Stochastic chaos in a Duffing oscillator and its control

Stochastic chaos discussed here means a kind of chaotic responses in a Duffing oscillator with bounded random parameters under harmonic excitations. A system with random parameters is usually called a stochastic system. The modifier ‘stochastic’ here implies dependent on some random parameter. As the system itself is stochastic, so is the response, even under harmonic excitations alone. In this paper stochastic chaos and its control are verified by the top Lyapunov exponent of the system. A non-feedback control strategy is adopted here by adding an adjustable noisy phase to the harmonic excitation, so that the control can be realized by adjusting the noise level. It is found that by this control strategy stochastic chaos can be tamed down to the small neighborhood of a periodic trajectory or an equilibrium state. In the analysis the stochastic Duffing oscillator is first transformed into an equivalent deterministic nonlinear system by the Gegenbauer polynomial approximation, so that the problem of controlling stochastic chaos can be reduced into the problem of controlling deterministic chaos in the equivalent system. Then the top Lyapunov exponent of the equivalent system is obtained by Wolf’s method to examine the chaotic behavior of the response. Numerical simulations show that the random phase control strategy is an effective way to control stochastic chaos.     

The chaos game revisited: Yet another, but a trivial proof of the algorithm’s correctness

The aim of this work is to explain why the most popular algorithm for approximating IFS fractals, the chaos game, works. Although there are a few proofs of the algorithm’s correctness in the relevant literature, the majority of them utilize notions and theorems of measure and ergodic theories. As a result, paradoxically, although the rules of the chaos game are very simple, the logic underlying the algorithm seems to be hard to comprehend for non-mathematicians. In contrast, the proof presented in this work uses only fundamentals of probability and can be understood by anyone interested in fractals.     

Suppressing chaos of a complex Duffing's system using a random phase

The effect of random phase for a complex Duffing's system is investigated. We show 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. Also Poincaré map analysis, phase plot and the time evolution are carried out to confirm the obtained results of Lyapunov exponent on dynamical behavior including the stability, bifurcation and chaos. Thus excellent agreement between these results is found.     

Effects of random noise in a dynamical model of love

This paper aims to investigate the stochastic model of love and the effects of random noise. We first revisit the deterministic model of love and some basic properties are presented such as: symmetry, dissipation, fixed points (equilibrium), chaotic behaviors and chaotic attractors. Then we construct a stochastic love-triangle model with parametric random excitation due to the complexity and unpredictability of the psychological system, where the randomness is modeled as the standard Gaussian noise. Stochastic dynamics under different three cases of “Romeo’s romantic style”, are examined and two kinds of bifurcations versus the noise intensity parameter are observed by the criteria of changes of top Lyapunov exponent and shape of stationary probability density function (PDF) respectively. The phase portraits and time history are carried out to verify the proposed results, and the good agreement can be found. And also the dual roles of the random noise, namely suppressing and inducing chaos are revealed.     

Messing about in transformations: Structured systemic planning for systemic solutions to systemic problems

The literature on multimethodology indicates that cognitive mapping is relevant to enriching the preliminary, information-gathering phase of Soft Systems Methodology (SSM), and especially the rich picture. By noting the structural complementarity evidenced between SSM’s transformations and the bipolar constructs used in the cognitive mapping methodology known as Strategic Options Development and Analysis (SODA), this paper shows how SODA can be applied beyond SSM’s initial stage, and much more analytically within the heart of SSM, to guide the manner in which a systemic resolution to a problematic situation may be approached. It is proposed, and illustrated through examples, that a SODA map of large numbers of transformations, resulting from an exercise in SSM, offers a methodological means for structuring what might otherwise be perceived as a mess. The paper discusses how in such cases ‘strategic options development and analysis’ of transformations, or SODA-T maps, serve, among other things, to identify relations between transformations, their hierarchies and priorities, problem epicenters and starting points for intervention. In addition, the arsenal of graph theory can be used to cut through what would otherwise appear as interlinked chaos requiring structured operationalization. In this respect, the SODA-T map offers a high-level connective orientation which can guide the interconnections between the respective human activity systems of the transformations, resulting in the final systemic plan. Ultimately, SODA-T mapping is shown to facilitate a structured approach toward systemic planning.     

A new method to control chaos in an economic system

In this paper, the method to control chaos by using phase space compression is applied to economic systems. Because of economic significance of state variable in economic dynamical systems, the values of state variables are positive due to capacity constraints and financial constraints, we can control chaos by adding upper bound or lower bound to state variables in economic dynamical systems, which is different from the chaos stabilization in engineering or physics systems. The knowledge about system dynamics and the exact variety of parameters are not needed in the application of this control method, so it is very convenient to apply this method. Two kinds of chaos in the dynamic duopoly output systems are stabilized in a neighborhood of an unstable fixed point by using the chaos controlling method. The results show that performance of the system is improved by controlling chaos. In practice, owing to capacity constraints, financial constraints and cautious responses to uncertainty in the world, the firm often restrains the output, advertisement expenses, research cost etc. to confine the range of these variables’ fluctuation. This shows that the decision maker uses this method unconsciously in practice.     

