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1.
Evolving to the edge of chaos: Chance or necessity? 总被引:2,自引:0,他引:2
We show that ecological systems evolve to edges of chaos (EOC). This has been demonstrated by analyzing three diverse model ecosystems using numerical simulations in combination with analytical procedures. It has been found that all these systems reside on EOC and display short-term recurrent chaos (strc). The first two are non-linear food chains and the third one is a linear food chain. The dynamics of first two is dictated by deterministic changes in system parameters. In contrast to this, dynamics of the third model system (the linear food chain) is governed by both deterministic changes in system parameters as well as exogenous stochastic perturbations (unforeseen changes in initial conditions) of these dynamical systems. 相似文献
2.
《Chaos, solitons, and fractals》2007,31(5):1074-1087
We show that ecological systems evolve to edges of chaos (EOC). This has been demonstrated by analyzing three diverse model ecosystems using numerical simulations in combination with analytical procedures. It has been found that all these systems reside on EOC and display short-term recurrent chaos (strc). The first two are non-linear food chains and the third one is a linear food chain. The dynamics of first two is dictated by deterministic changes in system parameters. In contrast to this, dynamics of the third model system (the linear food chain) is governed by both deterministic changes in system parameters as well as exogenous stochastic perturbations (unforeseen changes in initial conditions) of these dynamical systems. 相似文献
3.
M. Uleysky L. Konkov S. Prants 《Communications in Nonlinear Science & Numerical Simulation》2003,8(3-4):329
We study the coupled translational, electronic, and field dynamics of the combined system “a two-level atom + a single-mode quantized field + a standing-wave ideal cavity”. In the semiclassical approximation with a point-like atom, interacting with the classical field, the dynamics is described by the Heisenberg equations for the atomic and field expectation values which are known to produce semiclassical chaos under appropriate conditions. We derive Hamilton–Schrödinger equations for probability amplitudes and averaged position and momentum of a point-like atom interacting with the quantized field in a standing-wave cavity. They constitute, in general, an infinite-dimensional set of equations with an infinite number of integrals of motion which may be reduced to a dynamical system with four degrees of freedom if the quantized field is supposed to be initially prepared in a Fock state. This system is found to produce semiquantum chaos with positive values of the maximal Lyapunov exponent. At exact resonance, the semiquantum dynamics is regular. At large values of detuning |δ|1, the Rabi atomic oscillations are usually shallow, and the dynamics is found to be almost regular. The Doppler–Rabi resonance, deep Rabi oscillations that may occur at any large value of |δ| to be equal to |αp0|, is found numerically and described analytically (with α to be the normalized recoil frequency and p0 the initial atomic momentum). Two gedanken experiments are proposed to detect manifestations of semiquantum chaos in real experiments. It is shown that in the chaotic regime values of the population inversion zout, measured with atoms after transversing a cavity, are so sensitive to small changes in the initial inversion zin that the probability of detecting any value of zout in the admissible interval [−1,1] becomes almost unity in a short time. Chaotic wandering of a two-level atom in a quantized Fock field is shown to be fractal. Fractal-like structures, typical with chaotic scattering, are numerically found in the dependence of the time of exit of atoms from the cavity on their initial momenta. 相似文献
4.
In 1978, Marotto generalized Li–Yorke’s results on the criterion for chaos from one-dimensional discrete dynamical systems to n-dimensional discrete dynamical systems, showing that the existence of a non-degenerate snap-back repeller implies chaos in the sense of Li–Yorke. This theorem is very useful in predicting and analyzing discrete chaos in multi-dimensional dynamical systems. Yet, besides it is well known that there exists an error in the conditions of the original Marotto Theorem, and several authors had tried to correct it in different way, Chen, Hsu and Zhou pointed out that the verification of “non-degeneracy” of a snap-back repeller is the most difficult in general and expected, “almost beyond reasonable doubt,” that the existence of only degenerate snap-back repeller still implies chaotic, which was posed as a conjecture by them. In this paper, we shall give necessary and sufficient conditions of chaos in the sense of Li–Yorke for planar monotone or competitive discrete dynamical systems and solve Chen–Hsu–Zhou Conjecture for such kinds of systems. 相似文献
5.
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. 相似文献
6.
We study a continuous time model describing gene-for-gene, host–parasite interactions among self-replicating macromolecules evolving in both neutral and rugged fitness landscapes. Our model considers polymorphic genotypic populations of sequences with 3 bits undergoing mutation and incorporating a “type II” non-linear functional response in the host–parasite interaction. We show, for both fitness landscapes, a wide range of chaotic coevolutionary dynamics governed by Red Queen strange attractors. The analysis of a rugged fitness landscape for parasite sequences reveals that fittest genotypes achieve lower stationary concentration values, as opposed to the flattest ones, which undergo a higher stationary concentration. Our model also shows that the increase of parasites pressure (higher self-replication and mutation rates) generically involves a simplification of the host–parasite dynamical behavior, involving the transition from a chaotic to an ordered coevolutionary phase. Moreover, the same transition can also be found when hosts “run” faster through the hypercube. Our results, in agreement with previous studies in host–parasite coevolution, suggest that chaos might be common in coevolutionary dynamics of changing self-replicating entities undergoing a host–parasite ecology. 相似文献
7.
