Dynamics of an ecological model living on the edge of chaos

We present a new ecological model, which displays “edge of chaos” (EoC) in parameter space. This suggests that ecological systems are not chaotic, instead, their dynamics can be characterized as short-term recurrent chaos. The system’s dynamics is unpredictable and admits bursts of short-term predictability. We also provide results, which suggest that fully developed chaos will rarely be observed in natural systems.     

An analysis of the impact of chaotic dynamics on management information flow models

Defined very broadly, Chaos Theory is the study of the behavior of dynamic, nonlinear, feedback equations which, with certain parameters, produce random-appearing output, although all parts of the equation system are deterministic. In this research we use the insights provided by the study of Chaos Theory to investigate how chaos can impact management dynamics and thus influence managerial decision-making.It is common to use dynamic mathematical models as aids to management. If model formulation is such that the model produces chaotic output under certain circumstances, decisions based on the use of that model are seriously compromised. Further, when several models are used concurrently, the interactions between them may cause output to be chaotic even if no individual model exhibits such behavior. We provide an explanation of the reasons why this may happen, and illustrate the consequences through an example.     

Nonlinear dynamics of regenerative cutting processes—Comparison of two models

Understanding the nonlinear dynamics of cutting processes is essential for the improvement of machining technology. We study machine cutting processes by two different models, one has been recently introduced by Litak [Litak G. Chaotic vibrations in a regenerative cutting process. Chaos, Solitons & Fractals 2002;13:1531–5] and the other is the classic delay differential equation model. Although chaotic solutions have been found in both models, well known routes to chaos, such as period-doubling or quasi-periodic motion to chaos are not observed in either model. Careful analysis shows that the chaotic motion from the Litak’s model has sharper spectral peaks, a smaller correlation dimension and a smaller value for the largest positive Lyapunov exponent. Implications to the control of chaos in cutting processes are discussed.     

Interpreting supply chain dynamics: A quasi-chaos perspective

Chaotic phenomena, chaos amplification and other interesting nonlinear behaviors have been observed in supply chain systems. Chaos can be defined theoretically if the dynamics under study are produced only by deterministic factors. However, deterministic settings rarely present themselves in reality. In fact, real data are typically unknown. How can the chaos theory and its related methodology be applied in the real world? When the demand is stochastic, the interpretation and distribution of the Lyapunov exponents derived from the effective inventory at different supply chain levels are not similar to those under deterministic demand settings. Are the observed dynamics of the effective inventory random, chaotic, or simply quasi-chaos? In this study, we investigate a situation whereby the chaos analysis is applied to a time series as if its underlying structure, deterministic or stochastic, is unknown. The result shows clear distinction in chaos characterization between the two categories of demand process, deterministic vs. stochastic. It also highlights the complexity of the interplay between stochastic demand processes and nonlinear dynamics. Therefore, caution should be exercised in interpreting system dynamics when applying chaos analysis to a system of unknown underlying structure. By understanding this delicate interplay, decision makers have the better chance to tackle the problem correctly or more effectively at the demand end or the supply end.     

Dynamical consequences of predator interference in a tri-trophic model food chain

A model food chain involving a specialist and a generalist predator is proposed and studied. One of the salient features of this model food chain is that it combines both the schemes (Volterra and Leslie) of modeling predator–prey interaction in one system in such a way that the demerits of these individual formulations are suppressed and the resulting model system represents a common unit of real world food webs. The stability analysis of the proposed model is carried out. The Hopf bifurcation conditions of the positive equilibrium point are established. Our numerical computations show that chaotic dynamics is sensitive to changes in values of parameters measuring attributes of either interacting populations or their environments. Two dimensional parameter scans suggest that the model food chain displays short-term recurrent chaos. This can be regarded as a plausible explanation for why it has been so difficult to detect deterministic chaos in natural populations.     

Probabilistic control of chaos through small perturbations

We study the method of probabilistic control of chaotic dynamics (I. Antoniou, V. Basios, F. Bosco, Int. J. Bif., Chaos 6 (1996) 1563; I. Antoniou, V. Basios, F. Bosco, Absolute controllability condition for probablistic control of Chaos, Int. J. Bif., Chaos 8(2) (1998) 409) in connection with small random perturbations of the map parameter. By constructing a probabilistic dynamical system equivalent to the perturbed chaotic map we calculate a characteristic exponent which gives controllability conditions in terms of the applied perturbations.     