Chaos control by harmonic excitation with proper random phase

Chaos control may have a dual function: to suppress chaos or to generate it. We are interested in a kind of chaos control by exerting a weak harmonic excitation with random phase. The dual function of chaos control in a nonlinear dynamic system, whether a suppressing one or a generating one, can be realized by properly adjusting the level of random phase and determined by the sign of the top Lyapunov exponent of the system response. Two illustrative examples, a Duffing oscillator subject to a harmonic parametric control and a driven Murali-Lakshmanan-Chua (MLC) circuit imposed with a weak harmonic control, are presented here to show that the random phase plays a decisive role for control function. The method for computing the top Lyapunov exponent is based on Khasminskii's formulation for linearized systems. Then, the obtained results are further verified by the Poincare map analysis on dynamical behavior of the system, such as stability, bifurcation and chaos. Both two methods lead to fully consistent results.     

Spiking patterns of a minimal neuron to ELF sinusoidal electric field

Neuron as the main information carrier in neural systems is able to generate diverse spiking trains in response to different stimuli. Neuronal spiking patterns are related to the information processing in neural system. This paper investigates the dynamical behaviors of a two-dimensional minimal neuron model exposed to externally-applied extremely low frequency (ELF) sinusoidal electric field (EF). By numerical stimulation, it is found that neuron can exhibit different spiking patterns such as p:q mode-locking (i.e. a periodic oscillation defined as p action potentials generated by q cycle stimulations) and chaotic behaviors, depending on the values of stimulus frequencies. Transitions between different spiking patterns during exposure to the external EF are explored by interspike intervals (ISIs) and average firing rate. It is found that frequencies of the external EF can act as a bifurcation parameter, whose small change can cause the transition in neuronal behaviors. It is shown that a rich bifurcation structure including period-adding without chaos and mode-locking alternated with chaos suggests frequency discrimination of the neuronal firing patterns. Our results can provide a useful insight into the organization of similar bifurcation structures in excitable systems such as neurons under periodic forcing. Through detail analysis of the spiking patterns, we have explained how neuron’s information (spiking patterns) encodes the stimulus information (frequency), and vice versa.     

Chaos control in duffing system

Analytical and numerical results concerning the inhibition of chaos in Duffing’s equation with two weak forcing excitations are presented. We theoretically give parameter-space regions by using Melnikov’s function, where chaotic states can be suppressed. The intervals of initial phase difference between the two excitations for which chaotic dynamics can be eliminated are given. Meanwhile, the influence of the phase difference on Lyapunov exponents for different frequencies is investigated. Numerical simulation results show the consistence with the theoretical analysis and the chaotic motions can be controlled to period-motions by adjusting parameter of suppressing excitation.     

Complexity analysis of dynamic noncooperative game models for closed-loop supply chain with product recovery

In this paper, we consider a closed-loop supply chain (CLSC) with product recovery, which is composed of one manufacturer and one retailer. The retailer is in charge of recollecting and the manufacturer is responsible for product recovery. The system can be regarded as a coupling dynamics of the forward and reverse supply chain. Under different decision criteria, two noncooperative game models: Stackelberg game model and peer-to-peer game model are developed. The dynamic phenomena, such as the bifurcation, chaos and sensitivity to initial values are analyzed through bifurcation diagrams and the largest Lyapunov exponent (LLE). The influences of decision parameters on the complex nonlinear dynamics behaviors of the two models are further analyzed by comparing parameter basin plots, and the results show that with the improvement of retailer’s competitive position, the CLSC system will be more easier to enter into chaos.     

Uniform asymptotic methods for integrals

We give an overview of basic methods that can be used for obtaining asymptotic expansions of integrals: Watson’s lemma, Laplace’s method, the saddle point method, and the method of stationary phase. Certain developments in the field of asymptotic analysis will be compared with De Bruijn’s book Asymptotic Methods in Analysis. The classical methods can be modified for obtaining expansions that hold uniformly with respect to additional parameters. We give an overview of examples in which special functions, such as the complementary error function, Airy functions, and Bessel functions, are used as approximations in uniform asymptotic expansions.     

Analysis of period-doubling and chaos of a non-symmetric oscillator with piecewise-linearity

This paper presents an analysis of the dynamical behaviour of a non-symmetric oscillator with piecewise-linearity. The Chen–Langford (C–L) method is used to obtain the averaged system of the oscillator. Using this method, the local bifurcation and the stability of the steady-state solutions are studied. A Runge–Kutta method, Poincaré map and the largest Lyapunov’s exponent are used to detect the complex dynamical phenomena of the system. It is found that the system with piecewise-linearity exhibits periodic oscillations, period-doubling, period-3 solution and then chaos. When chaos is found, it is detected by examining the phase plane, bifurcation diagram and the largest Lyapunov’s exponent. The results obtained in this paper show that the vibration system with piecewise-linearity do exhibit quite similar dynamical behaviour to the discrete system given by the logistic map.     

