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In this paper we consider a five-parameter family of planar vector fields $${X_{\mu}: \, \begin{array}{ll} \frac{dx}{dt} = y,\\ \frac{dy}{dt} = x^4+\mu_3x^2+\mu_2x+\mu_1 +y(\mu_4+\mu_5x\\ \quad+\,c(\mu )x^2+x^3h(x,\mu))+y^2Q(x,y,\mu), \end{array}}$$ where μ = (μ 1, μ 2, μ 3, μ 4, μ 5), which is a small parameter vector, and c(0) ≠ 0. The family X μ represents the generic unfolding of a class of nilpotent cusp of codimension five. We discuss the local bifurcations of X μ, which exhibits numerous kinds of bifurcation phenomena including Bogdanov-Takens bifurcations of codimension four in Li and Rousseau (J. Differ. Eq. 79, 132–167, 1989) and Dumortier and Fiddelaers (In: Global analysis of dynamical systems, 2001), and Bogdanov-Takens bifurcations of codimension three in Dumortier et al. (Ergodic Theory Dynam. Syst. 7, 375–413, 1987) and Dumortier et al. (Bifurcations of planar vector fields. Nilpotent singularities and Abelian integrals, 1991). After making some rescalings, we obtain the truncated systems of X μ . For a truncated system, all possible bifurcation sets and related phase portraits are obtained. When the truncated system is a Hamiltonian system, the bifurcation diagram and the related phase portraits are given too. Hopf bifurcations are studied for another truncated system. And it shows that the system has the Hopf bifurcations of codimension at most three, and at most three limit cycles occur in the small neighborhood of the Hopf singularity.

In the analysis of traveling waves it is common that coupled parabolic-hyperbolic problems occur, where the hyperbolic part is not strictly hyperbolic. For example, this happens whenever a reaction diffusion equation with more than one non-diffusing component is considered in a co-moving frame. In this paper we analyze the stability of traveling waves in nonstrictly hyperbolic PDEs by reformulating the problem as a partial differential algebraic equation (PDAE). We prove uniform resolvent estimates for the original PDE problem and for the PDAE by using exponential dichotomies. It is shown that the zero eigenvalue of the linearization is removed from the spectrum in the PDAE formulation and, therefore, the PDAE problem is better suited for the stability analysis. This is rigorously done via the vector valued Laplace transform which also leads to optimal rates. The linear stability result presented here is a major step in the proof of nonlinear stability.

Conditions are given which preclude the existence of a nontrivial periodic orbit for a difference equation in ℝn. The conditions are analogous to those of Bendixson and Dulac for autonomous planar differential equations.

This paper is concerned with the spectrum the Hill operator L(y) = −y′′ + Q(x) y in $${L^{2}_{\rm per}[0, \pi]}$$ . We show that the eigenvalues of L can be characterized by knowing one of its eigenfunctions. Applications are given to nonlinear stability of a class of periodic problems.

This paper is devoted to analyze certain dynamical aspects of the planar mechanical system associated to the potential V = ax 4 + cx 2 y 2 as a function of the real parameters a and c.

In this paper, the stochastic SIR epidemic model with Beddington–DeAngelis incidence rate is investigated. We classify the model by introducing a threshold value $$\lambda $$ . To be more specific, we show that if $$\lambda <0$$ then the disease-free is globally asymptotic stable i.e., the disease will eventually disappear while the epidemic is persistence provided that $$\lambda >0$$ . In this case, we derive that the model under consideration has a unique invariant probability measure. We also depict the support of invariant probability measure and prove the convergence in total variation norm of transition probabilities to the invariant measure. Some of numerical examples are given to illustrate our results.

This paper is a study of chain recurrence and attractors for maps and semiflows on arbitrary metric spaces. The main results are as follows. (i) C. Conley's characterization of chain recurrence in terms of attractors holds for maps and semiflows on any metric space. (ii) An alternative definition of chain recurrence for semiflows is given and is shown to be equivalent to the usual definition. The alternative definition uses chains formed of orbit segments whose lengths are at least 1, while in the usual definition these lengths are required to be arbitrarily long. (iii) The chain recurrent set of a continuous semiflow is the same as the chain recurrent set of its time-one map. (iv) Conditions on a real-valued function are given that ensure that the semiflow generated by its gradient has only equilibria in its chain recurrent set. An example is given (onR 3) showing that a gradient flow may have nonequilibrium chain recurrent points if these conditions are violated.

We present a variant of Newton’s method for computing travelling wave solutions to scalar bistable lattice differential equations. We prove that the method converges to a solution, obtain existence and uniqueness of solutions to such equations with a small second order term and study the limiting behaviour of such solutions as this second order term tends to zero. The robustness of the algorithm will be discussed using numerical examples. These results will also be used to illustrate phenomena like propagation failure, which are encountered when studying lattice differential equations. We finish by discussing the broad application range of the method and illustrate that higher dimensional systems exhibit richer behaviour than their scalar counterparts.

Consideration is devoted to traveling N-front wave solutions of the FitzHugh–Nagumo equations of the bistable type. Especially, stability of the N-front wave is proven. In the proof, the eigenvalue problem for the N-front wave bifurcating from coexisting simple front and back waves is regarded as a bifurcation problem for projectivised eigenvalue equations, and a topological index is employed to detect eigenvalues.