Existence Of Chaos Research Articles

Qualitative behavior of a discrete predator–prey system under fear effects

Abstract Numerous field data and experiments on the perching birds or songbirds show that the fear of predators can cause significant changes in the prey population. Fear of predatory populations increases the chances of survival of the prey population, and this can greatly reduce the reproduction of the prey population. The influence of fear has contributed a leading role in both the environmental biology and theoretical ecology. Taking into account the interaction of predator–prey with non-overlapping generations, a discrete-time model is proposed and studied. Keeping in mind the biological feasibility of species, the existence of fixed points is studied along with the local asymptotic behavior of the proposed model around these fixed points. Furthermore, taking into account the oscillatory behavior of the model, various types of bifurcations are analyzed about biologically feasible fixed points with an application of center manifold theory and bifurcation theory of normal forms. Existence of chaos is discussed, and fluctuating and chaotic behavior of the system is controlled through implementation of different chaos control procedures. The illustration of theoretical discussion is carried out via validation of observed experimental field data and appropriate numerical simulation.

Conservative dynamics in a novel class of 3D generalized thermostatted systems.

This paper reports a method to derive a novel class of 3D generalized thermostatted oscillators from a simple damped harmonic oscillator. Its detailed procedure is obtained through a mathematical derivation. Then, we propose an example system to show the effectiveness of the method. Furthermore, the numerical analysis is performed to investigate its rich conservative dynamics, including chaotic sea, hierarchical invariant tori, and the coexistence of chaotic seas and invariant tori with islands-around-islands hierarchy, even if there are three dissipative terms in the example system. To verify the existence of conservative chaos at the physical level, an analog circuit is presented to observe the existing conservative chaotic flows in National Instruments multisim. We finally provide two systems, which can produce conservative chaotic flows with more complicated topologies, and draw our conclusions.

Chaos of the 2D linear hyperbolic equation with general van der Pol type boundary condition

This paper is concerned with the complex dynamics of the initial-boundary value problem of a 2D linear hyperbolic partial differential equation (PDE), where the parameter α that appeared in the general van der Pol type boundary condition is given by α∈R. The whole real line is divided into three intervals of the parameter to discuss the dynamics. The existence of chaos is first established in the sense of the exponential growth of total variation when the parameter locates in the central interval, which allows it to be positive, negative, or zero. By analyzing the chaotic dynamics of the piecewise continuous map induced by the hyperbolic PDE, such a PDE is further rigorously proved to be chaos in the interval of the positive parameter that is to the right of the central interval. Finally, the asymptotic behaviors of the hyperbolic PDE are systematically presented in the rest of the whole real line; more precisely, the hyperbolic PDE is either globally stable or unbounded.

The Dynamics of the Aquatic Food Chain System in the Contaminated Environment

In this paper, the aquatic food chain model, consisting of Phytoplankton, Zooplankton, and Fish, in the contaminated environment is proposed and studied. Modified Leslie–Gower model with Holling type IV functional response are used to describe the growth of Fish and the food transition throughout the food chain, respectively. The toxic substance affects directly the Phytoplankton and indirectly the other species. The local stability analysis of all possible equilibrium points is done. The persistence conditions of the model are established. The basin of attraction for each point is specified using the Lyapunov function. Bifurcation analysis near the coexistence equilibrium point is investigated. Detecting the existence of chaos is carried out using bifurcation diagrams. Numerical simulation shows that the food chain has rich dynamics including chaos. Moreover, the existence of toxic substances works as a stabilizing factor in the model.

Chaos Theory of 0-1 Test and Logistic Map in New Confirmed COVID-19 Cases

This study aims to forecast the new number of positive cases of COVID-19 using the logistic map, where the presence of chaos is determined using the 0-1 Test. There are a small number of investigations on chaos utilizing the 0-1 Test and logistic map in the literature. This study provides a straightforward technique of forecasting and a current way of determining chaos. Information on new confirmed COVID-19 cases and population of ten countries involving China, Vietnam, Korea, Malaysia, Singapore, New Zealand, India, USA, Brazil, and Mexico for six months from January 20 to June 15, 2020, are selected as samples. The logistic map runs through three stages: data training, forecasting, and validation using the Mean Absolute Error method (MAE). The data training procedure is critical for determining the best growth rate, r, for the logistic map. In chaotic investigations of the 0-1 Test, there appears to be an inverse expectation toward a logistic map. The 0-1 Test in the data of new confirmed COVID-19 cases in all the selected countries reveals the presence of non-chaotic. This contrasts with the existence of chaos in the logistic map forecasts for the USA, Brazil, and Mexico. Regardless, the logistic map was found to be capable of forecasting new COVID-19 positive cases with low error instances. Beginning in the middle of May 2020, new COVID-19 positive cases are forecasted to be on the rise in the USA, Brazil, and Mexico.

