Periodic Orbits and Chaos in Nonsmooth Delay Differential Equations

In this paper, we present a method to find periodic solutions for certain types of nonsmooth differential equations or nonsmooth delay differential equations. We apply the method to three examples, the first is a second-order differential equation with a nonsmooth term, in this case the method allows us to find periodic orbits in a nonlinear center. The two remaining examples are first-order nonsmooth delay differential equations. In the first one, there is a stable periodic solution and in the second, the presence of a chaotic attractor was detected. In the latter, the method allows us to obtain unstable periodic orbits within the attractor. For large values of the delay, both examples can be seen as singularly perturbed delay differential equations. For them, an analysis is performed with an associated discrete map which is obtained in the limit of large delays.

DYNAMICS OF BACTERIA-PHAGE INTERACTIONS WITH IMMUNE RESPONSE IN A CHEMOSTAT

A mathematical model of bacteria-phage interaction in the chemostat is formulated, which incorporates the host immune response with an aim to mimic phage therapy in vivo. It is shown that the host immune response induces the backward bifurcation. Thus, there exists the bistability of phage-free equilibrium with the phage-infection equilibrium. More importantly, it is found that the model exhibits the coexistence of a stable phage-infection equilibrium with a stable periodic solution. The crucial implication of these phenomena is that phage infection fails both from the smaller dose of initial injection and from the larger dose of initial injection. Thus, a proper design of phage dose is necessary for phage therapy. Further analysis indicate that the inhibition effects of bacteria and phages can induce periodic oscillations and modulated oscillation.

Analytical and Numerical Investigations of Stable Periodic Solutions of the Impacting Oscillator With a Moving Base and Two Fenders

A vibrating system with impacts, which can be applied to model the cantilever beam with a mass at its end and two-sided impacts against a harmonically moving frame, is investigated. The objective of this study is to determine in which regions of parameters characterizing system, the motion of the oscillator is periodic and stable. An analytical method to obtain stable periodic solutions to the equations of motion on the basis of Peterka's approach is presented. The results of analytical investigations have been compared to the results of numerical simulations. The ranges of stable periodic solutions determined analytically and numerically with bifurcation diagrams of spectra of Lyapunov exponents show a very good conformity. The locations of stable periodic solution regions of the system with a movable frame and two-sided impacts differ substantially from the locations of stable periodic solution regions for the system: (i) with a movable frame and one-sided impacts and (ii) with an immovable frame and two-sided impacts.

Dynamics for a discrete competition and cooperation model of two enterprises with multiple delays and feedback controls

This paper is concerned with a competition and cooperation model of two enterprises with multiple delays and feedback controls. With the aid of the difference inequality theory, we have obtained some sufficient conditions which guarantee the permanence of the model. Under a suitable condition, we prove that the system has global stable periodic solution. The paper ends with brief conclusions.

Twist Periodic Solutions in the Relativistic Driven Harmonic Oscillator

We study the one-dimensional forced harmonic oscillator with relativistic effects. Under some conditions of the parameters, the existence of a unique stable periodic solution is proved which is of twist type. The results depend on a Twist Theorem for nonlinear Hill’s equations which is established and proved here.

Dynamic Behaviors of a Discrete Periodic Predator-Prey-Mutualist System

A nonautonomous discrete predator-prey-mutualist system is proposed and studied in this paper. Sufficient conditions which ensure the permanence and existence of a unique globally stable periodic solution are obtained. We also investigate the extinction property of the predator species; our results indicate that if the cooperative effect between the prey and mutualist species is large enough, then the predator species will be driven to extinction due to the lack of enough food. Two examples together with numerical simulations show the feasibility of the main results.

Research on Walking Gait of Biped Robot Based on a Modified CPG Model

The neurophysiological studies of animals locomotion have verified that the fundamental rhythmic movements of animals are generated by the central pattern generator (CPG). Many CPG models have been proposed by scientific researchers. In this paper, a modified CPG model whose output function issin(x)is presented. The paper proves that the modified model has stable periodic solution and characteristics of the rhythmic movement using the Lyapunov judgement theorem and the phase diagram. A modified locomotion model is established in which the credit-assignment cerebellar model articulation controller (CA-CMAC) algorithm is used to realize the pattern mapping between the CPG output and the musculoskeletal system. And a seven-link biped robot is employed to simulate cyclic walking gait in order to test the validity of the locomotion model. The main findings include the following. (1) The modified CPG model can generate spontaneous oscillations which correspond to biological signals. (2) The analysis of the modified locomotion model reveals that the CA-CMAC algorithm can be used to realize the pattern mapping between the CPG output and the musculoskeletal system.