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.     

Designing torus-doubling solutions to discrete time systems by hybrid projective synchronization

Doubling of torus occurs in high dimensional nonlinear systems, which is related to a certain kind of typical second bifurcations. It is a nontrivial task to create a torus-doubling solution with desired dynamical properties based on the classical bifurcation theories. In this paper, dead-beat hybrid projective synchronization is employed to build a novel method for designing stable torus-doubling solutions into discrete time systems with proper properties to achieve the purpose of utilizing bifurcation solutions as well as avoiding the possible conflict of physical meaning of the created solution. Although anti-controls of bifurcation and chaos synchronizations are two different topics in nonlinear dynamics and control, the results imply that it is possible to develop some new interdisciplinary methods between chaos synchronization and anti-controls of bifurcations.     

Complex nonlinear dynamics in subdiffusive activator-inhibitor systems

In this article we analyze the linear stability of nonlinear time-fractional reaction-diffusion systems. As an example, the reaction-subdiffusion model with cubic nonlinearity is considered. By linear stability analysis and computer simulation, it was shown that fractional derivative orders can change substantially an eigenvalue spectrum and significantly enrich nonlinear system dynamics. A overall picture of nonlinear solutions in subdiffusive reaction-diffusion systems is presented.     

A remark on focal points of recessive solutions of discrete symplectic systems

We investigate nonoscillatory and controllable symplectic difference systems. We show that the recessive solution of such a system at +∞ has the same number of focal points (counting multiplicities) as the recessive solution at −∞.     

Positive periodic solutions in delayed Gause-type predator-prey systems

By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of positive periodic solutions in delayed Gause-type predator-prey systems. Some known results are shown to be special cases of the presented paper.     

A Sturmian separation theorem for symplectic difference systems

We establish a Sturmian separation theorem for conjoined bases of 2n-dimensional symplectic difference systems. In particular, we show that the existence of a conjoined basis without focal points in a discrete interval (0,N+1] implies that any other conjoined basis has at most n focal points (counting multiplicities) in this interval.     

Bifurcation of limit cycles at the equator

This paper studies center conditions and bifurcation of limit cycles from the equator for a class of polynomial differential system of order seven. By converting real planar system into complex system, we established the relation of focal values of a real system with singular point quantities of its concomitant system, and the recursion formula for the computation of singular point quantities of a complex system at the infinity. Therefore, the first 14 singular point quantities of a complex system at the infinity are deduced by using computer algebra system Mathematica. What’s more, the conditions for the infinity of the real system to be a center or 14 degree fine focus are derived, respectively. A system of order seven that bifurcates 12 limit cycles from the infinity is constructed for the first time.     

Geometric criteria for the nonexistence of cycles in Gause-type predator-prey systems

The global stability of a multi-species interacting system has apparently important biological implications. In this paper we study the global stability of Gause-type predator-prey models by providing new criteria for the nonexistence of cycles and limit cycles. Our criteria have clear geometrical interpretations and are easier to apply than other methods employed in recent studies. Using these criteria and related techniques we are able to develop new results on the existence and uniqueness of cycles in Gause-type models with various growth and response functions.

     

Oscillation theorems for symplectic difference systems

We consider symplectic difference systems involving a spectral parameter, together with the Dirichlet boundary conditions. The main result of the paper is a discrete version of the so-called oscillation theorem which relates the number of finite eigenvalues less than a given number to the number of focal points of the principal solution of the symplectic system. In two recent papers the same problem was treated and an essential ingredient was to establish the concept of the multiplicity of a focal point. But there was still a rather restrictive condition needed, which is eliminated here by using the concept of finite eigenvalues (or zeros) from the theory of matrix pencils.     

一类滞后型退化系统的周期解

本文考虑一类滞后型退化高维周期系统B(t)x′(t)=A(t)x(t) f(t,xt),利用线性系统指数型二分性理论和Krasnoselskii不动点定理得到了保证其周期解存在的充分性条件.     

