Preservation of Hyperbolic Equilibrium Points and Synchronization in Dynamical Systems C. Miranda-Reyes † , G. Fern´ andez-Anaya † and J.J. Flores-Godoy † † Departamento de F´ ısica y Matem´ aticas,Universidad Iberoamericana, Prolongaci´ on Paseo de la Reforma 880, Lomas de Santa Fe, M´ exico,D.F. C.P. 01219, M´ exico. Abstract—Classic results of the dynamical systems theory are extended and used to study the preservation of synchronization in chaotical dynamical systems. This results show that synchroniza- tion can be preserved after changes are made to the linear part of the dynamical system. When the jacobian matrix of the system is evaluated in the hyperbolic points, the structure of the signs of the eigenvalues of this matrix determine if the system is stable or unstable. In this work, we establish the sufficient conditions to preserve the structure of this hyperbolic points. Also, control tools are used to achieve synchronization in dynamical systems. Numerical simulations to very the effectiveness of the method are presented. Index Terms—Chaotic Systems, control theory, convergence and stability. I. INTRODUCTION Chaotic dynamic systems and chaos control is a theme that has been widely developed. The study of systems that present this kind of behavior is well documented and there are a lot of papers where applications are presented, as described in [1]. Also, there exist several papers where chaos control and synchronization of systems are managed. We can find different sources that deal with this kind of problems [2], [3]. But, there are a few ones that already solve the problem of preservation of synchronization and chaotic structure, we can see in [4], [5] some examples. In [4], mathematical tools to ensure synchronization and preservation of hyperbolic points are developed, in this paper this tools are extended, in particular the stable-unstable mani- fold theorem, and applied to dynamical chaotic systems. The structure of the signs of the Jacobian Matrix evaluated over the hyperbolic point of the system, plays a fundamental role in this work. Preserving this structure is the base for preserving synchronization. The control section of this work is based on the one developed in [6], we have extended this work and we apply this new results to a master-slave system, to prove numerically that after making the system change under the conditions stated in our results, the dynamical system preserves synchronization and its chaotic structure. In the last section, two dynamical system are modeled with numerical methods to show that the results we elaborate are correct. We use two different master-slave systems to show this. II. PRELIMINARIES As we know, the local stability of a dynamical system is completely related to the sign of the real part of the eigenvalues of the matrix that represents the linear part of the system. The main idea of this work is to preserve the stability of a dynamical system after we apply a specific transformation, to achieve this change into the system, we will apply a function that acts over the linear part of the system, represented by the matrix A, but designed to preserve the characteristics of the original system. First, we state the following definitions. Definition 1: Given a function h : C → C, we say that h(λ) is defined on the spectrum of A if there exist the numbers h(λ k ),h ′ (λ k ), ..., h (m k −1) (λ k ), k =1, 2, ..., s, (1) where h (m k −1) (λ k )= d m k −1 h(x) dx m k −1 | x=λ k . (2) Definition 2: An hyperbolic equilibrium point x 0 of a dynamical system ˙ x = f (x) (3) is an equilibrium point of the system (3) if the jacobian matrix ∂f/∂x = Df (x 0 ) evaluated in the point, has eigenvalues with strictly positive real part and eigenvalues with strictly negative real part. Now, we present an example of this kind of functions, to show a possible transformation that can be made to a system preserving either stability or instability of it. Example 1 - Let h(s) = p/s be a function where h(s): C → C and p> 0. This function preserves the sign of the real part of the number. let s = a + bi h(s)= p a + bi = ap − bpi a 2 + b 2 (4) where the real part of h(s) is: Re(h(s)) = ap a 2 + b 2 (5)