Multiple attractors and crisis route to chaos in a model food-chain

An attempt has been made to identify the mechanism, which is responsible for the existence of chaos in narrow parameter range in a realistic ecological model food-chain. Analytical and numerical studies of a three species food-chain model similar to a situation likely to be seen in terrestrial ecosystems has been carried out. The study of the model food chain suggests that the existence of chaos in narrow parameter ranges is caused by the crisis-induced sudden death of chaotic attractors. Varying one of the critical parameters in its range while keeping all the others constant, one can monitor the changes in the dynamical behaviour of the system, thereby fixing the regimes in which the system exhibits chaotic dynamics. The computed bifurcation diagrams and basin boundary calculations indicate that crisis is the underlying factor which generates chaotic dynamics in this model food-chain. We investigate sudden qualitative changes in chaotic dynamical behaviour, which occur at a parameter value a₁=1.7804 at which the chaotic attractor destroyed by boundary crisis with an unstable periodic orbit created by the saddle-node bifurcation. Multiple attractors with riddled basins and fractal boundaries are also observed. If ecological systems of interacting species do indeed exhibit multiple attractors etc., the long term dynamics of such systems may undergo vast qualitative changes following epidemics or environmental catastrophes due to the system being pushed into the basin of a new attractor by the perturbation. Coupled with stochasticity, such complex behaviours may render such systems practically unpredictable.     

Heteroclinic and Periodic Cycles in a Perturbed Convection Model

Vivancos and Minzoni (New Choatic behaviour in a singularly perturbed model, preprint) proposed a singularly perturbed rotating convection system to model the Earth's dynamo process. Numerical simulation shows that the perturbed system is rich in chaotic and periodic solutions. In this paper, we show that if the perturbation is sufficiently small, the system can only have simple heteroclinic solutions and two types of periodic solutions near the simple heteroclinic solutions. One looks like a figure “Delta” and the other looks like a figure “Eight”. Due to the fast - slow characteristic of the system, the reduced slow system has a relay nonlinearity (“Asymptotic Method in Singularly Perturbed Systems,” Consultants Bureau, New York and London, 1994) - solutions to the slow system are continuous but their derivative changes abruptly at certain junction surfaces. We develop new types of Melnikov integral and Lyapunov-Schmidt reduction methods which are suitable to study heteroclinic and periodic solutions for systems with relay nonlinearity.     

Structure stability of maps with snap-back repellers in Banach spaces

In this paper, we study small perturbations of a class of chaotic discrete systems in Banach spaces induced by snap-back repellers. If a map has a regular and non-degenerate snap-back repeller, then it still has a regular and non-degenerate snap-back repeller under a sufficiently small perturbation. Consequently, the perturbed system is still chaotic in the sense of both Devaney and Li–Yorke as the original one. Furthermore, in order to study structural stability of maps with regular and non-degenerate snap-back repellers, we first discuss structural stability of strictly A-coupled-expanding maps in Banach spaces. Applying this result, we show that a map with a regular and non-degenerate snap-back repeller in a Banach space is C ¹ structurally stable on its chaotic invariant set.     

Evolving chaos: Identifying new attractors of the generalised Lorenz family

In a recent paper, we presented an intelligent evolutionary search technique through genetic programming (GP) for finding new analytical expressions of nonlinear dynamical systems, similar to the classical Lorenz attractor's which also exhibit chaotic behaviour in the phase space. In this paper, we extend our previous finding to explore yet another gallery of new chaotic attractors which are derived from the original Lorenz system of equations. Compared to the previous exploration with sinusoidal type transcendental nonlinearity, here we focus on only cross-product and higher-power type nonlinearities in the three state equations. We here report over 150 different structures of chaotic attractors along with their one set of parameter values, phase space dynamics and the Largest Lyapunov Exponents (LLE). The expressions of these new Lorenz-like nonlinear dynamical systems have been automatically evolved through multi-gene genetic programming (MGGP). In the past two decades, there have been many claims of designing new chaotic attractors as an incremental extension of the Lorenz family. We provide here a large family of chaotic systems whose structure closely resemble the original Lorenz system but with drastically different phase space dynamics. This advances the state of the art knowledge of discovering new chaotic systems which can find application in many real-world problems. This work may also find its archival value in future in the domain of new chaotic system discovery.     

A Verified Optimization Technique to Locate Chaotic Regions of Hénon Systems

We present a new verified optimization method to find regions for Hénon systems where the conditions of chaotic behaviour hold. The present paper provides a methodology to verify chaos for certain mappings and regions. We discuss first how to check the set theoretical conditions of a respective theorem in a reliable way by computer programs. Then we introduce optimization problems that provide a model to locate chaotic regions. We prove the correctness of the underlying checking algorithms and the optimization model. We have verified an earlier published chaotic region, and we also give new chaotic places located by the new technique.     

