We study a generic piecewise monotone map f : [0, 1] → [0, 1] with a particular type of entropy-carrying set X. We show under certain conditions that maps sufficiently C-close to f with more topological entropy have entropy-carrying sets containing X. New homoclinic orbits are created to every periodic point of X. 2000 Mathematics Subject Classification (MSC2000). 37E05, 37B40, 37Gxx

We consider systems of partial differential equations equivariant under the Euclidean group E(n) and undergoing steady-state bifurcation (with nonzero critical wavenumber) from a fully symmetric equilibrium. A rigorous reduction procedure is presented that leads locally to an optimally small system of equations. In particular, when n = 1 and n = 2 and for reaction-diffusion equations with gener...

The response of an excitable neuron to trains of electrical spikes is relevant to the understanding of the neural code. In this paper, we study a neurobiologically motivated relaxation oscillator, with appropriately identified fast and slow coordinates, that admits an explicit mathematical analysis. An application of geometric singular perturbation theory shows the existence of an attracting in...

Our paper takes into account a new bifurcation case of the cycle length and a fifth-order difference equation dynamics of ym+1 = ymy α m−2y β m−4 + ym + y α m−2 + y β m−4 + γ ymy α m−2 + y α m−2y β m−4 + ymy β m−4 + γ+ 1 , m = 0, 1, 2, 3, . . . , where γ ∈ [0,∞) , α,β ∈ Z+, and y−4,y−3,y−1,y−2,y0 ∈ (0, ∞) is took into consideration. The disturbance of initials lead to a distinction of cycle len...

A theory for noise-driven subcritical instabilities in spatially extended systems is put forward. The theory allows one to calculate the critical bifurcation parameter for a first-order phase transition in such non-equilibrium systems in the thermodynamic limit and analyse the mechanism of phase transition. Two examples with distinctive features are studied in detail to demonstrate the usefulne...

New models for schistosomiasis are developed. These models incorporate several realistic features including drug treatment for human hosts, an infection age in snail hosts, density-dependent birth rate of snails, distribution of schistosomes within human hosts, and disease-induced mortality in both human and snail hosts. The qualitative and quantitative mathematical properties of the models are...

In this paper, we improve and extend the approach of Wang and Xia for stability analysis of biological systems by making use of Gröbner bases, (CAD-based) quantifier elimination, and discriminant varieties, as well as the stability criterion of Liénard and Chipart, and showing how to analyze the stability of Hopf bifurcation points. The stability and bifurcations for a class of self-assembling ...

In this paper, we consider the indefinite scalar curvature problem on R. We propose new conditions on the prescribing scalar curvature function such that the scalar curvature problem on R (similarly, on S ) has at least one solution. The key observation in our proof is that we use the bifurcation method to get a large solution and then after establishing the Harnack inequality for solutions nea...

Under the influence of long-range attractive and short-range repulsive forces, thin liquid films rupture and form complex dewetting patterns. This paper studies this phenomenon in one space dimension within the framework of fourth-order degenerate parabolic equations of lubrication type. We derive the global structure of the bifurcation diagram for steady-state solutions. A stability analysis o...

A method of developing students’ algorithmic thinking is demonstrated in the paper first. This is followed by an example of a physical problem step by step modeling and several approaches to its solution. These approaches demonstrate possible interdisciplinary learning, which is considered to be a very important part of future teachers’ education. Especially the relationship among physics, comp...