国际贸易系统中的分数维和最大Lyapunov指数

混沌是一种确定性的非线性运动,它不是随机的但对初始条件敏感依赖,许多领域的研究证实了混沌的存在.应用G-P方法和W o lf算法检验中国进出口贸易的月度序列,数值结果表明国际贸易市场具有非线性和低维混沌,实证为基于系统的混沌特征量建立国际贸易的系统动力学和预报模型提供了理论基础.     

SLAC: A tool for addressing chaos in the ecology classroom

Until the early 1970s, ecologists generally assumed that erratic fluctuations observed in natural populations were a product of stochastic noise. It is now known that extremely complex dynamics can arise from basic deterministic processes. This field of study is generally called chaos theory. Here, a computer program, SLAC (Stability, Limits, And Chaos), is described, which facilitates the study of a simple deterministic model known as the logistic difference equation. It is designed to familiarize the biology student, who may not be mathematically inclined, with the fundamental concepts of population dynamics, especially the not-so-intuitive notion that complexity can evolve from deterministic mechanics. In addition to the program, pedagogically significant issues associated with the derivation of the equation and its parameters, and population dynamics in general, are highlighted.     

High level chaos in the exchange and index markets

Many studies were inconclusive about the presence of chaos in financial markets due to test misspecification. Chaos tests present in the literature need noise-free time series, since any measurement error will induce the rejection of chaos. Moreover, chaos was merely tested on a low-level basis. This paper investigates the presence of a high-level noisy chaos in financial data; simulations were conclusive about the power of the test. When applied to six stock indexes and six exchange rates, the hypothesis of chaotic dynamics was rejected for all data.     

Complex dynamics in a discrete-time predator-prey system without Allee effect

In this paper, complex dynamics of the discrete-time predator-prey system without Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory and checked up by numerical simulations. Chaos, in the sense of Marotto, is also proved by both analytical and numerical methods. Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and richer dynamics behaviors. More specifically, this paper presents the finding of period-one orbit, period-three orbits, and chaos in the sense of Marotto, complete period-doubling bifurcation and invariant circle leading to chaos with a great abundance period-windows, simultaneous occurrance of two different routes (invariant circle and inverse period- doubling bifurcation, and period-doubling bifurcation and inverse period-doubling bifurcation) to chaos for a given bifurcation parameter, period doubling bifurcation with period-three orbits to chaos, suddenly appearing or disappearing chaos, different kind of interior crisis, nice chaotic attractors, coexisting (2,3,4) chaotic sets, non-attracting chaotic set, and so on, in the discrete-time predator-prey system. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding is given of the discrete-time predator-prey systems with Allee effect and without Allee effect.     

Chaos anticontrol and synchronization of three time scales brushless DC motor system

Chaos anticontrol of three time scale brushless dc motors and chaos synchronization of different order systems are studied. Nondimensional dynamic equations of three time scale brushless DC motor system are presented. Using numerical results, such as phase diagram, bifurcation diagram, and Lyapunov exponent, periodic and chaotic motions can be observed. By adding constant term, periodic square wave, the periodic triangle wave, the periodic sawtooth wave, and kx|x| term, to achieve anticontrol of chaotic or periodic systems, it is found that more chaotic phenomena of the system can be observed. Then, by coupled terms and linearization of error dynamics, we obtain the partial synchronization of two different order systems, i.e. brushless DC motor system and rate gyroscope system.     

Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system

To estimate the ultimate bound and positively invariant set for a dynamic system is an important but quite challenging task in general. In this paper, we attempt to investigate the ultimate bound and positively invariant set for two specific systems, the Lorenz system and a unified chaotic system. We derive an ellipsoidal estimate of the ultimate bound and positively invariant set for the Lorenz system, for all the positive values of its parameters a, b and c, and obtain the minimum value of volume for the ellipsoid. Comparing with the best results in the current literature [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534; X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419], our new results fill up the gap of the estimate for the cases of 0<a<1 and 0<b<2 [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419]. Furthermore, the estimation derived here contains the results given in [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534] and [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419] as special cases. Along the same line, we also provide estimates of cylindrical and ellipsoidal bounds for a unified chaotic system, for its parameter range , and obtain the minimum value of volume for the ellipsoid. The estimate is more accurate than and also extends the result of [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534] and [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419].     

