Thus, on the way to modeling such systems, understanding the related concepts and using appropriate tools and methods close to the desired natural system's behavior are highly needed.Ĭonsidering the required perception of complex dynamical systems, many studies focus on wave propagation and pattern recognition in a dynamical system. Therefore, there are several studies on complex systems using some useful concepts like synchronization, cooperative and synergic evolutions, nonlinear dynamics, chaotic behavior, self-organization, self-adaptation, and so on. This phenomenon is basically possible due to the capability of representing collective dynamical behaviors. An interesting characterization of complex systems is their representation of organized behaviors with no central organizing unit. In fact, complexity is the undeniable part of the natural phenomena including in various physical, chemical, and biological systems. That is why we need to focus on the fundamental dynamical representations while we accept a specific range of reductions in order to release from these complexities. Accordingly, it is not only because of our tremendous lack of knowledge about an individual neuron as the basic block of an excitable tissue, but also the connections and interactions that govern the large arrangement of the neurons forming a greatly intricate network. Therefore, generally, the neuronal system is a dynamical system with serious complexities. In fact, the electrical signal holding the specified information results in mechanical contraction of the cardiac muscle. It is also a good example to mention heartbeat variation and the required rhythmic contraction of the heart muscle. This processing is accomplished through information transference between the neurons under a dynamical law. The brain's great ability in information processing by the use of meaningful interactions between the sensory neurons, motor neurons, and interneurons through electrical propagations have made it a perfect organizing station. Any tissue that can have electrical activities can be considered an excitable tissue, like the heart and the brain tissue. An excitable tissue consisting of a large number of neurons is a good example for complex dynamical systems. Fundamentally, the concepts from qualitative theory of nonlinear differential equations accompanied by basics of bifurcation theory are employed in these models.Ī neuron is the basic nonlinear unit of the neuronal system that is responsible for all the biological behaviors in the body. These qualitative models focus on the qualitative features of oscillatory dynamics or excitable fluctuations of the neuron's membrane. Actually, these two factors have to do with neuromodulator concentrations and applied currents or synaptic inputs. Considering the studies in the biological and neuronal fields of science, there have been some qualitative dynamical neuronal models proposed for the purpose of getting closer to both a neuron's activities and its physiological conditions. In other words, a dynamical system consists of some possible states and their evolution over the time, which is governed by a specific law, that may not be known to us, necessarily. As a simple definition, a system is dynamical if its state change in time. It is believed that dynamical models are the best tools to explain the real world. Therefore, these features provide us an acceptable range of estimation for chaos measurement. Additionally, one positive Lyapunov exponent can denote chaotic demonstrations. One of the main characteristics of chaotic behavior is sensitive dependence on initial conditions. In fact, we need appropriate tools to recognize chaotic behavior both qualitatively and quantitatively. Therefore, it is important to develop our knowledge in this field of science and be able to distinguish chaotic behavior from other behaviors in such dynamical systems. For example, it is confirmed that chaotic heartbeat rate or brain signals guarantee the heart and brain respectively being in their healthy state. Especially in biological systems, it is very important that chaotic demonstrations play an essential role. Determinism means the changes in the state of the system is determined by a specific rule. In chaos theory, we can study how highly complex demonstrations which may seem to be random arise from determinism. Sajad Jafari, in Recent Advances in Chaotic Systems and Synchronization, 2019 1 IntroductionĬhaos theory application ranges from pattern formation in physical, chemical, and even biological systems, space programs, and the discovered chaotic superhighways between the planets, weather forecasting having to do with chaotic demonstrations first uncovered by Edward Norton Lorenz with his famous paper on deterministic nonperiodic flow, etc.
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