Period adding bifurcation in a logistic map with memory

We introduce a single step memory dependence in the fully chaotic logistic map. This makes it a two dimensional system in general. However, we show that by using composite functions to define two one dimensional maps, it is possible to obtain some analytic results for the bifurcation structure. Numerical results support the calculated bifurcation scheme and, in addition, yield a further insight which allows the calculation of the convergence ratio for a new period adding scenario.     

Two-component nonlinear Schrödinger models with a double-well potential

We introduce a model motivated by studies of Bose-Einstein condensates (BECs) trapped in double-well potentials. We assume that a mixture of two hyperfine states of the same atomic species is loaded in such a trap. The analysis is focused on symmetry-breaking bifurcations in the system, starting at the linear limit and gradually increasing the nonlinearity. Depending on values of the chemical potentials of the two species, we find numerous states, as well as symmetry-breaking bifurcations, in addition to those known in the single-component setting. These branches, which include all relevant stationary solutions of the problem, are predicted analytically by means of a two-mode approximation, and confirmed numerically. For unstable branches, outcomes of the instability development are explored in direct simulations.     

Local bifurcations of critical periods for cubic Liénard equations with cubic damping

Continuing Chicone and Jacobs’ work for planar Hamiltonian systems of Newton’s type, in this paper we study the local bifurcation of critical periods near a nondegenerate center of the cubic Liénard equation with cubic damping and prove that at most 2 local critical periods can be produced from either a weak center of finite order or the linear isochronous center and that at most 1 local critical period can be produced from nonlinear isochronous centers.     

The spectral method and numerical continuation algorithm for the von Kármán problem with postbuckling behaviour of solutions

In this paper a spectral method and a numerical continuation algorithm for solving eigenvalue problems for the rectangular von Kármán plate with different boundary conditions (simply supported, partially or totally clamped) and physical parameters are introduced. The solution of these problems has a postbuckling behaviour. The spectral method is based on a variational principle (Galerkin’s approach) with a choice of global basis functions which are combinations of trigonometric functions. Convergence results of this method are proved and the rate of convergence is estimated. The discretized nonlinear model is treated by Newton’s iterative scheme and numerical continuation. Branches of eigenfunctions found by the algorithm are traced. Numerical results of solving the problems for polygonal and ferroconcrete plates are presented. Communicated by A. Zhou.     

Erratum to “Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey” [Appl. Math. Modell. (2011) 366-381]

The aim of this paper is to correct two mistakes in [Appl. Math. Modell. (2011) 366-381], which are: the function defining the time rescaling given and the inclusion of a parameter outside of model.For a modified Leslie-Gower type predator-prey model considering the Allee effect on prey, a change of variables and a new time rescaling generating a diffeomorphism is proved; a topologically equivalent system to the original one is obtained, which is the same studied in the mentioned paper; we claim that the results and conclusions obtained are correct and the errors have not further implications.     

Spiking patterns of a minimal neuron to ELF sinusoidal electric field

Neuron as the main information carrier in neural systems is able to generate diverse spiking trains in response to different stimuli. Neuronal spiking patterns are related to the information processing in neural system. This paper investigates the dynamical behaviors of a two-dimensional minimal neuron model exposed to externally-applied extremely low frequency (ELF) sinusoidal electric field (EF). By numerical stimulation, it is found that neuron can exhibit different spiking patterns such as p:q mode-locking (i.e. a periodic oscillation defined as p action potentials generated by q cycle stimulations) and chaotic behaviors, depending on the values of stimulus frequencies. Transitions between different spiking patterns during exposure to the external EF are explored by interspike intervals (ISIs) and average firing rate. It is found that frequencies of the external EF can act as a bifurcation parameter, whose small change can cause the transition in neuronal behaviors. It is shown that a rich bifurcation structure including period-adding without chaos and mode-locking alternated with chaos suggests frequency discrimination of the neuronal firing patterns. Our results can provide a useful insight into the organization of similar bifurcation structures in excitable systems such as neurons under periodic forcing. Through detail analysis of the spiking patterns, we have explained how neuron’s information (spiking patterns) encodes the stimulus information (frequency), and vice versa.     

Bifurcation set and number of limit cycles in Z2-equivariant planar vector fields of degree 5

An Z₂-equivariant polynomial Hamiltonian system of degree 5 with two perturbation terms is considered in this paper. The phase plane (a, b) is divided into 15 different regions which give the bifurcation set of the system. Using the bifurcation theory of planar dynamical system and the method of detection function, we obtain the bifurcation set and the configurations of compound eyes of the system with 21 or 23 limit cycles.     

三维系统中一族闭轨在周期扰动下的分支

讨论一类三维系统在周期扰动下的分支问题.假设此三维系统有一族闭轨,利用 Poincar\'e映射及积分流形定理,得到了在周期扰动下由这族闭轨产生次调和解和不变环面的条件,并讨论了次调和解的鞍结点分支.     

Bifurcation of periodic orbits emanated from a vertex in discontinuous planar systems

We discuss bifurcation of periodic orbits in discontinuous planar systems with discontinuities on finitely many straight lines intersecting at the origin and the unperturbed system has either a limit cycle or an annulus of periodic orbits. Assume that the unperturbed periodic orbits cross every switching line transversally exactly once. For the first case we give a condition for the persistence of the limit cycle. For the second case, we obtain the expression of the first order Melnikov function and establish sufficient conditions on the number of limit cycles bifurcate from the periodic annulus. Then we generalize our results to systems with discontinuities on finitely many smooth curves. As an application, we present a piecewise cubic system with 4 switching lines and show that the maximum number of limit cycles bifurcate from the periodic annulus can be affected by the position of the switching lines.