共查询到20条相似文献,搜索用时 15 毫秒
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Symmetry breaking, bifurcations, periodicity and chaos in the Euler method for a class of delay differential equations 总被引:1,自引:0,他引:1
A discrete model is proposed to explore the rich dynamics of nonlinear delayed systems under Euler discretization, such as multiple steady states, multiple bifurcations, complex periodic oscillations, and chaos. 相似文献
3.
In a natural ecosystem, specialist predators feed almost exclusively on one species of prey. But generalist predators feed on many types of species. Consequently, their dynamics is not coupled to the dynamics of a specific prey population. However, the defense of prey formed by congregating made the predator tend to move in the direction of lower concentration of prey species. This is described by cross-diffusion in a generalist predator–prey model. First, the positive equilibrium solution is globally asymptotically stable for the ODE system and for the reaction–diffusion system without cross-diffusion, respectively, hence it does not belong to the classical Turing instability scheme. But it becomes linearly unstable only when cross-diffusion also plays a role. This implies that cross–diffusion can lead to the occurrence and disappearance of the instability. Our results exhibit some interesting combining effects of cross-diffusion, predations and intra-species interactions. Furthermore, we consider the existence and non-existence results concerning non-constant positive steady states (patterns) of the system. We demonstrate that cross-diffusion can create non-constant positive steady-state solutions. 相似文献
4.
John A. D. Appleby 《随机分析与应用》2013,31(4):813-826
Abstract In this note, we address the question of how large a stochastic perturbation an asymptotically stable linear functional differential system can tolerate without losing the property of being pathwise asymptotically stable. In particular, we investigate noise perturbations that are either independent of the state or influenced by the current and past states. For perturbations independent of the state, we prove that the assumed rate of fading for the noise is optimal. 相似文献
5.
This paper is concerned about the S–K–T competition model with spatially heterogeneous interactions, in which the two competing species are identical in all aspects except for their interspecific competitions. Under small variations of interspecific competitions, a good understanding of the dynamics and the structure of the set of steady states is gained by the Lyapunov–Schmidt procedure and detailed analysis. The results reveal that the role that the spatial heterogeneity plays in the coexistence of the competing species can be rather interesting and complicated. Either stable or unstable coexistence state can be produced by suitable spatial heterogeneity, so the spatial heterogeneity can be either favorable or unfavorable for the coexistence of competing species. 相似文献
6.
The effect of small constantly acting random perturbations of white noise type on a dynamical system with locally stable fixed point is studied. The perturbed system is considered in the form of Itô stochastic differential equations, and it is assumed that the perturbation does not vanish at a fixed point. In this case, the trajectories of the stochastic system issuing from points near the stable fixed point exit from the neighborhood of equilibrium with probability 1. Classes of perturbations such that the equilibrium of a deterministic system is stable in probability on an asymptotically large time interval are described. 相似文献
7.
《Chaos, solitons, and fractals》2001,12(2):321-332
In contrast to the single species models that were extensively studied in the 1970s and 1980s, predator–prey models give rise to long-period oscillations, and even systems with stable equilibria can display oscillatory transients with a regular frequency. Many fluctuating populations appear to be governed by such interactions. However, predator–prey models have been poorly studied with respect to the interaction of nonlinear dynamics, noise, and system identification. I use simulated data from a simple host–parasitoid model to investigate these issues. The addition of even a modest amount of noise to a stable equilibrium produces enough structured variation to allow reasonably accurate parameter estimation. Despite the fact that more-or-less regular cycles are generated by adding noise to any of the classes of deterministic attractor (stable equilibrium, periodic and quasiperiodic orbits, and chaos), the underlying dynamics can usually be distinguished, especially with the aid of the mechanistic model. However, many of the time series can also be fit quite well by a wrong model, and the fitted wrong model usually misidentifies the underlying attractor. Only the chaotic time series convincingly rejected the wrong model in favor of the true one. Thus chaotic population dynamics offer the best chance for successfully identifying underlying regulatory mechanisms and attractors. 相似文献
8.
