Dynamical Analysis and Periodic Solution of a Chaotic System with Coexisting Attractors

Chaotic attractors with no equilibria, with an unstable node, and with stable node-focus are presented in this paper. The conservative solutions are investigated by the semianalytical and seminumerical method. Furthermore, multiple coexisting attractors are investigated, and circuit is carried out. To study the system’s global structure, dynamics at infinity for this new chaotic system are studied using Poincaré compactification of polynomial vector fields in R 3 . Meanwhile, the dynamics near the infinity of the singularities are obtained by reducing the system’s dimensions on a Poincaré ball. The averaging theory analyzes the periodic solution’s stability or instability that bifurcates from Hopf-zero bifurcation.

Ancient solutions of the homogeneous Ricci flow on flag manifolds

For any flag manifold M=G/K of a compact simple Lie group G we describe non-collapsing ancient invariant solutions of the homogeneous unnormalized Ricci flow. Such solutions pass through an invariant Einstein metric on M, and by [13] they must develop a Type I singularity in their extinction finite time, and also to the past. To illustrate the situation we engage ourselves with the global study of the dynamical system induced by the unnormalized Ricci flow on any flag manifold M=G/K with second Betti number b2(M) = 1, for a generic initial invariant metric. We describe the corresponding dynamical systems and present non-collapsed ancient solutions, whose α-limit set consists of fixed points at infinity of MG. Based on the Poincaré compactification method, we show that these fixed points correspond to invariant Einstein metrics and we study their stability properties, illuminating thus the structure of the system’s phase space.

Dynamical Analysis and Simulation of a New Lorenz-Like Chaotic System

This work presents and investigates a new chaotic system with eight terms. By numerical simulation, the two-scroll chaotic attractor is found for some certain parameters. And, by theoretical analysis, we discuss the dynamical behavior of the new-type Lorenz-like chaotic system. Firstly, the local dynamical properties, such as the distribution and the local stability of all equilibrium points, the local stable and unstable manifolds, and the Hopf bifurcations, are all revealed as the parameters varying in the space of parameters. Secondly, by applying the way of Poincaré compactification in ℝ 3 , the dynamics at infinity are clearly analyzed. Thirdly, combining the dynamics at finity and those at infinity, the global dynamical behaviors are formulated. Especially, we have proved the existence of the infinite heteroclinic orbits. Furthermore, all obtained theoretical results in this paper are further verified by numerical simulations.

Unbounded solutions of models for glycolysis

The Selkov oscillator, a simple description of glycolysis, is a system of two ordinary differential equations with mass action kinetics. In previous work the authors established several properties of the solutions of this system. In the present paper we extend this to prove that this system has solutions which diverge to infinity in an oscillatory manner at late times. This is done with the help of a Poincaré compactification of the system and a shooting argument. This system was originally derived from another system with Michaelis–Menten kinetics. A Poincaré compactification of the latter system is carried out and this is used to show that the Michaelis–Menten system, like that with mass action, has solutions which diverge to infinity in a monotone manner. It is also shown to admit subcritical Hopf bifurcations and thus unstable periodic solutions. We discuss to what extent the unbounded solutions cast doubt on the biological relevance of the Selkov oscillator and compare it with other models for the same biological system in the literature.

Dynamics at Infinity and Existence of Singularly Degenerate Heteroclinic Cycles in Maxwell–Bloch System

In this paper, global dynamics of the Maxwell–Bloch system is discussed. First, the complete description of its dynamic behavior on the sphere at infinity is presented by using the Poincaré compactification in R3. Second, the existence of singularly degenerate heteroclinic cycles is investigated. It is proved that for a suitable choice of the parameters, there is an infinite set of singularly degenerate heteroclinic cycles in Maxwell–Bloch system. Specially, the chaotic attractors are found nearby singularly degenerate heteroclinic cycles in Maxwell–Bloch system by combining theoretical and numerical analyses for a special parameter value. It is hoped that these theoretical and numerical value results are given a contribution in an understanding of the physical essence for chaos in the Maxwell–Bloch system.

Modeling and analysis of dynamics for a 3D mixed Lorenz system with a damped term

The work in this paper consists of two parts. The one is modelling. After a method of classification for three dimensional (3D) autonomous chaotic systems and a concept of mixed Lorenz system are introduced, a mixed Lorenz system with a damped term is presented. The other is the analysis for dynamical properties of this model. First, its local stability and bifurcation in its parameter space are in detail considered. Then, the existence of its homoclinic and heteroclinic orbits, and the existence of singularly degenerate heteroclinic cycles, are studied by rigorous theoretical analysis. Finally, by using the Poincaré compactification for polynomial vector fields in R 3 ${\mathbb{R}}^{3}$ , a global analysis of this system near and at infinity is presented, including the complete description of its dynamics on the sphere near and at infinity. Simulations corroborate corresponding theoretical results. In particular, a possibly new mechanism for the creation of chaotic attractors is proposed. Some different structure types of chaotic attractors are correspondingly and numerically found.

Detecting Hidden Chaotic Regions and Complex Dynamics in the Self-Exciting Homopolar Disc Dynamo

In 1979, Moffatt pointed out that the conventional treatment of the simplest self-exciting homopolar disc dynamo has inconsistencies because of the neglect of induced azimuthal eddy currents, which can be resolved by introducing a segmented disc dynamo. Here we return to the simple dynamo system proposed by Moffatt, and demonstrate previously unknown hidden chaotic attractors. Then we study multistability and coexistence of three types of attractors in the autonomous dynamo system in three dimensions: equilibrium points, limit cycles and hidden chaotic attractors. In addition, the existence of two homoclinic orbits is proved rigorously by the generalized Melnikov method. Finally, by using Poincaré compactification of polynomial vector fields in three dimensions, the dynamics near infinity of singularities is obtained.