A Research on Nonendangered Population Protection Facing Biological Invasion

In nature, a biological invasion is a common phenomenon that often threatens the existence of local species. Getting rid of the invasive species is hard to achieve after its survival and reproduction. At present, killing some invasive ones and putting artificial local species are usual methods to prevent local species from extinction. An ODE model is constructed to simulate the invasive procedure, and the protection policy is depicted as a series of impulses depending on the state of the variables. Both the ODEs and the impulses form a state feedback impulsive model which describes the invasion and protection together. The existence of homoclinic cycle and bifurcation of order-1 periodic solution of the impulsive model are discussed, and the orbitally asymptotical stability of the order-1 periodic solution is certificated with a novel method. Finally, the numerical simulation result is listed to confirm the theoretical work.

State-Dependent Pulse Vaccination and Therapeutic Strategy in an SI Epidemic Model with Nonlinear Incidence Rate

In this paper, the state-dependent pulse vaccination and therapeutic strategy are considered in the control of the disease. A pulse system is built to model this process based on an SI ordinary differential equation model. At first, for the system neglecting the impulse effect, we give the classification of singular points. Then for the pulse system, by using the theory of the semicontinuous dynamic system, the dynamics is analyzed. Our analysis shows that the pulse system exhibits rich dynamics and the system has a unique order-1 homoclinic cycle, and by choosing p as the control parameter, the order-1 homoclinic cycle disappears and bifurcates an orbitally asymptotical stable order-1 periodic solution when p changes. Numerical simulations by maple 18 are carried out to illustrate the theoretical results.

Homoclinic Cycle Bifurcations in Planar Maps

In this study, bifurcations of an invariant closed curve (ICC) generated from a homoclinic connection of a saddle fixed point are analyzed in a planar map. Such bifurcations are called homoclinic cycle (HCC) bifurcations of the saddle fixed point. We examine the HCC bifurcation structure and the properties of the generated ICC. A planar map that can accurately control the stable and unstable manifolds of the saddle fixed point is designed for this analysis and the results indicate that the HCC bifurcation depends upon a product of two eigenvalues of the saddle fixed point, and the generated ICC is a chaotic attractor with a positive Lyapunov exponent.

BIFURCATIONS OF 2-2-1 HETERODIMENSIONAL CYCLES UNDER TRANSVERSALITY CONDITION

Bifurcations of generic 2-2-1 heterodimensional cycles connecting to three saddles, in which two of them have two-dimensional unstable manifolds, are studied by setting up a local moving frame. Under a certain transversal condition, we firstly present the existence, uniqueness and noncoexistence of a 3-point heterodimensional cycle, 2-point heterodimensional or equidimensional cycle, 1-homoclinic cycle and 1-periodic orbit bifurcated from the 3-point heterodimensional cycle, and the bifurcation surfaces and bifurcation regions are located when the u-component [Formula: see text] of the vector [Formula: see text] under the Poincaré mapping [Formula: see text] is nonzero. Conversely, we obtain some sufficient conditions such that the bifurcation of a 2-fold 1-periodic orbit occurs and a 1-periodic orbit coexists with the surviving heterodimensional cycle, showing some new bifurcation behaviors different from the well-known equidimensional cycles.

INTERMITTENCY NEAR A CODIMENSION THREE STEADY-STATE BIFURCATION

We study the existence and stability of heteroclinic connections near "hopping" cellular flame patterns. These are dynamic patterns in which individual cells make sequential, and abrupt, changes in their angular positions while they rotate nonuniformly about the center of a circular domain. Normal form analysis and experimental works have shown that these patterns are associated with a homoclinic cycle connecting group related equilibria. In fact, they emerge through a codimension three steady-state bifurcation of three modes with wave numbers in a 2:3:4 ratio. While cycles are known to exist in the mode-2 and mode-4 interactions, here we show that mode-3 destabilizes the connection so that only remnants, i.e. intermittent flame patterns of the cycles can be observed.

Product dynamics for homoclinic attractors

Heteroclinic cycles may occur as structurally stable asymptotically stable attractors if there are invariant subspaces or symmetries of a dynamical system. Even for cycles between equilibria, it may be difficult to obtain results on the generic behaviour of trajectories converging to the cycle. For more–complicated cycles between chaotic sets, the non–trivial dynamics of the ‘nodes’ can interact with that of the ‘connections’. This paper focuses on some of the simplest problems for such dynamics where there are direct products of an attracting homoclinic cycle with various types of dynamics. Using a precise analytic description of a general planar homoclinic attractor, we are able to obtain a number of results for direct product systems. We show that for flows that are a product of a homoclinic attractor and a periodic orbit or a mixing hyperbolic attractor, the product of the attractors is a minimal Milnor attractor for the product. On the other hand, we present evidence to show that for the product of two homoclinic attractors, typically only a small subset of the product of the attractors is an attractor for the product system.

Chaotic dynamics in ${\mathbb Z}_2$-equivariant unfoldings of codimension three singularities of vector fields in ${\mathbb R}^3$

We study the most generic nilpotent singularity of a vector field in ${\mathbb R}^3$ which is equivariant under reflection with respect to a line, say the $z$-axis. We prove the existence of eight equivalence classes for $C^0$-equivalence, all determined by the 2-jet. We also show that in certain cases, the ${\mathbb Z}_2$-equivariant unfoldings generically contain codimension one heteroclinic cycles which are comparable to the Shil'nikov-type homoclinic cycle in non-equivariant unfoldings. The heteroclinic cycles are accompanied by infinitely many horseshoes and also have a reasonable possibility of generating suspensions of Hénon-like attractors, and even Lorenz-like attractors.