solving Second-Order Linear Differential Equations with Random Analytic Coefficients about Regular-Singular Points

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 230
Author(s):  
Juan-Carlos Cortés ◽  
Ana Navarro-Quiles ◽  
José-Vicente Romero ◽  
María-Dolores Roselló
Keyword(s):  
Differential Equations ◽  
Initial Conditions ◽  
Random Variable ◽  
Second Order ◽  
Singular Points ◽  
Linear Differential Equations ◽  
Transformation Technique ◽  
Order Linear ◽  
Bessel Differential Equation ◽  
Linear Differential

In this contribution, we construct approximations for the density associated with the solution of second-order linear differential equations whose coefficients are analytic stochastic processes about regular-singular points. Our analysis is based on the combination of a random Fröbenius technique together with the random variable transformation technique assuming mild probabilistic conditions on the initial conditions and coefficients. The new results complete the ones recently established by the authors for the same class of stochastic differential equations, but about regular points. In this way, this new contribution allows us to study, for example, the important randomized Bessel differential equation.

UNCERTAINTY IN CHAOS SYNCHRONIZATION

2001 ◽  
Vol 11 (06) ◽  
pp. 1723-1735 ◽  
Author(s):  
GUO-QUN ZHONG ◽  
KIM-FUNG MAN ◽  
KING-TIM KO
Keyword(s):  
Coupling Strength ◽  
Chaos Synchronization ◽  
Phase Synchronization ◽  
Initial Conditions ◽  
Coupled Systems ◽  
Sensitive Dependence ◽  
Initial States ◽  
Synchronization Regime

In this paper a variety of uncertainty phenomena in chaos synchronization, which are caused by the sensitive dependence on initial conditions and coupling strength, are numerically investigated. Two identical Chua's circuits are considered for both mutually- and unidirectionally-coupled systems. It is found that initial states of the system play an important role in chaos synchronization. Depending on initial conditions, distinct behaviors, such as in-phase synchronization, anti-phase synchronization, oscillation-quenching, and bubbling of attractors, may occur. Based on the findings, we clarify that the systems, which satisfy the standard synchronization criterion, do not necessarily operate in a synchronization regime.

scholarly journals Chimera States in Star Networks

2016 ◽  
Vol 26 (09) ◽  
pp. 1630023 ◽  
Author(s):  
Chandrakala Meena ◽  
K. Murali ◽  
Sudeshna Sinha
Keyword(s):  
Analog Circuit ◽  
Initial Conditions ◽  
Mean Field ◽  
Coupled Systems ◽  
Dynamical Equations ◽  
Chimera States ◽  
Diffusive Coupling ◽  
Star Networks ◽  
Random Initial Conditions ◽  
Coupling Strengths

We consider star networks of chaotic oscillators, with all end-nodes connected only to the central hub node, under diffusive coupling, conjugate coupling and mean-field diffusive coupling. We observe the existence of chimeras in the end-nodes, which are identical in terms of the coupling environment and dynamical equations. Namely, the symmetry of the end-nodes is broken and coexisting groups with different synchronization features and attractor geometries emerge. Surprisingly, such chimera states are very wide-spread in this network topology, and large parameter regimes of moderate coupling strengths evolve to chimera states from generic random initial conditions. Further, we verify the robustness of these chimera states in analog circuit experiments. Thus it is evident that star networks provide a promising class of coupled systems, in natural or engineered contexts, where chimeras are prevalent.

A Study of Noise Impact on the Stability of Electrostatic MEMS

2020 ◽  
Vol 15 (11) ◽  
Author(s):  
Yan Qiao ◽  
Wei Xu ◽  
Hongxia Zhang ◽  
Qin Guo ◽  
Eihab Abdel-Rahman
Keyword(s):  
Noise Intensity ◽  
Initial Conditions ◽  
Basins Of Attraction ◽  
Low Noise ◽  
Intensity Level ◽  
Lumped Mass ◽  
Mass Model ◽  
Restoring Force ◽  
Electrostatic Mems ◽  
The Stability

Abstract Noise-induced motions are a significant source of uncertainty in the response of micro-electromechanical systems (MEMS). This is particularly the case for electrostatic MEMS where electrical and mechanical sources contribute to noise and can result in sudden and drastic loss of stability. This paper investigates the effects of noise processes on the stability of electrostatic MEMS via a lumped-mass model that accounts for uncertainty in mass, mechanical restoring force, bias voltage, and AC voltage amplitude. We evaluated the stationary probability density function (PDF) of the resonator response and its basins of attraction in the presence noise and compared them to that those obtained under deterministic excitations only. We found that the presence of noise was most significant in the vicinity of resonance. Even low noise intensity levels caused stochastic jumps between co-existing orbits away from bifurcation points. Moderate noise intensity levels were found to destroy the basins of attraction of the larger orbits. Higher noise intensity levels were found to destroy the basins of attraction of smaller orbits, dominate the dynamic response, and occasionally lead to pull-in. The probabilities of pull-in of the resonator under different noise intensity level are calculated, which are sensitive to the initial conditions.

The multitype continuous-time Markov branching process in a periodic environment

1980 ◽  
Vol 12 (01) ◽  
pp. 81-93 ◽  
Author(s):  
B. Klein ◽  
P. D. M. MacDonald
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Cell Kinetics ◽  
Initial Conditions ◽  
Random Variable ◽  
Diurnal Rhythms ◽  
Biological Applications ◽  
Regular Case ◽  
Markov Branching Process ◽  
Periodic Environment

The multitype continuous-time Markov branching process has many biological applications where the environmental factors vary in a periodic manner. Circadian or diurnal rhythms in cell kinetics are an important example. It is shown that in the supercritical positively regular case the proportions of individuals of various types converge in probability to a non-random periodic vector, independent of the initial conditions, while the absolute numbers of individuals of various types converge in probability to that vector multiplied by a random variable whose distribution depends on the initial conditions. It is noted that the proofs are straightforward extensions of the well-known results for a constant environment.

SYNCHRONIZING CHAOTIC ATTRACTORS OF CHUA'S CANONICAL CIRCUIT: THE CASE OF UNCERTAINTY IN CHAOS SYNCHRONIZATION

2006 ◽  
Vol 16 (07) ◽  
pp. 1961-1976 ◽  
Author(s):  
I. M. KYPRIANIDIS ◽  
A. N. BOGIATZI ◽  
M. PAPADOPOULOU ◽  
I. N. STOUBOULOS ◽  
G. N. BOGIATZIS ◽  
...  
Keyword(s):  
Coupling Strength ◽  
Chaos Synchronization ◽  
Phase Synchronization ◽  
Initial Conditions ◽  
Chaotic Synchronization ◽  
Chaotic Attractors ◽  
Coexisting Attractors

In this paper, we have studied the dynamics of two identical resistively coupled Chua's canonical circuits and have found that it is strongly affected by initial conditions, coupling strength and the presence of coexisting attractors. Depending on the coupling variable, chaotic synchronization has been observed both numerically and experimentally. Anti-phase synchronization has also been studied numerically clarifying some aspects of uncertainty in chaos synchronization.

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