Strange nonchaotic attractors in a fifth-order amplitude equation of Rayleigh–Bénard system near the codimension-two point

We consider a fifth-order amplitude equation for a codimension-two bifurcation point in the presence of a periodically modulated Rayleigh number. It is found, by analysis of Poincaré surfaces and a construction of the bifurcation diagram, that the system exhibits strange nonchaotic behaviour close to the codimension-two point. The Lyapunov exponents associated with these trajectories are calculated using a new method that exploits the underlying symplectic structure of Hamiltonian dynamics.     

Autonomous strange nonchaotic oscillations in a system of mechanical rotators

We investigate strange nonchaotic self-oscillations in a dissipative system consisting of three mechanical rotators driven by a constant torque applied to one of them. The external driving is nonoscillatory; the incommensurable frequency ratio in vibrational-rotational dynamics arises due to an irrational ratio of diameters of the rotating elements involved. It is shown that, when losing stable equilibrium, the system can demonstrate two- or three-frequency quasi-periodic, chaotic and strange nonchaotic self-oscillations. The conclusions of the work are confirmed by numerical calculations of Lyapunov exponents, fractal dimensions, spectral analysis, and by special methods of detection of a strange nonchaotic attractor (SNA): phase sensitivity and analysis using rational approximation for the frequency ratio. In particular, SNA possesses a zero value of the largest Lyapunov exponent (and negative values of the other exponents), a capacitive dimension close to 2 and a singular continuous power spectrum. In general, the results of this work shed a new light on the occurrence of strange nonchaotic dynamics.     

Strange nonchaotic attractors in a fifth-order amplitude equation of Rayleigh–Bénard system near the codimension-two point

We consider a fifth-order amplitude equation for a codimension-two bifurcation point in the presence of a periodically modulated Rayleigh number. It is found, by analysis of Poincaré surfaces and a construction of the bifurcation diagram, that the system exhibits strange nonchaotic behavior close to the codimension-two point. The Lyapunov exponents associated with these trajectories are calculated using a new method that exploits the underlying symplectic structure of Hamiltonian dynamics.     

Application of multistage homotopy-perturbation method for the solutions of the Chen system

In this paper, a new reliable algorithm based on an adaptation of the standard homotopy-perturbation method (HPM) is applied to the Chen system which is a three-dimensional system of ODEs with quadratic nonlinearities. The HPM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions to the Chen system. We shall call this technique as the multistage HPM (for short MHPM). In particular we look at the accuracy of the HPM as the Chen system changes from a nonchaotic system to a chaotic one. Numerical comparisons between the MHPM and the classical fourth-order Runge–Kutta (RK4) numerical solutions reveal that the new technique is a promising tool for the nonlinear chaotic and nonchaotic systems of ODEs.     

Feedback control of bursting and multistability in chaotic systems

With a modulated CO₂ laser as a standard model of periodically driven multistable systems, we experimentally demonstrate that a small-amplitude optoelectronic feedback perturbation can efficiently transform a bursting chaotic system to a nonchaotic one. Numerical simulations are in excellent agreement with the experimental results. The control has also been equally effective in the case of a driven FitzHugh-Nagumo model of Neuroscience.     

Dynamics of a two-frequency parametrically driven duffing oscillator

Summary We investigate the transition from two-frequency quasiperiodicity to chaotic behavior in a model for a quasiperiodically driven magnetoelastic ribbon. The model system is a two-frequency parametrically driven Duffing oscillator. As a driving parameter is increased, the route to chaos takes place in four distinct stages. The first stage is a torus-doubling bifurcation. The second stage is a transition from the doubled torus to a strange nonchaotic attractor. The third stage is a transition from the strange nonchaotic attractor to a geometrically similar chaotic attractor. The final stage is a hard transition to a much larger chaotic attractor. This latter transition arises as the result of acrisis, the characterization of which is one of our primary concerns. Numerical evidence is given to indicate that the crisis arises from the collision of the chaotic attractor with the stable manifold of a saddle torus. Intermittent bursting behavior is present after the crisis with the mean time between bursts scaling as a power law in the distance from the critical control parameter; τ ∼ (A-A_c)^-α. The critical exponent is computed numerically, yielding the value α=1.03±0.01. Theoretical justification is given for the computed critical exponent. Finally, a Melnikov analysis is performed, yielding an expression for transverse crossings of the stable and unstable manifolds of the crisis-initiating saddle torus.     

Some characteristics of complex behavior of orbits in dynamical systems

We present a brief review of mathematical notions of complexity based on instability of orbits. We show that the complexity as a function of time may grow exponentially in chaotic situations or polynomially for systems with zero topological entropy. At the end we discuss the class of nonchaotic systems for which all orbits are stable but nevertheless behavior of orbits is complex. We introduce a new notion of complexity for such a kind of systems.     