The aim of the paper is to study systems with one-and-a-half degrees of freedom generated by a Hamiltonian with a quartic unperturbed part and broad perturbation spectrum. To this end, an approximate interpolating Hamiltonian system is firstly studied. Behaviour of the Poincaré–Birkhoff or dimerised chains in their routes to reconnection when the perturbation parameter varies is particularly presented. In the second step, a discrete system associated to the full Hamiltonian system is constructed and studied. We point out interesting properties of the dynamics of the Poincaré–Birkhoff or dimerised chains, such as pairs of homoclinic orbits to the same equilibrium point (sandglass) and triple reconnection. Then we use the scenario of reconnections to explain the destruction of transport barriers in the non-autonomous system. 相似文献
9.
Irina Bashkirtseva Ekaterina Ekaterinchuk 《Journal of Difference Equations and Applications》2016,22(3):376-390
We consider the time-delayed logistic model under the influence of random perturbations. A parametric analysis of stochastically forced regular attractors (equilibria, closed invariant curves, discrete cycles) of this model is performed using the stochastic sensitivity functions technique. A spatial arrangement of random states in stochastic attractors is described by confidence domains. The phenomenon of noise-induced transitions in a zone of discrete cycles is discussed. 相似文献
10.
In the present work, we report the induction of rhythms in electrochemical systems. Limit cycle oscillations are provoked by reversing the stability of a previously stable fixed point on which the system’s dynamics were originally settled. Superimposing an appropriate perturbation term, sign of the local eigenvalues (real part) of the fixed point was changed. This alters the stability of the fixed point without varying its location. This protocol was tested both in a numerical model simulating electrochemical corrosion as well as an experimental electrochemical cell. Both numerical results and experimental observations indicate that, employing suitable perturbations, it is indeed possible to generate oscillatory behavior in nonlinear systems whose autonomous dynamics exhibit steady state behavior. 相似文献
11.
Yunshyong Chow 《Journal of Difference Equations and Applications》2013,19(9):1350-1371
We study a discrete host–parasitoid system where the host population follows the classical Ricker functional form and is also subject to Allee effects. We determine basins of attraction of the local attractors of the single population model when the host intrinsic growth rate is not large. In this situation, existence and local stability of the interior steady states for the host–parasitoid interaction are completely analysed. If the host's intrinsic growth rate is large, then the interaction may support multiple interior steady states. Linear stability of these steady states is provided. 相似文献
12.
D. G. Rakhimov 《Differential Equations》2017,53(5):607-616
We use the reduction method, which allows one to reduce the study of perturbations of multiple eigenvalues to perturbations of simple eigenvalues, to analyze the general perturbation problem for Fredholm points of the discrete spectrum of linear operator functions analytically depending on the spectral parameter. The same method is used to study a perturbation of multiple Fredholm points of the discrete Schmidt spectrum (s-numbers) of a linear operator. We present an example of a problem on a perturbation of the domain of the Sturm–Liouville problem for a second-order differential operator. 相似文献
13.
This paper is concerned with stability analysis of biological networks modeled as discrete and finite dynamical systems. We
show how to use algebraic methods based on quantifier elimination, real solution classification and discriminant varieties
to detect steady states and to analyze their stability and bifurcations for discrete dynamical systems. For finite dynamical
systems, methods based on Gr?bner bases and triangular sets are applied to detect steady states. The feasibility of our approach
is demonstrated by the analysis of stability and bifurcations of several discrete biological models using implementations
of algebraic methods. 相似文献
14.
Antoni Ferragut Jaume Llibre Marco Antonio Teixeira 《Rendiconti del Circolo Matematico di Palermo》1932,56(1):101-115
We studyC 1 perturbations of a reversible polynomial differential system of degree 4 in\(\mathbb{R}^3 \). We introduce the concept of strongly reversible vector field. If the perturbation is strongly reversible, the dynamics of the perturbed system does not change. For non-strongly reversible perturbations we prove the existence of an arbitrary number of symmetric periodic orbits. Additionally, we provide a polynomial vector field of degree 4 in\(\mathbb{R}^3 \) with infinitely many limit cycles in a bounded domain if a generic assumption is satisfied. 相似文献
15.
