New Developments in the Partial Control of Chaotic Systems

Abstract

This thesis has been developed during the past years in the Research Group on Nonlinear Dynamics, Chaos and Complex Systems of the URJC. All this work is devoted to new developments of the partial control method. The main goal of this technique is to control chaotic dynamics with escapes and affected by external disturbances. This thesis is organized as follows. Chapter 1. Introduction This chapter is a brief introduction to the main topics of our work. We describe the first steps of chaos theory and how the need of control arose in that field. Then we analize the main features of transient chaotic behaviour and the first attempts to control it. Finally, we show the evolution of the partial control method from the first ideas until the point this thesis was started. Chapter 2. Description of the partial control method The partial control method is used under different approaches along this thesis. In this chapter a general introduction to this method is given. The motivation to apply this method and the main dynamical conditions to apply it are presented. An algorithm to compute safe sets and how this set is used to control the system, is briefly described. Chapter 3. Partial control to avoid a species extinction In this chapter we present the first application of the partial control method in this thesis. Here, we have worked with an ecological model that describes the interaction between 3-species: resources, consumers and predators. The interest of this model lies in the fact that, for a choice of parameters, transient chaos involves the extinction of one of the species. Taking into account that the system is affected by external disturbances, we implement the partial control with the goal of avoiding the extinction. Chapter 4. Controlling chaos in the Lorenz system The Lorenz system is one of the most well-known systems in Nonlinear Dynamics. This makes it an excellent candidate to show how the partial control method can be applied in different ways depending on our requirements. For a certain choice of parameters, trajectories of this system eventually converge to two fixed points attractors via transient chaos. In order to avoid this escape, we describe three different ways based on building maps of one, two and three dimensions, respectively. Pros and cons of each one are analized, and for the first time a three-dimensional safe set is shown. Chapter 5. A different application of partial control In all the previous works, the computed safe set were used to keep the trajectories in the region of interest. Here we consider a new application of the safe set. Without any extra computation, we show in this chapter how this set can be also used to accelerate the escape of the trajectories if necessary. This fact, allows the controller a great flexibility to avoid or force the escape when it is required. Chapter 6. When disturbance affects a parameter Random maps are discrete dynamical systems where one or several of their parameters vary randomly at every iteration. It is possible to find in these maps a transient chaotic behaviour, however few methodologies have been proposed to control them. Here, we propose an extension of the partial control method, that we call parametric partial control. To do that, we consider the scenario where the disturbances and the control terms are affecting directly some parameter of the system. To illustrate how the method works, we have applied it to three paradigmatic models in Nonlinear Dynamics, the logistic map, the H´enon map and the Duffing oscillator. Chapter 7. Controlling time-delay coordinate maps Delay-coordinate maps are a family of discrete maps where the dynamics have certain dependence on past states of the system. We consider these maps specially relevant because they can appear in the delay reconstruction technique of time series from experimental data. The main obstacle of these maps is that only the present state of the system can be modified. In this chapter, we study the convenience of the application of partial control under this constraint. To do that, a modified version of the partial control method is presented and some examples are illustrated. For the first time, it is treated a system that exhibits Hamiltonian chaos, and also a system that presents hyperchaos. Chapter 8. A new approach: the safety functions With the aim of dealing with more general systems and new circumstances where the control is needed, a new approach of partial control is proposed. Instead of using the information given by the safe sets, we have developed a new tool called the safety function. This tool allows us to know how safe is each point and also enable us to deal with more general situations where the system is affected by disturbances in different ways. In this chapter, we have designed an algorithm to compute these functions. Furthermore, we also show how safe sets and the safety functions are closely connected. To illustrate this new approach some examples are treated with special emphasis in the time series example. We believe that this work will open a door to new and stimulating applications in the field of control of chaotic systems. Chapter 9. Discussion A brief overview of the main results of this thesis and the possible research lines for a future work, is given in this chapter. Chapter 10. Conclusions In this chapter we summarize the main conclusions of the research work done during this thesis.