Narrow Results

The paper devotes to the synthesis of local and global analysis of a discrete model describing the second-order digital filter with nonlinear signal processors. The discrete model gives rise to a two-dimensional non-invertible map, whose basins of attraction have complicated topological structures due to the intrinsic multi-stability. The influences of joint parameters on the local dynamics are presented in great details. Both theoretical and numerical results are plotted on the two-dimensional parametric planes. To show more detailed bifurcation structure, the isoclines are extended to higher periodic orbits for detecting the cusps of resonant entrainments. Invariant manifolds and critical curves are employed to illustrate the global dynamics of the model vividly. The tangency and intersections of invariant manifolds expound the process of erosions of basins of attraction. The global bifurcations of basins of attraction are deduced dynamically by critical curves.

Basins generated by a noninvertible mapping formed by two symmetrically coupled logistic maps are studied when the only parameter λ of the system is modified. Complex patterns on the plane are visualized as a consequence of the basins' bifurcations. According to the already established nomenclature in the literature, we present the relevant phenomenology organized in different scenarios: fractal islands disaggregation, finite disaggregation, infinitely disconnected basin, infinitely many converging sequences of lakes, countable self-similar disaggregation and sharp fractal boundary. By use of critical curves, we determine the influence of zones with different number of first rank preimages in the mechanisms of basin fractalization.

Invariant areas generated by two-dimensional endomorphisms are studied using the method of critical curves. The invariant areas considered in this paper are obtained by iterating the noninvariant set constituted by the connected basin of an attracting set or the immediate basin of a nonconnected basin. This new kind of invariant area is of mixed type in the sense that its boundary is made up of critical curves arcs and arcs of saddle manifolds. The presentation is illustrated by three examples. A bifurcation changing the degree of connexity of an invariant area is described.

This paper is an attempt to give new results, by a computer-assisted study, on some global bifurcations that change the structure of the domain of feasible trajectories (bounded discrete trajectories having an ecological sense) which can be obtained by the union of all rank preimages of axes. Three two-dimensional recurrence equations (or maps) are analyzed. The two first maps are degenerated invertible maps (i.e. the inverses of them are well defined except a set of zero lebergue measure) for which the basins of attractor are obtained by the backward iteration of a stable manifold of a saddle fixed point belonging to the basin boundary, and the interior domains of feasible trajectories are given by the intersection between the basin of attractor and the first quadrant. The other is a noninvertible map which is investigated by the use of critical curves, a powerful tool for the analysis of global properties of two-dimensional maps.

Two-dimensional (Z₁–Z₃–Z₁) maps are such that the plane is divided into three unbounded open regions: a region Z₃, whose points generate three real rank-one preimages, bordered by two regions Z₁, whose points generate only one real rank-one preimage. This paper is essentially devoted to the study of the structures, and the global bifurcations, of the basins of attraction generated by such maps. In particular, the cases of fractal structure of such basins are considered. For the class of maps considered in this paper, a large variety of dynamic situations is shown, and the bifurcations leading to their occurrence are explained.

This paper aims to provide some new results, by a computer-assisted study, on some global bifurcations that change the domains of feasible trajectories (bounded discrete trajectories having an ecological sense). It is shown that the domain boundaries of two-dimensional maps can be generally obtained by the union of all rank preimages of two axes. A discrete predator–prey ecosystem is employed to demonstrate how the global properties of persistence in a biological model can be analyzed. The main results of this paper are from the study of some global bifurcations that change the stable manifold of a saddle fixed point belonging to the domain boundary. The basin fractalization is explained by using two types of nonclassical singular sets. The first one, critical curve, separates the plane into two regions having a different number of real inverses (here zero and two). The second one is a line of nondefinition for one of the two inverses of the map, i.e. this inverse has a vanishing denominator on this line.

We consider in-line and overlap geometry models of Josephson junctions with point or rectangular inhomogeneity and investigate the effect of their location on the Josephson vortices and the current. We analyze numerically the critical dependencies "current-magnetic field" caused by one- and two-point current injections. The obtained results elucidate the relation between these critical curves and the fractions of the injection current at the ends of the junction. We also find out similarities between the exponentially shaped junctions, and those with inhomogeneity at the end when a two-point current injection is present. We juxtapose the critical curves of the distinct junctions with inner inhomogeneity and discuss the similarity between them and the Josephson junctions with phase shifts.

The transitions of Josephson junctions from a superconducting mode to a resistive one as bifurcations of the static solutions of appropriately posed multiparametric compound boundary- and eigenvalue problems are interpreted and solved using the continuous analog of Newton method.

In this paper, a dynamical Cournot model with nonlinear demand and R&D spillovers is established. The system is symmetric when the duopoly firms have same economic environments, and it is proved that both the diagonal and the coordinate axes are the one-dimensional invariant manifolds of system. The results show that Milnor attractor of system can be found through calculating the transverse Lyapunov exponents. The synchronization phenomenon is verified through basins of attraction. The effects of adjusting speed and R&D spillovers on the dynamical behaviors of the system are discussed. The topological structures of basins of attraction are analyzed through critical curves, and the evolution process of “holes” in the feasible region is numerically simulated. In addition, various global bifurcation behaviors, such as two kinds of contact bifurcation and the blowout bifurcation, are shown.

In this paper we consider a two species population model based on the discretisation of the original Lotka-Volterra competition equations. We analyse the global dynamic properties of the resulting two-dimensional noninvertible dynamical system in the case where the interspecific competition is considered to be “mixed”. The main results of this paper are derived from the study of some global bifurcations that change the structure of the attractors and their basins. These bifurcations are investigated using the method of Critical Curves.