Periodicals of Engineering and Natural Sciences ISSN 2303-4521 Vol. 7, No. 3, September 2019, pp.1345-1353 Available online at: http://pen.ius.edu.ba 1345 Topological geometry analysis for complex dynamic systems based on adaptive control method M. Lellis Thivagar 1 , Abdulsattar Abdullah Hamad 2 1,2 School of Mathematics, Madurai Kamaraj University, Madurai, Tamil Nadu, India Article Info ABSTRACT Received, 2019 Several models had been proposed for dynamic systems, and different criteria had been analyzed for such models such as, Hamiltonian, synchronization, Lyapunov expansion, and stability. The geometry criteria play a significant part in analyzing dynamic systems and some study articles analyze the geometry of such topics. The synchronization and the complex-network control with specified topology, meanwhile, the exact topology may be unknown. In the present paper, and by making use of the adaptive control method, a proposed control method is developed to determine the actual topology. The basic idea in the proposed method is to receive evolution of the network-nodes. Keyword: Lu dynamic system model, Hamiltonian, synchronization, Lyapunov expansion, stability Corresponding Author: Abdulsattar Abdullah Hamad, 2 School of Mathematics, Madurai Kamaraj University, Madurai, Tamil Nadu, India Email: satar198700@gmail.com , al_kasrage@yahoo.com 1. Introduction Since the 1950s and 1960s, new sciences have begun to emerge and capture the curiosity and interest of scientists. Among them are Systems Theory, Chaos Theory, Cybernetics and Artificial Intelligence. Systems theory was founded by the German biologist Ludwig von Bertalanffy, the English economist Kenneth Boulding and others between 1940 and 1970, and is based on the principles of physics, biology and applied engineering. This science then continued its growth and intertwined with many other sciences including: philosophy, sociology, organizational theory, management, economics and other sciences. Systems theory seeks not only to present the world as systems and knowledge of how these systems work and the common links between all existing systems in the world. Are all theoretical, applied and human sciences. Therefore, knowledge and understanding of the principles of systems theory can gain one comprehensive understanding of science and then the whole world that science seeks to read and discover [1]. The whole world is mobile systems or living machines that exist and continue in a certain medium and interact with other systems (machines) and have decay factors inside them and decay factors outside them remain resistant until the collapse of resistance in the end, decomposing its elements to join other systems still work. It applies to everything from galaxies, stars, planets and not to humans and other living things. Systems in the world can be divided into open systems, closed systems (or) simple systems, complex systems (or) fixed systems and mobile systems [2]. 1.1 Open System A system that is defined by a basic connection between it and its surroundings. A closed system is a system that tends to be confined to itself and avoids interaction with environmental data, needs, expectations and aspirations. A closed system tends to ignore external considerations. Closed systems are defined by being caught within boundaries that restrict their autonomy and communication with the setting as the structures very nature does not allow it to separate from that system factors from the surroundings.