Traveling Wave Solutions In Coupled Chua'S Circuits,Part I: Periodic Solutions

We study a singularly perturbed system of partial di erential equations that models a one-dimensional array of coupled Chua's circuits. The PDE system is a natural generalization to the FitzHugh-Nagumo equation. In part I of the paper, we show that similar to the FitzHugh-Nagumo equation, the system has periodic traveling wave solutions formed alternatively by fast and slow flows. First, asymptotic method is used on the singular limit of the fast/slow systems to construct a formal periodic solution. Then, dynamical systems method is used to obtain an exact solution near the formal periodic soluion. In part II, we show that the system can have more complicated periodic and chaotic traveling wave solutions that do not exist in the FitzHugh-Nagumos equation.     

Chaotic dynamics of homogeneous Yang-Mills fields with two degrees of freedom

In the present paper, we consider a scenario of transition to chaotic dynamics in the Hamiltonian system of homogeneous Yang-Mills fields with two degrees of freedom in the case of the Higgs mechanism. We show that in such a system, as well as in other Hamiltonian and conservative systems of equations, the nonlocal effect of multiplication of hyperbolic and elliptic cycles and tori around elliptic cycles in neighborhoods of the separatrix surfaces of hyperbolic cycles plays a key role on the initial stage of transition from a regular motion to a chaotic one. We observe that the new elliptic and hyperbolic cycles of the Hamiltonian system are generated as stable and saddle cycles of the extended dissipative system of equations not only as a result of saddle-node bifurcations but also as a result of fork-type bifurcations.     

The Effects of Continuously Varying the Fractional Differential Order of Chaotic Nonlinear Systems

We consider nonlinear third order differential equations which are known to exhibit chaotic behaviour, and amend their order using fractional calculus techniques. By doing this we demonstrate that by continuously increasing the order of differentiation for those systems from 2 to 3, a period doubling route to chaos ensues. This period doubling begins at a system specific order value between 2 and 3.     

Suppressing grazing chaos in impacting system by structural nonlinearity

In this note we investigate the influence of structural nonlinearity of a simple cantilever beam impacting system on its dynamic responses close to grazing incidence by a means of numerical simulation. To obtain a clear picture of this effect we considered two systems exhibiting impacting motion, where the primary stiffness is either linear (piecewise linear system) or nonlinear (piecewise nonlinear system). Two systems were studied by constructing bifurcation diagrams, basins of attractions, Lyapunov exponents and parameter plots. In our analysis we focused on the grazing transitions from no impact to impact motion. We observed that the dynamic responses of these two similar systems are qualitatively different around the grazing transitions. For the piecewise linear system, we identified on the parameter space a considerable region with chaotic behaviour, while for the piecewise nonlinear system we found just periodic attractors. We postulate that the structural nonlinearity of the cantilever impacting beam suppresses chaos near grazing.     

On the fuzzy minimum entropy control to stabilize the unstable fixed points of chaotic maps

This paper presents a fuzzy algorithm for controlling chaos in nonlinear systems via minimum entropy approach. The proposed fuzzy logic algorithm is used to minimize the Shannon entropy of a chaotic dynamics. The fuzzy laws are determined in such a way that the entropy function descends until the chaotic trajectory of the system is replaced by a regular one. The Logistic and the Henon maps as two discrete chaotic systems, and the Duffing equation as a continuous one are used to validate the proposed scheme and show the effectiveness of the control method in chaotic dynamical systems.     

Diagnostics of the degree of noise influence on a nonlinear system using relative metric entropy

In this paper we summarize and substantiate the relative metric entropy approach introduced in our previous papers [1, 2]. Using this approach we study the mixing influence of noise on both regular and chaotic systems. We show that the synchronization phenomenon as well as stochastic resonance decrease, the degree of mixing is caused by white Gaussian noise.     

Estimates for correlations in billiards with large arcs

Bunimovich billiards are ergodic and mixing. However, if the billiard table contains very large arcs on its boundary then if there exist trajectories experience infinitely many collisions in the vicinity of periodic trajectories on the large arc. The hyperbolicity is nonuniform and the mixing rate is very slow. The corresponding dynamics are intermittent between regular and chaotic, which makes them particularly interesting in physical studies. The study of mixing rates in intermittent chaotic systems is mo...     

