Globally Stable Synchronization Conditions in Total Diffusive Linear Bidirectional Coupling Between Continuous Dynamical Systems and Partial Replacement (Rössler and Lorenz) - Laureano, M.; Mendes, D. A.; Ferreira, M. A. M. (EasyChair Preprint Nº 9873, 2025)
In order to obtain asymptotical synchronization, we combine diffusive linear bidirectional coupling with partial replacement on the nonlinear terms of the second system, a coupling version that was less explored. All these bidirectional coupling schemes are established between Lorenz systems or Rössler systems with chaotic behavior/with control parameters that lead to chaotic behavior. The sufficient conditions of global stable synchronization are obtained from a different approach of the Lyapynov direct method for the transversal system. In one coupling we apply a result based on classification of the symmetric matrix AT +A as negative definite, where A is the matrix characterizing the transversal system. In the remaining couplings the sufficient conditions are based on (the) increase/accretion of derivative (quero dizer majoração da derivada) of an appropriate Lyapunov function assuming yet the limitation of certain variables. In fact, the effectiveness of a coupling between systems with equal dimension follows of the analysis of the synchronization error e(t) and, if the system variables can be bounded by positive constants, the derivative of an appropriate Lyapunov function can be increased.(quero dizer majorada) as required by the Lyapynov direct method. In what follows we will always consider two chaotic dynamical systems, since they are sufficient to study the essential in the proposed coupling schemes. Our motivation for researching chaos synchronization methods is to explore their practical application in various scientific areas, such as physics, biology or economics.