True random number generators are in general more secure than pseudo random number generators. In this paper, we propose a novel true random number generator which generates a 256-bit random number by computer mouse movement. It is cheap, convenient and universal for personal computers. To eliminate the effect of similar movement patterns generated by the same user, three chaos-based approaches, namely, discretized 2D chaotic map permutation, spatiotemporal chaos and “MASK” algorithm, are adopted to post-process the captured mouse movements. Random bits generated by three users are tested using NIST statistical tests. Both the spatiotemporal chaos approach and the “MASK” algorithm pass the tests successfully. However, the latter has a better performance in terms of efficiency and effectiveness and so is more practical for common personal computer applications. 相似文献
8.
We introduce the stochastic dynamics in the phase space that corresponds to the Boltzmann equation and hierarchy and is the Boltzmann–Grad limit of the Hamiltonian dynamics of systems of hard spheres. By the method of averaging over the space of positions, we derive from it the stochastic dynamics in the momentum space that corresponds to the space-homogeneous Boltzmann equation and hierarchy. Analogous dynamics in the mean-field approximation was postulated by Kac for the explanation of the phenomenon of propagation of chaos and derivation of the Boltzmann equation. 相似文献
9.
We consider three different dynamic systems. The first runs “smoothly” during a certain finite time interval, undergoes an abrupt change in the dynamics during the next (finite) time interval and is governed by the second system. The solution of this second system lies on a surface for a finite amount of time and becomes invisible. At the beginning of the third phase, the system is subjected to an impulse which causes the solution to leave the surface and we have the new hybrid impulsive system. In this paper, we employ two measures to find suitable conditions so that the new system again runs as smoothly as the first. 相似文献
10.
Gamal M. Mahmoud Emad E. Mahmoud Ahmed A. Farghaly Shaban A. Aly 《Chaos, solitons, and fractals》2009,42(5):2858-2864
Synchronization is an important phenomenon commonly observed in nature. It is also often artificially induced because it is desirable for a variety of applications in physics, applied sciences and engineering. In a recent paper [Mahmoud GM, Mohamed AA, Aly SA. Strange attractors and chaos control in periodically forced complex Duffing’s oscillators. Physica A 2001;292:193–206], a system of periodically forced complex Duffing’s oscillators was introduced and shown to display chaotic behavior and possess strange attractors. Such complex oscillators appear in many problems of physics and engineering, as, for example, nonlinear optics, deep-water wave theory, plasma physics and bimolecular dynamics. Their connection to solutions of the nonlinear Schrödinger equation has also been pointed out.In this paper, we study the remarkable phenomenon of chaotic synchronization on these oscillator systems, using active control and global synchronization techniques. We derive analytical expressions for control functions and show that the dynamics of error evolution is globally stable, by constructing appropriate Lyapunov functions. This means that, for a relatively large set initial conditions, the differences between the drive and response systems vanish exponentially and synchronization is achieved. Numerical results are obtained to test the validity of the analytical expressions and illustrate the efficiency of these techniques for inducing chaos synchronization in our nonlinear oscillators. 相似文献
11.
Antonio Palacios 《Journal of Difference Equations and Applications》2013,19(7):671-686
Cycling behavior, in which solution trajectories linger around steady-states and periodic solutions, is known to be a generic feature of coupled cell systems of differential equations. In this type of systems, cycling behavior can even occur independently of the internal dynamics of each cell. This conclusion has lead to the discovery of "cycling chaos", in which solution trajectories cycle around symmetrically related chaotic sets. In this work, we demonstrate that cycling behavior also occurs in coupled systems of difference equations. More specifically, we prove the existence of structurally stable cycles between fixed points, and use numerical simulations to illustrate that the resulting cycles can also persist independently of the internal dynamics of each cell. Consequently, we demonstrate that cycles involving periodic orbits as well as cycling chaos also occur in systems of difference equations. 相似文献
12.
《Chaos, solitons, and fractals》2001,12(2):205-218
We review our recent efforts to understand why chaotic dynamics is rarely observed in natural populations. The study of two-model ecosystems considered in this paper suggests that chaos exists in narrow parameter ranges. This dynamical behaviour is caused by the crisis-induced sudden death of chaotic attractors. The computed bifurcation diagrams and basin boundary calculations reinforce our earlier conclusion [Chaos, Solitons & Fractals 8 (12) (1997) 1933; Int J Bifurc Chaos 8 (6) (1998) 1325] that the reason why chaos is rarely observed in natural populations is hidden within the mathematical structure of the ecological interactions and not with the problem associated with the data (insufficient length, precision, noise, etc.) and its analysis. We also argue that crisis-limited chaotic dynamics can be commonly found in model terrestrial ecosystems. 相似文献
13.