The chaos and control of a food chain model supplying additional food to top-predator

The control and management of chaotic population is one of the main objectives for constructing mathematical model in ecology today. In this paper, we apply a technique of controlling chaotic predator–prey population dynamics by supplying additional food to top-predator. We formulate a three species predator–prey model supplying additional food to top-predator. Existence conditions and local stability criteria of equilibrium points are determined analytically. Persistence conditions for the system are derived. Global stability conditions of interior equilibrium point is calculated. Theoretical results are verified through numerical simulations. Phase diagram is presented for various quality and quantity of additional food. One parameter bifurcation analysis is done with respect to quality and quantity of additional food separately keeping one of them fixed. Using MATCONT package, we derive the bifurcation scenarios when both the parameters quality and quantity of additional food vary together. We predict the existence of Hopf point (H), limit point (LP) and branch point (BP) in the model for suitable supply of additional food. We have computed the regions of different dynamical behaviour in the quantity–quality parametric plane. From our study we conclude that chaotic population dynamics of predator prey system can be controlled to obtain regular population dynamics only by supplying additional food to top predator. This study is aimed to introduce a new non-chemical chaos control mechanism in a predator–prey system with the applications in fishery management and biological conservation of prey predator species.     

Running waves and dynamical chaos in active media: A numerical study

In the framework of a self-similar problem, we numerically analyze the onset of solutions in the form of running waves in excitable and oscillating active media. We show that the passage to the solution in the form of a running wave occurs through cascades of bifurcations responsible for the development of chaotic dynamics in nonlinear dissipative systems of differential equations.     

Routes to chaos in continuous mechanical systems: Part 2. Modelling transitions from regular to chaotic dynamics

In second part of the paper both classical and novel scenarios of transition from regular to chaotic dynamics of dissipative continuous mechanical systems are studied. A detailed analysis allowed us to detect the already known classical scenarios of transition from periodic to chaotic dynamics, and in particular the Feigenbaum scenario. The Feigenbaum constant was computed for all continuous mechanical objects studied in the first part of the paper. In addition, we illustrate and discuss different and novel scenarios of transition of the analysed systems from regular to chaotic dynamics, and we show that the type of scenario depends essentially on excitation parameters.     

Nonadiabatic couplings and chaos in a dynamical potential well model

The mechanism of nonadiabatic couplings between quantum states of a potential well model with finite heights and a dynamical width coordinate is investigated in detail. The system is described in a mixed quantum-classical approach in which the oscillations of the classical width coordinate induce transitions between the quantum states of a particle trapped inside the well. The dynamics of the system is considered in detail for transitions between two quantum states and resulting coupled Bloch-oscillator equations. Poincaré sections showing a mixed phase space with chaotic and regular behaviour are found by a numerical investigation. In particular, chaos results for high energies of the well width oscillations when the mixing between the adiabatic reference states is strong. The inclusion of relaxation is considered and shown that in this case the regimes of chaotic and regular dynamics are not separated as in the relaxation free case. In particular, for some initial conditions chaos can become a transient phenomena placed in a time window between regular oscillations of the system.     

Resonance dynamics of nonlinear flutter systems

We consider a special class of nonlinear systems of ordinary differential equations, namely, the so-called flutter systems, which arise in Galerkin approximations of certain boundary value problems of nonlinear aeroelasticity and in a number of radiophysical applications. Under the assumption of small damping coefficient, we study the attractors of a flutter system that arise in a small neighborhood of the zero equilibrium state as a result of interaction between the 1: 1 and 1: 2 resonances. We find that, first, these attractorsmay be both regular and chaotic (in the latter case, we naturally deal with numerical results); and second, for certain parameter values, they coexist with the stable zero solution; i.e., the phenomenon of hard excitation of self-oscillations is observed.     

A model proposal for the chaotic structure of Istanbul stock exchange

Chaos theory is considered a novel way of understanding the behaviour of nonlinear dynamic systems. It is well known that the evaluation of chaotic systems is dependent on initial conditions since exponential growth error is a common characteristic. This present paper evaluates the effects of a nonlinear dynamic system in Istanbul Stock Exchange, based on time series. The reliability of predicting stock behaviour depends on this fact. In other words, the aim is to prove that if ISE daily index return shows chaotic behaviour.     

Nature of chaos in conservative and dissipative systems of the Duffing-Holmes oscillator

We show that the chaotic dynamics of the conservative Duffing-Holmes oscillator obeys the universal Feigenbaum-Sharkovskii-Magnitskii theory of passage to chaos in dynamical systems of ordinary differential equations. Moreover, the cascades of bifurcations of the conservative and dissipative oscillators are continuously related to each other. Our study uses the stable control method, which permits rapidly stabilizing nearly any periodic solution and dynamically changing system parameters without moving far away from that periodic solution.