Flows of Continuous-Time Dynamical Systems with No Periodic Orbit as an Equivalence Class under Topological Conjugacy Relation

Ahmad Ahmad

doi:10.3844/jmssp.2011.207.215

Abstract

Problem statement: Flows of continuous-time dynamical systems with the same number of equilibrium points and trajectories, and which has no periodic orbit form an equivalence class under the topological conjugacy relation. Approach: Arbitrarily, two trajectories resulting from two distinct flows of this type of dynamical systems were written as a set of points (orbit). A homeomorphism which maps between these two sets is then built. Using the notion of topological conjugacy, they were shown to conjugate topologically. By the arbitrariness in selection of flows and their respective initial states, the results were extended to all the flows of dynamical system of that type. Results: Any two flows of such dynamical systems were shown to share the same dynamics temporally along with other properties such as order isomorphic and homeomorphic. Conclusion: Topological conjugacy serves as an equivalence relation in the set of flows of continuous-time dynamical systems which have same number of equilibrium points and trajectories, and has no periodic orbit.

Highlights

Dynamical system is a system where its temporal evolution from some initial state is dictated by a set of rules (Eduard et al, 1999)
We have shown that the flows of any two continuous-time dynamical systems with the same number of equilibrium points and trajectories, and has no periodic orbit can be conjugated topologically temporally
Their trajectories are linearly ordered and order isomorphic to each other by the relation induced from their flow Fig. 1

Summary

INTRODUCTION

Dynamical system is a system where its temporal evolution from some initial state is dictated by a set of rules (Eduard et al, 1999). On the other hand, (Krishan et al, 2010) shows that by adopting neurofuzzy system, the design of robust controllers for uncertain non-linear dynamical systems can be done without resorting to system model simplifications and linearization and without imposing structural conditions on the system uncertainties. This shows that the use of concept of dynamical system is vast. The type of dynamical system that we will be discussing in this study will be of continuous-time dynamical system non-periodic orbit where the evolution rule is a flow. For some survey on periodic system, readers may read (Baryarama et al, 2005) where the periodicity of the HIV/AIDS epidemic in a mathematical model that incorporates complacency is discussed or (Ibrahim et al, 2007) for the periodic and non periodic (complex) behavior of a model of bioreactor with cell recycling

MATERIALS AND METHODS

RESULTS

DISCUSSION

CONCLUSION