Mathematics

We generalize the greedy and lazy β -transformations for a real base β to the setting of alternate bases β=( β 0 ,?? β p?? ) , which were recently introduced by the first and second authors as a particular case of Cantor bases. As in the real base case, these new transformations, denoted T β and L β respectively, can be iterated in order to generate the digits of the greedy and lazy β -expansions of real numbers. The aim of this paper is to describe the dynamical behaviors of T β and L β . We first prove the existence of a unique absolutely continuous (with respect to an extended Lebesgue measure, called the p -Lebesgue measure) T β -invariant measure. We then show that this unique measure is in fact equivalent to the p -Lebesgue measure and that the corresponding dynamical system is ergodic and has entropy 1 p log( β p?? ??β 0 ) . We then express the density of this measure and compute the frequencies of letters in the greedy β -expansions. We obtain the dynamical properties of L β by showing that the lazy dynamical system is isomorphic to the greedy one. We also provide an isomorphism with a suitable extension of the β -shift. Finally, we show that the β -expansions can be seen as ( β p?? ??β 0 ) -representations over general digit sets and we compare both frameworks.

This study evaluates data-driven models from a dynamical system perspective, such as unstable fixed points, periodic orbits, chaotic saddle, Lyapunov exponents, manifold structures, and statistical values. We find that these dynamical characteristics can be reconstructed much more precisely by a data-driven model than by computing directly from training data. With this idea, we predict the laminar lasting time distribution of a particular macroscopic variable of chaotic fluid flow, which cannot be calculated from a direct numerical simulation of the Navier-Stokes equation because of its high computational cost.

We study the discrete-time dynamical systems of a model of wild mosquito population with distinct birth (denoted by β ) and death (denoted by μ ) rates. The case μ=β was considered in our previous work. In this paper we prove that for β<μ the mosquito population will die and for β>μ the population will survive, namely, the number of the larvaes goes to infinite and the number of adults has finite limit α μ , where α>0 is the maximum emergence rete.

One of the most common hypotheses on the theory of non-smooth dynamical systems is a regular surface as switching manifold, at which case there is at least well-defined and established Filippov dynamics. However, systems with singular switching manifolds still lack such well-established dynamics, although present in many relevant models of phenomena where multiple switches or multiple abrupt changes occur. At this work, we leverage a methodology that, through blow-ups and singular perturbation, allows the extension of Filippov dynamics to the singular case. Specifically, tridimensional systems whose switching manifold consists of an algebraic manifold with transversal self-intersection are considered. This configuration, known as double discontinuity, represents systems with two switches and whose singular part consists of a straight line, where ordinary Filippov dynamics is not directly applicable. For the general, non-linear case, beyond defining the so-called fundamental dynamics over the singular part, general theorems on its qualitative behavior are provided. For the affine case, however, theorems fully describing the fundamental dynamics are obtained. Finally, this fine-grained control over the dynamics is leveraged to derive Peixoto like theorems characterizing semi-local structural stability.

We study global dynamics of an SIR model with vaccination, where we assume that individuals respond differently to dynamics of the epidemic. Their heterogeneous response is modeled by the Preisach hysteresis operator. We present a condition for the global stability of the infection-free equilibrium state. If this condition does not hold true, the model has a connected set of endemic equilibrium states characterized by different proportion of infected and immune individuals. In this case, we show that every trajectory converges either to an endemic equilibrium or to a periodic orbit. Under additional natural assumptions, the periodic attractor is excluded, and we guarantee the convergence of each trajectory to an endemic equilibrium state. The global stability analysis uses a family of Lyapunov functions corresponding to the family of branches of the hysteresis operator.

A predator-prey model with functional response Holling type II, Allee effect in the prey and a generalist predator is considered. It is shown that the model with strong Allee effect has at most two positive equilibrium point in the first quadrant, one is always a saddle point and the other exhibits multi-stability phenomenon since the equilibrium point can be stable or unstable. While the model with weak Allee effect has at most three positive equilibrium point in the first quadrant, one is always a saddle point and the other two can be stable or unstable node. In addition, when the parameters vary in a small neighbourhood of system parameters the model undergoes to different bifurcations, such as saddle-node, Hopf and Bogadonov-Takens bifurcations. Moreover, numerical simulation is used to illustrate the impact in the stability of positive equilibrium point(s) by adding an Allee effect and an alternative food source for predators.

Delayed neural field models can be viewed as a dynamical system in an appropriate functional analytic setting. On two dimensional rectangular space domains, and for a special class of connectivity and delay functions, we describe the spectral properties of the linearized equation. We transform the characteristic integral equation for the delay differential equation (DDE) into a linear partial differential equation (PDE) with boundary conditions. We demonstrate that finding eigenvalues and eigenvectors of the DDE is equivalent with obtaining nontrivial solutions of this boundary value problem (BVP). When the connectivity kernel consists of a single exponential, we construct a basis of the solutions of this BVP that forms a complete set in L 2 . This gives a complete characterization of the spectrum and is used to construct a solution to the resolvent problem. As an application we give an example of a Hopf bifurcation and compute the first Lyapunov coefficient.

We study the dynamics of a circadian oscillator model which was proposed by Tyson, Hong, Thron and Novak. This model indicates a molecular mechanism for the circadian rhythm in Drosophila. After giving a detailed study of the equilibria, we further investigate the effects of the rates of mRNA degradation and synthesis. When the rate of mRNA degradation is rather fast, we prove that there are no periodic orbits in this model. When the rate of mRNA degradation is slow enough, this model is transformed into a slow-fast system. Then based on the geometric singular perturbation theory, we prove the existence of canard explosion, relaxation oscillations, homoclinic/heteroclinic orbits and saddle-node bifurcations as the rates of mRNA degradation and synthesis change. Finally, we give the biological interpretation of the obtained results and point out that this model can be transformed into a Liénard-like equation, which could be helpful to investigate the dynamics of the general case.

Very little is currently known about the dynamics of non-polynomial entire maps in several complex variables. The family of transcendental Hénon maps offers the potential of combining ideas from transcendental dynamics in one variable, and the dynamics of polynomial Hénon maps in two. Here we show that these maps all have infinite topological and measure theoretic entropy. The proof also implies the existence of infinitely many periodic orbits of any order greater than two.

In this paper we develop an SIR model for coinfection. We discuss how the underlying dynamics depends on the carrying capacity K : from a simple dynamics to a more complicated. This can help in understanding of appearance of more complicated dynamics, for example, chaos etc. The density dependent population growth is also considered. It is presented that pathogens can invade in population and their invasion depends on the carrying capacity K which shows that the progression of disease in population depends on carrying capacity. Our approach is based on a bifurcation analysis which allows to generalize considerably the previous Lotka-Volterra type models.