Discrete models are proposed to delve into the rich dynamics of nonlinear delayed systems under Euler discretization, such as backwards bifurcations, stable limit cycles, multiple limit-cycle bifurcations and chaotic behavior. The effect of breaking the special symmetry of the system is to create a wide complex operating conditions which would not otherwise be seen. These include multiple steady states, complex periodic oscillations, chaos by period doubling bifurcations. Effective computation of multiple bifurcations, stable limit cycles, symmetrical breaking bifurcations and chaotic behavior in nonlinear delayed equations is developed. 相似文献
16.
G.W. Patrick 《Mathematische Zeitschrift》1999,232(4):747-788
An interesting situation occurs when the linearized dynamics of the shape of a formally stable Hamiltonian relative equilibrium
at nongeneric momentum 1:1 resonates with a frequency of the relative equilibrium's generator. In this case some of the shape
variables couple to the group variables to first order in the momentum perturbation, and the first order perturbation theory
implies that the relative equilibrium slowly changes orientation in the same way that a charged particle with magnetic moment moves on a sphere under the influence of a radial magnetic monopole. In the course of showing this a normal form is constructed
for linearizations of relative equilibria and for Hamiltonians near group orbits of relative equilibria.
Received August 27, 1998; in final form February 20, 1999 相似文献
17.
Fordyce A. Davidson 《Applicable analysis》2013,92(6-7):717-734
In this article we consider the spectral properties of a class of non-local operators that arise from the study of non-local reaction-diffusion equations. Such equations are used to model a variety of physical and biological systems with examples ranging from Ohmic heating to population dynamics. The operators studied here are bounded perturbations of linear (local) differential operators. The non-local perturbation is in the form of an integral term. It is shown here that the spectral properties of these non-local operators can differ considerably from those of their local counterpart. Multiplicities of eigenvalues are studied and new oscillation results for the associated eigenfunctions are presented. These results highlight problems with certain similar results and provide an alternative formulation. Finally, the stability of steady states of associated non-local reaction-diffusion equations is discussed. 相似文献
18.
The paper deals with periodic orbits in three systems of ordinarydifferential equations. Two of the systems, the Falkner–Skanequations and the Nosé equations, do not possess fixedpoints, and yet interesting dynamics can be found. Here, periodicorbits emerge in bifurcations from heteroclinic cycles, connectingfixed points at infinity. We present existence results for suchperiodic orbits and discuss their properties using careful asymptoticarguments. In the final part results about the Nosé equationsare used to explain the dynamics in a dissipative perturbation,related to a system of dynamo equations. 相似文献
19.
Kurt Frischmuth Ekaterina S. Kovaleva Vyacheslav G. Tsybulin 《Nonlinear Analysis: Real World Applications》2011,12(1):146-155
The system of nonlinear parabolic equations that models the dynamics of three populations is studied. The strong nonuniqueness in the form of continuous cosymmetric family of steady states is detected for some parameter values. Collapse of the family of equilibria under perturbation of boundary conditions is analyzed numerically. 相似文献
20.
Hongbin Wang 《Journal of Mathematical Analysis and Applications》2010,368(1):9-1126
First, we identify the critical values for Hopf-pitchfork bifurcation. Second, we derive the normal forms up to third order and their unfolding with original parameters in the system near the bifurcation point, by the normal form method and center manifold theory. Then we give a complete bifurcation diagram for original parameters of the system and obtain complete classifications of dynamics for the system. Furthermore, we find some interesting phenomena, such as the coexistence of two asymptotically stable states, two stable periodic orbits, and two attractive quasi-periodic motions, which are verified both theoretically and numerically. 相似文献