Transient and asymptotic dynamics of a prey–predator system with diffusion

In this paper, we study a prey–predator system associated with the classical Lotka–Volterra nonlinearity. We show that the dynamics of the system are controlled by the ODE part. First, we show that the solution becomes spatially homogeneous and is subject to the ODE part as t → ∞ . Next, we take the shadow system to approximate the original system as D → ∞ . The asymptotics of the shadow system are also controlled by those of the ODE. The transient dynamics of the original system approaches to the dynamics of its ODE part with the initial mean as D → ∞ . Although the asymptotic dynamics of the original system are also controlled by the ODE, the time periods of these ODE solutions may be different. Concerning this property, we have that any δ > 0 admits D₀ > 0 such that if , the time period of the ODE, satisfies , then the solution to the original system with D ≥ D₀ cannot approach the stationary state. Copyright © 2012 John Wiley & Sons, Ltd.     

Horseshoes in chromosome’s attractors

In this paper, we study dynamics of a class of chromosome’s attractors. We show that these chromosome sequences are chaotic by giving a rigorous verification for existence of horseshoes in these systems. We prove that the Poincaré maps derived from these chromosome’s attractors are semi-conjugate to the 2-shift map, and its entropy is no less than log 2. The chaotic behavior is robust in the following sense: chaos exists when one parameter varies from −5.5148 to −5.4988.     

Dynamics of particle trajectories in a Rayleigh–Bénard problem

Fluid particle trajectories for the Rayleigh–Bénard problem in a cube with perfectly conducting lateral walls have been investigated. The velocity and temperature fields of the stationary flow solutions have been obtained by means of a parameter continuation procedure based on a Galerkin spectral method. The rich dynamics of the resulting fluid particle paths has been studied for three branches of stationary solutions and different values of the Rayleigh number within the range10⁴Ra1.5×10⁵ at a Prandtl number equal to 130. The stability properties and bifurcations of fixed points, which play a key role in the global dynamics, have been analyzed. Main periodic orbits and their stability character have also been determined. Poincaré maps reveal that regions of chaotic motion and regions of regular motion coexist inside the cavity. The boundaries of these three-dimensional regions have been determined. The metric entropy gives an indication of the mixing properties of the large chaotic zone.     

Statistics of Poincaré recurrences for a class of smooth circle maps

Statistics of Poincaré recurrence for a class of circle maps, including sub-critical, critical, and super-critical cases, are studied. It is shown how the topological differences in the various types of the dynamics are manifested in the statistics of the return times.     

Different faces of chaos in FRW models with scalar fields—geometrical point of view

FRW cosmologies with conformally coupled scalar fields are investigated in a geometrical way by the means of geodesics of the Jacobi metric. In this model of dynamics, trajectories in the configuration space are represented by geodesics. Because of the singular nature of the Jacobi metric on the boundary set of the domain of admissible motion, the geodesics change the cone sectors several times (or an infinite number of times) in the neighborhood of the singular set .

We show that this singular set contains interesting information about the dynamical complexity of the model. Firstly, this set can be used as a Poincaré surface for construction of Poincaré sections, and the trajectories then have the recurrence property. We also investigate the distribution of the intersection points. Secondly, the full classification of periodic orbits in the configuration space is performed and existence of UPO is demonstrated. Our general conclusion is that, although the presented model leads to several complications, like divergence of curvature invariants as a measure of sensitive dependence on initial conditions, some global results can be obtained and some additional physical insight is gained from using the conformal Jacobi metric. We also study the complex behavior of trajectories in terms of symbolic dynamics.     

Evolutionary dynamics in a Lotka–Volterra competition model with impulsive periodic disturbance

In this paper, we develop a theoretical framework to investigate the influence of impulsive periodic disturbance on the evolutionary dynamics of a continuous trait, such as body size, in a general Lotka–Volterra‐type competition model. The model is formulated as a system of impulsive differential equations. First, we derive analytically the fitness function of a mutant invading the resident populations when rare in both monomorphic and dimorphic populations. Second, we apply the fitness function to a specific system of asymmetric competition under size‐selective harvesting and investigate the conditions for evolutionarily stable strategy and evolutionary branching by means of critical function analysis. Finally, we perform long‐term simulation of evolutionary dynamics to demonstrate the emergence of high‐level polymorphism. Our analytical results show that large harvesting effort or small impulsive harvesting period inhibits branching, while large impulsive harvesting period promotes branching. Copyright © 2015 John Wiley & Sons, Ltd.     

