scholarly journals Brownian Behavior in Coupled Chaotic Oscillators

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2503
Author(s):  
Francisco Javier Martín-Pasquín ◽  
Alexander N. Pisarchik
Keyword(s):  
Quantum Mechanics ◽  
Brownian Motion ◽  
Time Evolution ◽  
Phase Difference ◽  
Stochastic Systems ◽  
Phase Synchronization ◽  
Deterministic Model ◽  
Dynamical Behavior ◽  
Deterministic Models ◽  
Chaotic Oscillators

Since the dynamical behavior of chaotic and stochastic systems is very similar, it is sometimes difficult to determine the nature of the movement. One of the best-studied stochastic processes is Brownian motion, a random walk that accurately describes many phenomena that occur in nature, including quantum mechanics. In this paper, we propose an approach that allows us to analyze chaotic dynamics using the Langevin equation describing dynamics of the phase difference between identical coupled chaotic oscillators. The time evolution of this phase difference can be explained by the biased Brownian motion, which is accepted in quantum mechanics for modeling thermal phenomena. Using a deterministic model based on chaotic Rössler oscillators, we are able to reproduce a similar time evolution for the phase difference. We show how the phenomenon of intermittent phase synchronization can be explained in terms of both stochastic and deterministic models. In addition, the existence of phase multistability in the phase synchronization regime is demonstrated.

IMPERFECT PHASE SYNCHRONIZATION IN THE LOCOMOTOR BEHAVIOR OFHALOBACTERIUM SALINARIUM

2003 ◽  
Vol 13 (10) ◽  
pp. 3085-3092
Author(s):  
GIANCARLO TORRI ◽  
SERGIO RINALDI ◽  
CARLO PICCARDI
Keyword(s):  
Phase Synchronization ◽  
Deterministic Model ◽  
Input Sequence ◽  
Locomotor Behavior ◽  
Halobacterium Salinarium ◽  
Chaotic Oscillators ◽  
Periodic Sequences ◽  
Sequence Of Events ◽  
Output Time ◽  
Low Dimensional

A chaotic system can be used to transform a periodic input sequence of events, e.g. impulses, into a sequence of output events. Even after transient, the output sequence is often aperiodic. In particular, it might happen that the output events are basically synchronized with the input events, but that some of them are randomly skipped from the output time series (imperfect phase synchronization [Zacks et al., 1999, 2000]). This phenomenon has been neatly observed by Schimz and Hildebrand [1992] in a rich series of experiments on the locomotor behavior of Halobacterium salinarium, where the input event was a light impulse and the output event was the reversal in the swimming direction of the bacterium. In this paper, we show that the same phenomenon occurs when classical, low dimensional chaotic oscillators are forced by a periodic sequence of impulses. This proves, on one side, that imperfect phase synchronization is a common phenomenon in chaotic oscillators subjected to periodic sequences of impulses, and, on the other side, that there are high chances that the locomotor behavior of Halobacterium salinarium can be modeled by a low dimensional deterministic model.

scholarly journals Dynamics of Stochastically Perturbed SIS Epidemic Model with Vaccination

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Yanan Zhao ◽  
Daqing Jiang
Keyword(s):  
Epidemic Model ◽  
Death Rate ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Dynamical Behavior ◽  
Standard Technique ◽  
Deterministic Models ◽  
Sis Epidemic Model ◽  
Disease Free ◽  
Sis Epidemic

We introduce stochasticity into an SIS epidemic model with vaccination. The stochasticity in the model is a standard technique in stochastic population modeling. In the deterministic models, the basic reproduction numberR0is a threshold which determines the persistence or extinction of the disease. When the perturbation and the disease-related death rate are small, we carry out a detailed analysis on the dynamical behavior of the stochastic model, also regarding of the value ofR0. IfR0≤1, the solution of the model is oscillating around a steady state, which is the disease-free equilibrium of the corresponding deterministic model, whereas, ifR0>1, there is a stationary distribution, which means that the disease will prevail. The results are illustrated by computer simulations.

scholarly journals Suppressing Chaos of Duffing-Holmes System Using Random Phase

2011 ◽  
Vol 2011 ◽  
pp. 1-8
Author(s):  
Li Longsuo
Keyword(s):  
Lyapunov Exponent ◽  
Time Evolution ◽  
Stochastic Systems ◽  
Random Noise ◽  
Dynamical Behavior ◽  
Random Phase ◽  
Bifurcation And Chaos ◽  
Analysis Phase ◽  
Linear Stochastic Systems ◽  
Map Analysis

The effect of random phase for Duffing-Holmes equation is investigated. We show that as the intensity of random noise properly increases the chaotic dynamical behavior will be suppressed by the criterion of top Lyapunov exponent, which is computed based on the Khasminskii's formulation and the extension of Wedig's algorithm for linear stochastic systems. Then, the obtained results are further verified by the Poincaré map analysis, phase plot, and time evolution on dynamical behavior of the system, such as stability, bifurcation, and chaos. Thus excellent agrement between these results is found.