Jorge Duarte Cristina JanuárioNuno Martins Josep Sardanyés 《Nonlinear Analysis: Real World Applications》2012,13(4):1647-1661
Density-dependent effects, both positive or negative, can have an important impact on the population dynamics of species by modifying their population per-capita growth rates. An important type of such density-dependent factors is given by the so-called Allee effects, widely studied in theoretical and field population biology. In this study, we analyze two discrete single population models with overcompensating density-dependence and Allee effects due to predator saturation and mating limitation using symbolic dynamics theory. We focus on the scenarios of persistence and bistability, in which the species dynamics can be chaotic. For the chaotic regimes, we compute the topological entropy as well as the Lyapunov exponent under ecological key parameters and different initial conditions. We also provide co-dimension two bifurcation diagrams for both systems computing the periods of the orbits, also characterizing the period-ordering routes toward the boundary crisis responsible for species extinction via transient chaos. Our results show that the topological entropy increases as we approach to the parametric regions involving transient chaos, being maximum when the full shift R(L)∞ occurs, and the system enters into the essential extinction regime. Finally, we characterize analytically, using a complex variable approach, and numerically the inverse square-root scaling law arising in the vicinity of a saddle-node bifurcation responsible for the extinction scenario in the two studied models. The results are discussed in the context of species fragility under differential Allee effects. 相似文献
14.
A graph that can be isometrically embedded into a hypercube is called a partial cube (or binary Hamming graph). Klavžar, Gutman and Mohar [S. Klavžar, I. Gutman, B. Mohar, Labeling of benzenoid systems which reflects the vertex-distance relations, J. Chem. Inf. Comput. Sci. 35 (1995) 590–593] showed that all benzenoid systems are partial cubes. In this article we show that none of the coronoid systems (benzenoid systems with “holes”) is a partial cube. 相似文献
15.
The controversy surrounding the correctness of Marotto’s theorem continues over the last two decades, with many researchers claiming to have found an error in the proof. In this paper, we show that Marotto’s theorem is indeed correct for analyzing the existence of chaos in the sense of Li-Yorke even after relaxing certain assumptions in the proof. In addition, we extend the theory to derive the conditions for the existence of chaos in the sense of Devaney. We show that these results can be applied to study the chaotification of linear switching systems. 相似文献
16.
Akitaka Dohtani 《Chaos, solitons, and fractals》2011,44(4-5):290-297
In this study, we further develop the perturbation method of Marotto [6] and investigate the general mechanisms responsible for nonlinear dynamics, which are typical of multidimensional systems. We focus on the composites of interdependent relations between different variables. First, we prove a general result on chaos, which shows that the cyclic composites of nonlinear interdependent relations are sources of chaotic dynamics in multidimensional systems. By considering several examples, we conclude that the cyclic composites play an important role in detecting chaotic dynamics. 相似文献
17.
Juan Luis García Guirao Raquel García Rubio 《Journal of Computational and Applied Mathematics》2009,233(4):1091-1095
We introduced the so-called Cournot-like models, i.e. n-dimensional discrete dynamical systems which constitute the mathematical environment for modeling some economic and biological processes. The main aim of this work is to present a characterization of the dynamical simplicity for these types of systems through the property “to have zero topological entropy”. Cournot-like systems generalize the well-known economic situation of competition in a duopolistic market introduces by Cournot in 1838. 相似文献
18.
The chaos expansion of a general non-linear function of a Gaussian stationary increment process conditioned on its past realizations is derived. This work combines the Wiener chaos expansion approach to study the dynamics of a stochastic system with the classical problem of the prediction of a Gaussian process based on a realization of its past. This is done by considering special bases for the Gaussian space 𝒢 generated by the process, which allows us to obtain an orthogonal basis for the Fock space of 𝒢 such that each basis element is either measurable or independent with respect to the given samples. This allows us to easily derive the chaos expansion of a random variable conditioned on part of the sample path. We provide a general method for the construction of such basis when the underlying process is Gaussian with stationary increment. We evaluate the basis elements in the case of the fractional Brownian motion, which leads to a prediction formula for this process. 相似文献
19.
In this paper, the reachability realization of a switched linear discrete-time system, which is a collection of linear time-invariant discrete-time systems along with some maps for “switching” among them, is addressed. The main contribution of this paper is to prove that for a switched linear discrete-time system, there exists a basic switching sequence such that the reachable (controllable) state set of this basic switching sequence is equal to the reachable (controllable) state set of the system. Hence, the reachability (controllability) can be realized by using only one switching sequence. We also discuss the stabilizability of switched systems, and obtain a sufficient condition for stabilizability. Two numeric examples are given to illustrate the results. 相似文献
20.
We consider stationary multiscale systems as defined by Basseville, Benveniste, Nikoukhah and Willsky. We show that there are deep analogies with the discrete time non stationary setting as developed by the first author, Dewilde and Dym. Following these analogies we define a point evaluation with values in a C*–algebra and the corresponding “Hardy space” in which Cauchy’s formula holds. This point evaluation is used to define in this context the counterpart of classical notions such as Blaschke factors. 相似文献