The general equations of analytical dynamics

It is shown that the generalized Poincaré and Chetayev equations, which represent the equations of motion of mechanical systems using a certain closed system of infinitesimal linear operators, are related to the fundamental equations of analytical dynamics. Equations are derived in quasi-coordinates for the case of redundant variables; it is shown that when an energy integral exists the operator X₀ = ∂/∂t satisfies the Chetayev cyclic-displacement conditions. Using the energy integral the order of the system of equations of motion is reduced, and generalized Jacobi-Whittaker equations are derived from the Chetayev equations. It is shown that the Poincaré-Chetayev equations are equivalent to a number of equations of motion of non-holonomic systems, in particular, the Maggi, Volterra, Kane, and so on, equations. On the basis of these, and also of other previously obtained results, the Poincaré and Chetayev equations in redundant variables, applicable both to holonomic and non-holonomic systems, can be regarded as general equations of classical dynamics, equivalent to the well-known fundamental forms of the equations of motion, a number of which follow as special cases from the Poincaré and Chetayev equations.     

Neimark–Sacker bifurcation for the discrete-delay Kaldor–Kalecki model

Results of an extensive numerical study of the influence of additive white noise on the dynamics of a pair of delayed coupled FitzHugh–Nagumo neurons are presented. An intuitively clear simple method is utilized to predict the critical intensities of the noise. In general, the qualitative properties of the noiseless dynamics are stable under the influence of the noise of a reasonable magnitude. However, there are regions of the coupling and time-lag parameters where the noise of a physically acceptable magnitude does cause qualitative changes of the dynamics. These regions are studied in detail.     

Dynamic complexities in predator–prey ecosystem models with age-structure for predator

Natural populations, whose generations are non-overlapping, can be modelled by difference equations that describe how the populations evolve in discrete time-steps. In the 1970s ecological research detected chaos and other forms of complex dynamics in simple population dynamics models, initiating a new research tradition in ecology. However, in former studies most of the investigations of complex population dynamics were mainly concentrated on single populations instead of higher dimensional ecological systems. This paper reports a recent study on the complicated dynamics occurring in a class of discrete-time models of predator–prey interaction based on age-structure of predator. The complexities include (a) non-unique dynamics, meaning that several attractors coexist; (b) antimonotonicity; (c) basins of attraction (defined as the set of the initial conditions leading to a certain type of an attractor) with fractal properties, consisting of pattern of self-similarity and fractal basin boundaries; (d) intermittency; (e) supertransients; and (f) chaotic attractors.     

Zero‐zero‐Hopf bifurcation and ultimate bound estimation of a generalized Lorenz–Stenflo hyperchaotic system

This paper is devoted to the analysis of complex dynamics of a generalized Lorenz–Stenflo hyperchaotic system. First, on the local dynamics, the bifurcation of periodic solutions at the zero‐zero‐Hopf equilibrium (that is, an isolated equilibrium with double zero eigenvalues and a pair of purely imaginary eigenvalues) of this hyperchaotic system is investigated, and the sufficient conditions, which insure that two periodic solutions will bifurcate from the bifurcation point, are obtained. Furthermore, on the global dynamics, the explicit ultimate bound sets of this hyperchaotic system are obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

Nonlinear dynamics and solitons in the presence of rapidly varying periodic perturbations

Problems of nonlinear dynamics and soliton propagation in the presence of rapidly varying periodic perturbations are considered applying a rigorous analytical approach based on asymptotic expansions. The method we develop allows derivation of an effective nonlinear equation for the slowly varying field component in any order of the asymptotic procedure as expansions in the parameter ω⁻¹, ω being the frequency of the rapidly varying (direct or parametric) driving force. The general approach is demonstrated on several examples of different physical nature, including chaos suppression in the parametrically driven Duffing oscillator, dynamics of the sine-Gordon kinks in the presence of rapidly varying direct or parametric driving force, propagation of envelope (nonlinear Schrödinger) solitons in optical fibres with periodic amplification, stability of solitons on rapidly varying spatial periodic potential, and so on.     

SUSTAINABLE YIELDS IN FISHERIES: UNCERTAINTY,RISK‐AVERSION,AND MEAN‐VARIANCE ANALYSIS

Abstract We consider a model of a fishery in which the dynamics of the unharvested fish population are given by the stochastic logistic growth equation Similar to the classical deterministic analogon, we assume that the fishery harvests the fish population following a constant effort strategy. In the first step, we derive the effort level that leads to maximum expected sustainable yield, which is understood as the expectation of the equilibrium distribution of the stochastic dynamics. This replaces the nonzero fixed point in the classical deterministic setup. In the second step, we assume that the fishery is risk averse and that there is a tradeoff between expected sustainable yield and uncertainty measured in terms of the variance of the equilibrium distribution. We derive the optimal constant effort harvesting strategy for this problem. In the final step, we consider an approach that we call the mean‐variance analysis to sustainable fisheries. Similar as in the now classical mean‐variance analysis in finance, going back to Markowitz [1952] , we study the problem of maximizing expected sustainable yields under variance constraints, and with this, minimizing the variance, e.g., risk, under guaranteed minimum expected sustainable yields. We derive explicit formulas for the optimal fishing effort in all four problems considered and study the effects of uncertainty, risk aversion, and mean reversion speed on fishing efforts.