Stochastic Stability, Stochastic Attraction And Homoclinic And Heteroclinic Bifurcation

This paper is devoted to study the stochastic stability,stochastic attraction for the sto-chastic differential equations with additive noise and the homoclinic and heteroclinic bifur-cation.The whole paper is divided into two parts.In Part 1,firstly we consider an epidemiological model,which is a SIRS model existingloss of immunity with or without distributed time delay influenced by random perturbations.In the SIRS model,γis the per capita rate of loss of immunity.We obtain conditionsunder which the stochastic SIRS system with or without delay are stochastically stable.Ourcondition for the system without delay is 0＜β＜λ+μ-σ²/2 which improves that given in[89]forγ=0.For the theoretic results,we employ the undetermined Lyapunov Functions.Then,we do the computer simulation of our SIRS model,which agree well with our theoreticanalysis.What's more,we conjecture that the loss of immunity (i.e.γ≠0) does not modifythe stochastic stability threshold.Secondly,we prove that under the condition 3k^1/2≥|β|,there is a unique D-randomattractor in L²(Rⁿ) for the stochastic Ginzburg-Landau equation with additive noise on theentire n-dimensional space Rⁿ.Due to the difficulty that Sobolev embeddings are no longercompact and the compactness of solutions cannot be obtained using the standard method,the unboundedness of the domain is a great challenging for proving the existence of anattractor.In the case of deterministic equations,this difficulty has been overcome by theenergy equation approach.In the stochastic case,there has never been an effective method.In our paper,firstly we show that the stochastic Ginzburg-Landau equation with additivenoise can be recast as a random dynamical system.And then we demonstrate the systempossesses a unique D-random attractor,the asymptotic compactness is established by themethod of uniform estimates on the tails of its solutions.In Part 2,we study the homoclinic and heteroclinic bifurcation using the moving frameapproach.The dynamics of the system confined in the tubular neighborhood of the homoclinicloop is heavily relied on the behavior of the stable and unstable manifolds (or foliation)of the saddle,which are locally developed along the loop.So in the meaning of the firstapproximation,the tangent vector bundles,situated in the tangent space bundles confined onthe loop which is the intersection of the stable manifold and the unstable manifold,inherit andexhibit sufficiently the properties (such as the geometry,the invariance,the contractibility,the expansiveness,etc.) of the system near the loop.Thus,if we select carefully some tangentvector bundles along the loop and some others complement to them to form a moving frame,then there is no strange to see the resulting system will certainly have the simplest form.We first consider the bifurcation of non-resonance 3D homoclinic bifurcation with inclination-flip.When the homoclinic loop takes no strong inclination flip,we obtain that the bifurcationresult is unique such that either the homoclinic orbit persists or one unique periodic orbit bi-furcates.And then,under the condition of strong inclination flip,we also get the bifurcationresult is unique such that either the homoclinic orbit persists or one unique periodic orbit bi-furcates.For the homoclinic bifurcation with strong inclination flip of'weak'type,we obtainthe existence of 1-homoclinic bifurcation,2-fold periodic orbit bifurcation,a period-doublingbifurcation surface P²ⁿ of 2^n-1 periodic orbit and a 2ⁿ-homoclinic bifurcation surface H²ⁿfor (?) n∈N.And we figure out the complete bifurcation diagram based on the existenceregion of the corresponding bifurcation.And then,we investigate the bifurcation of twisted double homoclinic loops.That is,m≥0,n≥0,m+n＞0,l≥2,f(0)=0.Under the condition of one twisted orbit,weobtain the existence and uniqueness of the 1-1 double homoclinic loop,2-1 double homoclinicloop,2-1 right homoclinic loop,1-1 large homoclinic loop,2-1 large homoclinic loop and 2-1 large period orbit.For the case of double twisted orbits,we obtain the existence or non-existence of 1-1 double homoclinic loop,1-2 double homoclinic loop,2-1 double homoclinicloop,2-2 double homoclinic loop,2-1 large homoclinic loop,1-2 large homoclinic loop,2-2large homoclinic loop,2-2 right homoclinic loop,2-2 large homoclinic loop,2-2 left homoclinicloop and 2-2 large period orbit.Here,"left"or"right"means the corresponding orbit circu-lates in the small neighborhood of the original double homoclinic loops whereas it just takesinfinite time in the neighborhood of one orbit of the double homoclinic loops,eitherΓ₁ orΓ₂.And"large"means that the corresponding orbit moves around in the small neighborhood ofthe original double homoclinic loops and it takes infinite time in the neighborhood of eachhomoclinic orbit.In addition,"p-q"refers to the rounding number in each orbit's neighbor-hood.Precisely speaking,p-q loop will roundΓ₂ p cycles,while it has winding number q ina small neighborhood ofΓ₁.Moreover,the bifurcation surfaces and their existence regions are given.Besides,bifur-cation sets are presented on the 2 dimensional subspace spanned by the first two Melnikovvectors.Finally,we study the bifurcation of heterodimensional cycle with orbit-flip in its non-transversal orbit.The subject of heterodimensional cycle is very challenging and difficult.This is because:the codimension for each connecting orbit is always equal for equidimensionalcycle,which is codimension one,generally speaking.However,for the heterodimensionalcycle,the codimensions for connecting orbits are not equally distributed,which are evenconcentrated.Thus,it will lead to much more degenerate bifurcation equations.Under some generic hypotheses,we give conditions for the existence,uniqueness and non-coexistence of the homoclinic orbit,heteroclinic orbit and periodic orbit.And we also presentthe coexistence conditions for the homoclinic orbit and the periodic orbit.But it is impossiblefor the coexistence of the periodic orbit and the persistent heterodimensional cycle or thecoexistence of the homoclinic loop and the persistent heterodimensional cycle.Moreover,thedouble and triple periodic orbit bifurcation surfaces are established as well.Based on thebifurcation analysis,the bifurcation surfaces and the existence regions are located.