Chaotic advection in a recirculating flow: Effect of a fluid-flexible-solid interaction.

The present work is on laminar recirculating flow-induced deformation as well as motion of a neutrally buoyant flexible elliptical solid, resulting in Lagrangian chaos in a two-dimensional lid-driven cavity flow. Using a fully Eulerian and monolithic approach-based single-solver for the fluid flow and flexible-solid deformation, a chaotic advection study is presented for various aspect ratios β ( =0.5-1.0) and a constant volume fraction Φ=10% of an elliptical solid at a constant Ericksen number Er=0.05 and Reynolds number Re=100. Our initial analysis reveals maximum chaotic advection at β=0.5 for which a comprehensive nonlinear dynamical analysis is presented. The Poincaré map revealed elliptic islands and chaotic sea in the fluid flow. Three large elliptic islands, apart from certain smaller islands, were identified near the solid. Periodic point analysis revealed the lowest order hyperbolic/elliptic periodic points to be three. Adaptive material tracking gave a physical picture of a deforming material blob revealing its exponential stretch along with steep folds and demonstrated unstable/stable manifolds corresponding to lowest order hyperbolic points. Furthermore, adaptive material tracking demonstrates heteroclinic connections and tangles in the system that confirm the existence of chaos. For the transient as compared to the periodic flow, adaptive material tracking demonstrates a larger exponential increase of the blob's interfacial area. The finite-time Lyapunov exponent field revealed attracting/repelling Lagrangian coherent structures and entrapped fluid zones. Our work demonstrates an immersed deformable solid-based onset of chaotic advection, for the first time in the literature, which is relevant to a wide range of applications.

The Existence of Autonomous Chaos in EDM Process

The dynamical evolution of electrical discharge machining (EDM) has drawn immense research interest. Previous research on mechanism analysis has discussed the deterministic nonlinearity of gap states at pulse-on discharging duration, while describing the pulse-off deionization process separately as a stochastic evolutionary process. In this case, the precise model describing a complete machining process, as well as the optimum performance parameters of EDM, can hardly be determined. The main purpose of this paper is to clarify whether the EDM system can maintain consistency in dynamic characteristics within a discharge interval. A nonlinear self-maintained equivalent model is first established, and two threshold conditions are obtained by the Shilnikov theory. The theoretical results prove that the EDM system could lead to chaos without external excitation. The time series of the deionization process recorded in the EDM experiments are then analyzed to further validate this theoretical conclusion. Qualitative chaotic analyses verify that the autonomous EDM process has chaotic characteristics. Quantitative methods are used to estimate the chaotic feature of the autonomous EDM process. By comparing the quantitative results of the autonomous EDM process with the non-autonomous EDM process, a deduction is further made that the EDM system will evolve towards steady chaos under an autonomous state.

Chaos Induced by Heteroclinic Cycles Connecting Repellers for First-Order Partial Difference Equations

This paper studies chaos and chaotification for a class of first-order partial difference equations with a finite or infinite system size by using the theory of heteroclinic cycles connecting repellers. Three criteria of chaos induced by regular and nondegenerate heteroclinic cycles connecting repellers or regular heteroclinic cycles connecting repellers are established, respectively. Especially, one chaotification theorem for general discrete systems in the finite-dimensional space [Formula: see text] or the infinite-dimensional space [Formula: see text] is established, which is also based on heteroclinic cycles connecting repellers. Then, by using this result, two chaotification schemes for first-order partial difference equations are established. For illustrating the existence of chaos and the validity of chaotification schemes, three examples are provided with computer simulations.

General chaotic behaviors of heavy ion collisions at intermediate energy based on dynamical transport model

Motivated to explore the dynamical mechanism of multi-fragmentation phase transition and expand our knowledge on the deterministic chaos in nonlinear dynamic process, we study systematically the nonlinear dynamical behaviors of heavy ion collisions at intermediate energy. In order to highlight the general characteristics of the collision dynamics, in the present paper, we simulate a simple collision system, Ca+Ca, and obverse the performances of the typical nonlinear characteristic quantities such as the production of generalized entropy and the fractal structure of the semiclassical phase space during the reaction process. The multifragmentation entropy, information dimension and the dynamical fluctuations of fragment mass distribution in the final state of the reaction are also evaluated. Our results about the intermediate processes and the final products of the reaction are verified mutually so that confirm strongly the existence of nonlinear chaos in the heavy ion collisions at intermediate energy. Meanwhile, we find that the typical characteristics of the critical phase transition obtained in the analysis of the final production of the reaction are relatively more clear in the vicinity of the incident energies E=100 MeV.