Dominance of statistical fluctuation in the factorial-moment study of chaos in low multiplicity events of high energy collisions

Using Monte Carlo simulation it is shown that in low multiplicity events the single-event factorial moments are saturated by the statistical fluctuations. The diversification of the event-space momentsC _{p, q} of single-event moments with the diminishing of phase space scale, called “erraticity”, observed in experiment can readily be reproduced by a flat probability distribution with only statistical fluctuations and therefore it has nothing to do with chaos as suggested. The possibility of studying chaos in high multiplicity events using erraticity analysis is discussed.     

On the classical limit of quantum mechanics,fundamental graininess and chaos: Compatibility of chaos with the correspondence principle

The aim of this paper is to review the classical limit of Quantum Mechanics and to precise the well known threat of chaos (and fundamental graininess) to the correspondence principle. We will introduce a formalism for this classical limit that allows us to find the surfaces defined by the constants of the motion in phase space. Then in the integrable case we will find the classical trajectories, and in the non-integrable one the fact that regular initial cells become “amoeboid-like”. This deformations and their consequences can be considered as a threat to the correspondence principle unless we take into account the characteristic timescales of quantum chaos. Essentially we present an analysis of the problem similar to the one of Omnès (1994,1999), but with a simpler mathematical structure.     

基于相空间重构技术的金融系统混沌识别

借助工程技术领域内识别非线性系统混沌现象的相空间重构技术研究了金融混沌的识别问题.以我国金融系统历经本轮全球金融危机这个金融史上影响最为严重的金融混沌为研究对象;研究发现:在全球金融危机的冲击下,我国金融系统在运行过程中发生了确定性的失稳,出现了金融混沌现象.从而为进一步防范与控制金融混沌奠定基础.     

Avoiding non-optimal management decisions by applying a three-person inspection game

An inspection game models a conflict situation between an inspector and an inspectee. The mathematical analysis aims to generate optimal behavior of the inspectee under the assumption that an undesirable action of the inspectee could otherwise be carried out strategically. In this paper the controller’s (inspector’s) particular job is to audit a manager’s (inspectee’s) decision and to submit a report to the company’s top managers for examination. Thus, a conflict as regards the choice of behavioral actions of the manager, the controller and the top management impends. Based on Fandel and Trockel (2011a) this modified inspection game is discussed here for the first time as a three-person game in the context of a manager’s faulty decision that will unnecessarily add to the company’s costs and that the top management understandably wishes to minimize. We will first examine the conditions under which a Nash equilibrium occurs in this three-person game in which poor management, poor monitoring and poor revision coincide. We will then examine the effects that the penalties and bonuses exert on the Nash equilibrium solution. We will find that penalties and bonuses can neutralize each other in their effects on the improved decision making by the manager and the controller.     

Hopf bifurcation and topological horseshoe of a novel finance chaotic system

This paper deals with the existence of both Hopf bifurcation and topological horseshoe for a novel finance chaotic system. First, through rigorous mathematical analysis, we show that a Hopf bifurcation occurs at systems’ three equilibriums S_0,1,2 and Hopf bifurcation at equilibrium S₀ is non-degenerate and supercritical. Second, the computer-assisted verifications for horseshoe chaos in the system are given. Simulation results are presented to support the analysis.     

Reliability of Coupled Oscillators

We study the reliability of phase oscillator networks in response to fluctuating inputs. Reliability means that an input elicits essentially identical responses upon repeated presentations, regardless of the network’s initial condition. Single oscillators are well known to be reliable. We show in this paper that unreliable behavior can occur in a network as small as a coupled oscillator pair in which the signal is received by the first oscillator and relayed to the second with feedback. A geometric explanation based on shear-induced chaos at the onset of phase-locking is proposed. We treat larger networks as decomposed into modules connected by acyclic graphs, and give a mathematical analysis of the acyclic parts. Moreover, for networks in this class, we show how the source of unreliability can be localized, and address questions concerning downstream propagation of unreliability once it is produced.     

Chaos synchronization for master slave piecewise linear systems: Application to Chua’s circuit

This paper deals with chaos synchronization for master slave piecewise linear systems. The synchronization problem is formulated as a global stability problem of error synchronization dynamics. New sufficient conditions are provided using a Lyapunov approach and the so-called S-procedure. We show that the synchronization problem can be solved as an optimization problem subject to a set of Linear Matrix Inequalities (LMI) for which a state feedback controller is designed efficiently. The effectiveness of the proposed solution is verified via simulation results using the original Chua’s circuit model. Furthermore, it will be proven that the new sufficient conditions relaxed the conservatism of previous existing works.     

Bifurcations and chaos of a discrete mathematical model for respiratory process in bacterial culture

The discrete mathematical model for the respiratory process in bacterial culture obtained by Euler method is investigated. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory, condition of existence of chaos in the sense of Marotto＇s definition of chaos is proved. The bifurcation diagrams, Lyapunov exponents and phase portraits are given for different parameters of the model, and the fractal dimension of chaotic attractor was also calculated. The numerical simulation results confirm the theoretical analysis and also display the new and complex dynamical behaviors compared with the continuous model. In particular~ we found that the new chaotic attractor, and new types of two or four coexisting chaotic attractors, and two coexisting invariant torus.