Golden mean relevance for chaos inhibition in a system of two coupled modified van der Pol oscillators

In this work, we present a novel evidence of the importance of the golden mean criticality of a system of oscillators in agreement with El Naschie’s E-infinity theory. We focus on chaos inhibition in a system of two coupled modified van der Pol oscillators. Depending on the coupling between the two oscillators, the system shows chaotic behavior for different ranges of the coupling parameter. Chaos suppression, as a transition from irregular behavior to a periodical one, is induced by perturbing the system with a harmonic signal with amplitude considerably lower than the value which causes entrainment. The frequency of the perturbation is related to the main frequencies in the spectrum of the freely running system (without perturbation) by the golden mean. We demonstrate that this effect is also obtained for a perturbation with frequency such that the ratio of half the frequency of the first main component in the freely running chaotic spectrum over the frequency of the perturbation is very close (five digits coincidence) to the golden mean. This result is shown to hold for arbitrary values of the coupling parameter in the various ranges of chaotic dynamics of the free running system.     

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.     

Chaos in a generalized van der Pol system and in its fractional order system

In this paper, chaos of a generalized van der Pol system with fractional orders is studied. Both nonautonomous and autonomous systems are considered in detail. Chaos in the nonautonomous generalized van der Pol system excited by a sinusoidal time function with fractional orders is studied. Next, chaos in the autonomous generalized van der Pol system with fractional orders is considered. By numerical analyses, such as phase portraits, Poincaré maps and bifurcation diagrams, periodic, and chaotic motions are observed. Finally, it is found that chaos exists in the fractional order system with the order both less than and more than the number of the states of the integer order generalized van der Pol system.     

Chaos and chaos synchronization for a non-autonomous rotational machine systems

Chaos and chaos synchronization of the centrifugal flywheel governor system are studied in this paper. By mechanics analyzing, the dynamical equation of the centrifugal flywheel governor system is established. Because of the non-linear terms of the system, the system exhibits both regular and chaotic motions. The characteristic of chaotic attractors of the system is presented by the phase portraits and power spectra. The evolution from Hopf bifurcation to chaos is shown by the bifurcation diagrams and a series of Poincaré sections under different sets of system parameters, and the bifurcation diagrams are verified by the related Lyapunov exponent spectra. This letter addresses control for the chaos synchronization of feedback control laws in two coupled non-autonomous chaotic systems with three different coupling terms, which is demonstrated and verified by Lyapunov exponent spectra and phase portraits. Finally, numerical simulations are presented to show the effectiveness of the proposed chaos synchronization scheme.     

A chaotic attractor in ecology: theory and experimental data

Chaos has now been documented in a laboratory population. In controlled laboratory experiments, cultures of flour beetles (Tribolium castaneum) undergo bifurcations in their dynamics as demographic parameters are manipulated. These bifurcations, including a specific route to chaos, are predicted by a well-validated deterministic model called the “LPA model”. The LPA model is based on the nonlinear interactions among the life cycle stages of the beetle (larva, pupa and adult). A stochastic version of the model accounts for the deviations of data from the deterministic model and provides the means for parameterization and rigorous statistical validation. The chaotic attractor of the deterministic LPA model and the stationary distribution of the stochastic LPA model describe the experimental data in phase space with striking accuracy. In addition, model-predicted temporal patterns on the attractor are observed in the data. This paper gives a brief account of the interdisciplinary effort that obtained these results.     

混沌与拓扑强混合

讨论了拓扑强混合、Li—Yorke混沌和修改的Devaney混沌三者之间的关系,我们得到:Li—Yorke混沌与修改的Devaney混沌无蕴涵关系;Li—Yorke混沌和修改的Devaney混沌均不能蕴涵着拓扑强混合,这解决了文献[1]中提出的两个问题.