Euler integral as a source of chaos in the three–body problem

In this paper we address, from a purely numerical point of view, the question, raised in Pinzari (2019), Pinzari (2020), and partly considered in Pinzari (2020), Di Ruzza et al. (2020), Chen and Pinzari (2021), whether a certain function, referred to as “Euler Integral”, is a quasi-integral along the trajectories of the three-body problem. Differently from our previous investigations, here we focus on the region of the “unperturbed separatrix”, which turns to be complicated by a collision singularity. Concretely, we reduce the Hamiltonian to two degrees of freedom and, after fixing some energy level, we discuss in detail the resulting three-dimensional phase space around an elliptic and an hyperbolic periodic orbit. After measuring the strength of variation of the Euler Integral (which are in fact small), we detect the existence of chaos closely to the unperturbed separatrix. The latter result is obtained through a careful use of the machinery of covering relations, developed in Gierzkiewicz and Zgliczyński (2019), Zgliczynski and Gidea (2004), Wilczak and Zgliczynski (2003).

Chaos in a Simplest Cyclic Memristive Neural Network

Previous studies have shown that cyclic neural networks which have no autoexcitation and are unidirectional cannot generate chaos. Inspired by this finding, the present paper constructs a new memristive neural network composed of three nodes connected by the simplest circular loop, whose synaptic weights are replaced by hyperbolic memristors. The memristive neural network can generate chaos via period-doubling bifurcation, and generate different stable and periodic states with the variation of parameters. Another remarkable feature of the new memristive neural network is that it coexists with point and periodic attractors, periodic and chaotic attractors from different initial conditions. Detailed dynamic analysis and circuit implementation are given to illustrate the existence of chaos and coexisting attractors, which gives a positive answer to the interesting question whether chaos can occur in neural network with the simplest cyclic connections.

Li-Yorke chaos in a class of controlled delay difference equations

A class of controlled delay difference equations is considered. Proof of the existence of chaos in the sense of Li-Yorke in the controlled system is given by applying the snap-back repeller theorem. Conditions for the existence of chaos are weak and lots of functions can be taken as controllers, which facilitates the chaotification of delay difference equations. One illustrative example is provided with computer simulations.

On the dynamics of a Caputo-like discrete fractional Rössler system: chaos, stabilization and synchronization

This study examines the dynamics of a novel Caputo-like discrete fractional Rössler system. The dynamics of this discrete fractional system are numerically analyzed using phase portraits, bifurcation diagrams and Lyapunov exponents. The study confirmed the existence of chaos in the proposed system where one scroll chaotic attractors are displayed. Control laws are presented to force the states of the proposed system to converge asymptotically to zero and to exhibit complete synchronization of coupled Caputo-like discrete fractional Rössler systems. Numerical simulations are introduced to illustrate the findings of this study.

Matching geometric and expansion characteristics of wild chaotic attractors

Wild chaotic attractors exhibit chaotic dynamics with a robustness property that cannot be destroyed with small perturbations. We consider a discrete-time system with the smallest possible dimension, namely, defined by a non-invertible map on the complex plane. For this map, wild chaos has been proven to exist in a small parameter region. Recently, it was conjectured to exist in a much larger region of parameter space, past a so-called backward critical tangency, at which a sequence of pre-images of a critical point converges to a saddle fixed point. Geometrically, a backward critical tangency leads to an abundance of homoclinic and heteroclinic tangencies between invariant manifolds of different dimensions, generating precisely what are believed to be the necessary ingredients for wild chaos. In this paper, we present corroborating evidence for this conjecture by computing Lyapunov exponents associated with the attractor. When the sum of the two (largest) Lyapunov exponents is positive, the dynamics is wild chaotic for this non-invertible map. We find that the zero-sum locus matches the locus of backward critical tangency, confirming its role as a boundary of existence of wild chaos.

