Center Bifurcation in the Lozi Map

We consider the well-known Lozi map, which is a 2D piecewise linear map depending on two parameters. This map can be considered as a piecewise linear analog of the Hénon map, allowing to simplify the rigorous proof of the existence of a chaotic attractor. The related parameter values belong to a part of the parameter plane where the map has two saddle fixed points. In the present paper, we investigate a different part of the parameter plane, namely, the vicinity of the curve related to a center bifurcation of the fixed point. A distinguishing property of the Lozi map is that it is conservative at the parameter value corresponding to this bifurcation. As a result, the bifurcation structure close to the center bifurcation curve is quite complicated. In particular, an attracting fixed point (focus) can coexist with various attracting cycles, as well as with chaotic attractors, and the number of coexisting attractors increases as the parameter point approaches the center bifurcation curve. The main result of the present paper is related to the rigorous description of this bifurcation structure. Specifically, we obtain, in explicit form, the boundaries of the main periodicity regions associated with the pairs of complementary cycles with rotation number [Formula: see text]. Similar approach can be applied to other periodicity regions. Our study contributes also to the border collision bifurcation theory since the Lozi map is a particular case of the 2D border collision normal form.

Dynamics of the meromorphic families $f_\lambda=\lambda \tan^pz^q$

This paper continues our investigation of the dynamics of families of transcendental meromorphic functions with finitely many singular values all of which are finite. Here we look at a generalization of the family of polynomials $P_a(z)=z^{d-1}(z- \frac{da}{(d-1)})$, the family $f_{\lambda}=\lambda \tan^p z^q$. These functions have a super-attractive fixed point, and, depending on $p$, one or two asymptotic values. Although many of the dynamical properties generalize, the existence of an essential singularity and of poles of multiplicity greater than one implies that significantly different techniques are required here. Adding transcendental methods to standard ones, we give a description of the dynamical properties; in particular we prove the Julia set of a hyperbolic map is either connected and locally connected or a Cantor set. We also give a description of the parameter plane of the family $f_{\lambda}$. Again there are similarities to and differences from the parameter plane of the family $P_a$ and again there are new techniques. In particular, we prove there is dense set of points on the boundaries of the hyperbolic components that are accessible along curves and we characterize these points.

Multistability Regions and Related Basins of Multiattractors in Piecewise Linear Discontinuous Map from Financial Markets

By adding trend followers, we extend the model given by Tramontana et al. from one-dimensional ([Formula: see text]D) piecewise linear discontinuous (PWLD) map to a new 2D PWLD map. Using this map in financial markets, we describe the bifurcation mechanisms associated with the appearance/disappearance of cycles, which may be related to several cases: border collision bifurcations; Poincaré equator collision bifurcations; degenerate flip bifurcations in both supercritical and subcritical cases. We investigate the multistability regions in the parameter plane and related basins of multiattractors to uncover the reason for the unpredictability of the internal law of price fluctuations in financial market.

Bifurcation Characteristics of Fundamental and Subharmonic Impact Motions of a Mechanical Vibration System with Motion Limiting Constraints on a Two-Parameter Plane

A two-degree-of-freedom periodically forced system with multiple gaps and rigid constraints is studied. Multiple types of impact vibrations occur at each rigid constraint and interact with each other, which results in the emergence of some complex transitions in the system. Through the cosimulation of the key parameters gap value δ between the two masses and the excitation force frequency ω, the types, existence areas, and bifurcation regularities of the periodic and subharmonic motions can be obtained on the (ω, δ)-parameter plane. In the corresponding three-dimensional surface diagram of the maximum impact velocity, the distribution law of the maximum impact velocity at each constraint can be obtained. The transition laws of fundamental impact motions in the low-frequency parameter domain are studied, and two types of transition regions in the transitions of adjacent fundamental impact motions are found: tongue-like regions and hysteresis regions. Moreover, these two types of transition regions show some atypical partitioning and deformation due to the combined effects of impact vibrations at each constraint. By combining the two-parameter plane diagram and the three-dimensional surface diagram, the effect of changing the gap values between each mass and the fixed constraint and the damping coefficient ζ on the dynamic characteristics of the system is studied. Combining the existence areas of periodic motions and the distribution of maximum impact velocity can provide guidance for the reasonable selection of system parameters.

On the transverse size of hadrons at asymptotically high energies

We show that Gribov diffusion of the partons in the impact parameter plane, which leads to the square-root-of-logarithmic growth of the transverse size of the hadrons, can occur only simultaneously with a similar diffusion in the transverse-momentum space. At the same time, a restriction of the partons in the transverse momenta entails an increase in their propagation in the impact parameter plane. Ultimately this leads to a logarithmic growth of the transverse size of hadrons at asymptotically high energies.

Hopf Bifurcation and Stability Crossing Curve in a Planktonic Resource–Consumer System with Double Delays

In this paper, a planktonic resource–consumer system with two delays is investigated and the coefficients depend on [Formula: see text] one of the two delays. Firstly, the property of solution and the existence of equilibrium are given. The dynamical analysis of the system including stability and Hopf bifurcation by using the delays as parameters is carried out. Both the single delay and two delays can cause the system to produce Hopf bifurcation and the stable switching phenomena may exist. Furthermore, using the crossing curve methods, we obtain the stable changes of equilibrium in two-delay parameter plane, which generalizes the results of the system that the coefficients do not depend on delay. Furthermore, the numerical simulation results show that the theoretical analyses are correct when the delays change.

Three-Dimensional Torus Breakdown and Chaos With Two Zero Lyapunov Exponents in Coupled Radio-Physical Generators

Using an example a system of two coupled generators of quasi-periodic oscillations, we study the occurrence of chaotic dynamics with one positive, two zero, and several negative Lyapunov exponents. It is shown that such dynamic arises as a result of a sequence of bifurcations of two-frequency torus doubling and involves saddle tori occurring at their doublings. This transition is associated with typical structure of parameter plane, like cross-road area and shrimp-shaped structures, based on the two-frequency quasi-periodic dynamics. Using double Poincaré section, we have shown destruction of three-frequency torus.