Phase synchronization of chaotic oscillators by external driving

1997 ◽  
Vol 104 (3-4) ◽  
pp. 219-238 ◽  
Author(s):  
Arkady S. Pikovsky ◽  
Michael G. Rosenblum ◽  
Grigory V. Osipov ◽  
Jürgen Kurths
Keyword(s):  
Phase Synchronization ◽  
Chaotic Oscillators ◽  
External Driving

Phase synchronization analysis by assessment of the phase difference gradient

2009 ◽  
Vol 19 (2) ◽  
pp. 023120 ◽  
Author(s):  
Martin Vejmelka ◽  
Milan Paluš ◽  
W. T. Lee
Keyword(s):  
Phase Difference ◽  
Phase Synchronization

scholarly journals Determinism

2021 ◽  
Vol 20 (5) ◽  
pp. 1-34
Author(s):  
Edward A. Lee
Keyword(s):  
Quantum Physics ◽  
Deterministic Model ◽  
Physical World ◽  
Deterministic Models ◽  
Newtonian Physics ◽  
Physical Systems ◽  
Newton’S Laws ◽  
And Behavior

This article is about deterministic models, what they are, why they are useful, and what their limitations are. First, the article emphasizes that determinism is a property of models, not of physical systems. Whether a model is deterministic or not depends on how one defines the inputs and behavior of the model. To define behavior, one has to define an observer. The article compares and contrasts two classes of ways to define an observer, one based on the notion of “state” and another that more flexibly defines the observables. The notion of “state” is shown to be problematic and lead to nondeterminism that is avoided when the observables are defined differently. The article examines determinism in models of the physical world. In what may surprise many readers, it shows that Newtonian physics admits nondeterminism and that quantum physics may be interpreted as a deterministic model. Moreover, it shows that both relativity and quantum physics undermine the notion of “state” and therefore require more flexible ways of defining observables. Finally, the article reviews results showing that sufficiently rich sets of deterministic models are incomplete. Specifically, nondeterminism is inescapable in any system of models rich enough to encompass Newton’s laws.

Experimental Implementation of Networked Chaotic Oscillators Based on Cross-Coupled Inverter Rings in a CMOS Integrated Circuit

2015 ◽  
Vol 24 (09) ◽  
pp. 1550144 ◽  
Author(s):  
Ludovico Minati
Keyword(s):  
Experimental Data ◽  
Integrated Circuit ◽  
Phase Synchronization ◽  
Random Number Generation ◽  
Partial Amplitude ◽  
Chaotic Oscillator ◽  
Chaotic Oscillators ◽  
Cmos Integrated Circuit ◽  
Experimental Implementation ◽  
Small Clusters

A novel chaotic oscillator based on "cross-coupled" inverter rings is presented. The oscillator consists of a 3-ring to which higher odd n-rings are progressively coupled via diodes and pass gates; it does not contain reactive or resistive elements, and is thus suitable for area-efficient implementation on a CMOS integrated circuit. Numerical simulation based on piece-wise linear approximation predicted the generation of positive spikes having approximately constant periodicity but highly variable cycle amplitude. Simulation Program with Integrated Circuit Emphasis (SPICE) simulations and experimental data from a prototype realized on 0.7 μm technology confirmed this finding, and demonstrated increasing correlation dimension (D2) as 5-, 7- and 9-rings were progressively coupled to the 3-ring. Experimental data from a ring of 24 such oscillator cells showed phase synchronization and partial amplitude synchronization (formation of small clusters), emerging depending on DC gate voltage applied at NMOS transistors implementing diffusive coupling between neighboring cells. Thanks to its small area, simple synchronizability and digital controllability, the proposed circuit enables experimental investigation of dynamical complexity in large networks of coupled chaotic oscillators, and may additionally be suitable for applications such as broadband signal and random number generation.

scholarly journals Coarse, Medium or Fine? A Quantum Mechanics Approach to Single Species Population Dynamics

2020 ◽  
Author(s):  
Olcay Akman ◽  
Leon Arriola ◽  
Aditi Ghosh ◽  
Ryan Schroeder
Keyword(s):  
Population Dynamics ◽  
Quantum Mechanics ◽  
Single Species ◽  
Stable Equilibrium Point ◽  
Ode Model ◽  
Deterministic Models ◽  
Species Population ◽  
Quantum Approach ◽  
Single Species Population ◽  
Quantum Framework

AbstractStandard heuristic mathematical models of population dynamics are often constructed using ordinary differential equations (ODEs). These deterministic models yield pre-dictable results which allow researchers to make informed recommendations on public policy. A common immigration, natural death, and fission ODE model is derived from a quantum mechanics view. This macroscopic ODE predicts that there is only one stable equilibrium point . We therefore presume that as t → ∞, the expected value should be . The quantum framework presented here yields the same standard ODE model, however with very unexpected quantum results, namely . The obvious questions are: why isn’t , why are the probabilities ≈ 0.37, and where is the missing probability of 0.26? The answer lies in quantum tunneling of probabilities. The goal of this paper is to study these tunneling effects that give specific predictions of the uncertainty in the population at the macroscopic level. These quantum effects open the possibility of searching for “black–swan” events. In other words, using the more sophisticated quantum approach, we may be able to make quantitative statements about rare events that have significant ramifications to the dynamical system.

Super-Brownian Motion and Critical Spatial Stochastic Systems

2004 ◽  
Vol 47 (2) ◽  
pp. 280-297 ◽  
Author(s):  
Ed Perkins
Keyword(s):  
Brownian Motion ◽  
Limit Theorems ◽  
Stochastic Systems ◽  
Contact Process ◽  
Voter Model ◽  
Oriented Percolation ◽  
Short Introduction ◽  
Super Brownian Motion

AbstractThis article is a short introduction to super-Brownian motion. Some of its properties are discussed but our main objective is to describe a number of limit theorems which show super-Brownian motion is a universal limit for rescaled spatial stochastic systems at criticality above a critical dimenson. These systems include the voter model, the contact process and critical oriented percolation.

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