Chaotic behaviors of an in-plane tethered satellite system with elasticity

The existence and identification of chaos regarding in-plane pitch motions of tethered satellite systems are scientific problems of concern, and are directly related to normal system operations in the station-keeping phase. This paper studies the effect of microamplitude longitudinal oscillation on the occurrence of chaos in a tethered system subjected to atmospheric perturbations. Based on a simplified rod model that considers tether elasticity and satellite masses, the Melnikov method is used to identify a criterion that can predict chaos, and a chaotic zone is further proposed to recognize the relationship between the chaos and system parameters. Then the cell mapping method is used to describe the system's global dynamic behaviors, including chaos. A discrete model of the flexible system that consists of particles connected by massless springs and satellite rigid bodies is structured so that more accurate dynamic simulations can be used to verify the theoretical analysis. Finally, numerical examples demonstrate that the chaotic zone and criterion expression are useful tools for revealing chaos. The relationship between the dynamics and system parameters is also assessed. The results of the simplified model agree with those of the sophisticated model.

A Geometric Criterion for the Existence of Chaos Based on Periodic Orbits in Continuous-Time Autonomous Systems

A new geometric criterion is derived for the existence of chaos in continuous-time autonomous systems in three-dimensional Euclidean spaces, where a type of Smale horseshoe in a subshift of finite type exists, but the intersection of stable and unstable manifolds of two points on a hyperbolic periodic orbit does not imply the existence of a Smale horseshoe of the same type on any cross section of these two points. This criterion is based on the existence of a hyperbolic periodic orbit, differing from the classical equilibrium-based Shilnikov criterion and the condition of transversal homoclinic or heteroclinic orbit of a Poincaré map.

Fractional-Order Discrete-Time SIR Epidemic Model with Vaccination: Chaos and Complexity

This research presents a new fractional-order discrete-time susceptible-infected-recovered (SIR) epidemic model with vaccination. The dynamical behavior of the suggested model is examined analytically and numerically. Through using phase attractors, bifurcation diagrams, maximum Lyapunov exponent and the 0−1 test, it is verified that the newly introduced fractional discrete SIR epidemic model vaccination with both commensurate and incommensurate fractional orders has chaotic behavior. The discrete fractional model gives more complex dynamics for incommensurate fractional orders compared to commensurate fractional orders. The reasonable range of commensurate fractional orders is between γ = 0.8712 and γ = 1, while the reasonable range of incommensurate fractional orders is between γ2 = 0.77 and γ2 = 1. Furthermore, the complexity analysis is performed using approximate entropy (ApEn) and C0 complexity to confirm the existence of chaos. Finally, simulations were carried out on MATLAB to verify the efficacy of the given findings.

Analysis of particulate component PM10 over residential and commercial area in urban city of Chennai

In order to study the spatial and temporal material components of pollution data of a urban city of Chennai, particulate component PM10 is analysed using Hurst and Lyapunov exponents to identify the nature of trend and existence of chaos. PM10 particles are produced both naturally and by human activities. The rise of urban lifestyle has contributed to a greater extent the exponential increase in the pollution and hence PM10 concentration also. Hence there is a urgent need to study the particles. In this paper PM10 data recorded in two monitoring stations Anna Nagar and Kilpauk located in the residential and commercial area of Chennai during the year 2013 is chosen to be the study area. The Hurst exponent values obtained by using rescaled range analysis method are 0.6693 and 0.5768 for PM10 Anna Nagar and Kilpauk respectively. It is observed from Hurst values that the area under study is predictable and has a particular trend and not random as it confirms persistent behaviour. A chaotic indicator namely Maximum Lyapunov Exponent method is used in this research, to examine if there is any existence of chaotic behavior in PM10 time series data. The maximum Lyapunov exponent values obtained are 3.0820 for Anna Nagar and 2.9019 for Kilpauk area. The significant positive Lyapunov exponent obtained is an indication that the PM10 concentrations is very sensitive to initial conditions and there is the presence of chaotic behavior in the PM10 time series data.

Laminar chaos in coupled time-delay systems

The possibility of the existence of laminar chaos in coupled time-delayed feedback systems is investigated. The cases of unidirectional and mutual coupling of time-delay systems are considered. It is shown for the first time that laminar chaos can exist not only in a system with a variable delay time, but also in a system with a constant delay time, if it is coupled with a system in the regime of laminar chaos. Keywords: Time-delay systems, Laminar chaos, Coupled oscillators, Synchronization

Entropy Production and Fractal Dimensions in Heavy Ion Nuclear Reaction at Intermediate Energies

The characteristics of the nonlinear dynamics in the Heavy Ion Collision (HIC) at intermediate energies have been studied by evaluating the productions of the Generalized Entropy (GE) and the Multifragmentation Entropy (ME) as well as the features of the information and fractal dimensions within the Isospin Quantum Molecular Dynamical Model compensated by the lattice methods. Results demonstrate from various views that the existence of deterministic chaos in the dynamical process of reaction.