Solution of bordered singular systems in numerical continuation and bifurcation

In numerical continuation and bifurcation problems linear systems with coefficient matrices in the block form arise naturally. Here

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and n may be large but m is small. A usually has a special structure (banded, block banded, sparse,…) and B, C, D are dense, so that it is advisable to use a specialized solver for A and to solve with M by some block method. Unfortunately, A is often also a nearly singular matrix (in fact, made nonsingular only by roundoff and truncation errors). On the other hand, M is usually nonsingular but can be ill-conditioned and in certain situations will degenerate to singularity as well. We describe numerical tests for this problem using the mixed block elimination method of Govaerts and Pryce (1993) for solving bordered linear systems with possibly nearly singular blocks A. To this end, we compute by Newton's method a triple-point bifurcation point in a parameterized reaction—diffusion equation (the Brusselator). The numerical tests show that the linear systems are solved in a stable way, in spite of the use of a black-box solver (SGBTRS from LAPACK) for a nearly singular matrix.     

Heteroclinic cycles in lattice dynamical systems

A criterion of spatial chaos occurring in lattice dynamical systems—heteroclinic cycle—is discussed It is proved that if the system has asymptotically stable heteroclinic cycle, then it has asymptotically stable homoclinic point which implies spatial chaos. Project supported by the National Natural Science Foundation of China.     

Bifurcation of periodic orbits emanated from a vertex in discontinuous planar systems

We discuss bifurcation of periodic orbits in discontinuous planar systems with discontinuities on finitely many straight lines intersecting at the origin and the unperturbed system has either a limit cycle or an annulus of periodic orbits. Assume that the unperturbed periodic orbits cross every switching line transversally exactly once. For the first case we give a condition for the persistence of the limit cycle. For the second case, we obtain the expression of the first order Melnikov function and establish sufficient conditions on the number of limit cycles bifurcate from the periodic annulus. Then we generalize our results to systems with discontinuities on finitely many smooth curves. As an application, we present a piecewise cubic system with 4 switching lines and show that the maximum number of limit cycles bifurcate from the periodic annulus can be affected by the position of the switching lines.     

Quasilinear systems with memory kernels

We consider a quasilinear integrodifferential system in non-normal form. Such a system is a generalization of a phase-field model with memory and includes, as a particular case, the system describing the combustion of a material with memory. In this paper, we study both the direct and the inverse problems. Our fundamental tools are: the theory of analytic semigroups, optimal regularity results and fixed point arguments.     

Complex behavior in condensed matter: Morphological ordering in dissipative carrier systems

This article studies the dissipative thermodynamic regime of an electron system in bulk matter under the action of an external source of energy, which generates electron-hole pairs with a nonequilibrium distribution in energy space. It is shown that with increasing values of the source power (furthering the distance from equilibrium), and strictly in the case of a p-doped material, the carrier system displays complex behavior characterized by undergoing a succession of transitions between synergetically self-organized dissipative structures. The sequence goes from the homogeneous steady state (or stochastic thermal chaos), to sinusoidal spatial deviations (morphological ordering), to intricate ordered states (subharmonic bifurcations), to deterministic turbulent-like chaos (large amount of nonlinear periodic spatial organization of the Landau-Prigogine's type). The phenomenon may arise, for example, in semiconductor systems, molecular polymers, and protein molecular chains in biosystems. © 1997 John Wiley & Sons, Inc.     

Boundary controllability of integrodifferential systems in Banach spaces

Sufficient conditions for boundary controllability of integrodifferential systems in Banach spaces are established. The results are obtained by using the strongly continuous semigroup theory and the Banach contraction principle. Examples are provided to illustrate the theory.     

Partial extinction,permanence and global attractivity in nonautonomous <Emphasis Type="Italic">n</Emphasis>-species Lotka-Volterra competitive systems with impulses

In this paper, the qualitative properties of general nonautonomous Lotka-Volterra n-species competitive systems with impulsive effects are studied. Some new criteria on the permanence, extinction and global attractivity of partial species are established by used the methods of inequalities estimate and Liapunov functions. As applications, nonautonomous two species Lotka-Volterra systems with impulses are discussed.     

On (2+1)-dimensional hydrodynamic type systems possessing a pseudopotential with movable singularities

We consider a class of hydrodynamic type systems that have three independent and N ? 2 dependent variables and possess a pseudopotential. It turns out that systems having a pseudopotential with movable singularities can be described by some functional equation. We find all solutions of this equation, which permits constructing interesting examples of integrable systems of hydrodynamic type for arbitrary N.