Nonlinear dynamics of ion-acoustic waves in quantum plasmas with exchange-correlation effects

Nonlinear properties of ion-acoustic waves (IAWs) are studied in electron-ion (EI) degenerate plasma with the electron exchange-correlation effects by using the quantum hydrodynamic (QHD) model. To investigate arbitrary amplitude IAWs, we have reduced the model equations into a system of ordinary differential equations using a traveling wave transformation. Computational investigations have been performed to examine the combined effect of Bohm potential and exchange-correlation potential significantly modifies the dynamics of IAWs by employing the concept of dynamical systems. The equilibrium points of the model are determined and its stability natures are analyzed. The phase portrait and Poincaré return map of the dynamical system are displayed numerically. Quasiperiodic as well as chaotic dynamics of the system are confirmed through the Poincaré return map diagrams.

Dynamics Near the Heterodimensional Cycles with Nonhyperbolic Equilibrium

In this paper, bifurcations of heterodimensional cycle with one nonhyperbolic equilibrium and one saddle-focus in three-dimensional vector fields are investigated. We study the interaction of a transcritical bifurcation with a codimension-0/codimension-2 heteroclinic cycle. Based on the construction of a Poincaré return map, we obtain the expressions of parametric curves of homoclinic and heteroclinic connections around the heterodimensional cycle as well as periodic orbits. Furthermore, the configurations of the parametric curves corresponding to different bifurcations are illustrated.

Integrable zero-Hopf singularities and three-dimensional centres

In this paper we show that the well-known Poincaré–Lyapunov non-degenerate analytic centre problem in the plane and its higher-dimensional version, expressed as the three-dimensional centre problem at the zero-Hopf singularity, have a lot of common properties. In both cases the existence of a neighbourhood of the singularity in the phase space completely foliated by periodic orbits (including equilibria) is characterized by the fact that the system is analytically completely integrable. Hence its Poincaré–Dulac normal form is analytically orbitally linearizable. There also exists an analytic Poincaré return map and, when the system is polynomial and parametrized by its coefficients, the set of systems with centres corresponds to an affine variety in the parameter space of coefficients. Some quadratic polynomial families are considered.

REFLECTIONS ON THE HYPERBOLIC PLANE

The most general solution to the Einstein equations in 4 = 3 + 1 dimensions in the asymptotic limit close to the cosmological singularity under the BKL (Belinskii–Khalatnikov–Lifshitz) hypothesis can be visualized by the behavior of a billiard ball in a triangular domain on the Upper Poincaré Half Plane (UPHP). The billiard system (named "big billiard") can be schematized by dividing the successions of trajectories according to Poincaré return map on the sides of the billiard table, according to the paradigms implemented by the BKL investigation and by the CB–LKSKS (Chernoff–Barrow–Lifshitz–Khalatnikov–Sinai–Khanin–Shchur) one. Different maps are obtained, according to different symmetry-quotienting mechanisms used to analyze the dynamics. In the inhomogeneous case, new structures have been uncovered, such that, in this framework, the billiard table (named "small billiard") consists of 1/6 of the previous one. The connections between the symmetry-quotienting mechanisms are further investigated on the UPHP. The relation between the complete billiard and the small billiard are also further explained according to the role of Weyl reflections. The quantum properties of the system are sketched as well, and the physical interpretation of the wave function is further developed. In particular, a physical interpretation for the symmetry-quotienting maps is proposed.

Resonant Homoclinic Flips Bifurcation in Principal Eigendirections

A codimension-4 homoclinic bifurcation with one orbit flip and one inclination flip at principal eigenvalue direction resonance is considered. By introducing a local active coordinate system in some small neighborhood of homoclinic orbit, we get the Poincaré return map and the bifurcation equation. A detailed investigation produces the number and the existence of 1-homoclinic orbit, 1-periodic orbit, and double 1-periodic orbits. We also locate their bifurcation surfaces in certain regions.

Limit Cycle Bifurcations from a Nilpotent Focus or Center of Planar Systems

We study analytic properties of the Poincaré return map and generalized focal values of analytic planar systems with a nilpotent focus or center. We use the focal values and the map to study the number of limit cycles of this kind of systems and obtain some new results on the lower and upper bounds of the maximal number of limit cycles bifurcating from the nilpotent focus or center. The main results generalize the classical Hopf bifurcation theory and establish the new bifurcation theory for the nilpotent case.

HORSESHOE IN THE HYPERCHAOTIC MCK CIRCUIT

The well-known Matsumoto–Chua–Kobayashi (MCK) circuit is of significance for studying hyperchaos, since it was the first experimental observation of hyperchaos from a real physical system. In this paper, we discuss the existence of hyperchaos in this circuit by virtue of topological horseshoe theory. The two disjoint compact subsets producing a horseshoe found in a specific 3D cross-section, both expand in two directions under the fourth Poincaré return map, this fact means that there exists hyperchaos in the circuit.

Feedback control of an underactuated planar bipedal robot with impulsive foot action

A planar underactuated bipedal robot with an impulsive foot model is considered. The analysis extends previous work on a model with unactuated point feet of Westervelt et al. to include the actuator model of Kuo. The impulsive actuator at each leg end is active only during the double support phase, which results in the model being identical to the model with unactuated point feet for the single support phase. However, the impulsive foot actuation results in a different model for the double support map. Conditions for the existence of a hybrid zero dynamics for the robot with foot actuation are studied. A feedback design method is proposed that integrates actuation in the single and double support phases. A stability analysis is performed using a Poincaré return map. As in Kuo's model, a more efficient gait is demonstrated with an impulsive foot action.

PERIOD-DOUBLING SCENARIO WITHOUT FLIP BIFURCATIONS IN A ONE-DIMENSIONAL MAP

In this work a one-dimensional piecewise-smooth dynamical system, representing a Poincaré return map for dynamical systems of the Lorenz type, is investigated. The system shows a bifurcation scenario similar to the classical period-doubling one, but which is influenced by so-called border collision phenomena and denoted as border collision period-doubling bifurcation scenario. This scenario is formed by a sequence of pairs of bifurcations, whereby each pair consists of a border collision bifurcation and a pitchfork bifurcation. The mechanism leading to this scenario and its characteristic properties, like symmetry-breaking and symmetry-recovering as well as emergence of coexisting attractors, are investigated.