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Overview on Complex Systems Theory and Chaos Theory
Published in Sergey V. Samoilenko, Kweku-Muata Osei-Bryson, Creating Theoretical Research Frameworks Using Multiple Methods, 2017
Sergey V. Samoilenko, Kweku-Muata Osei-Bryson
As demonstrated by the Lorenz attractor, a chaotic system does not have to be complex, for a chaotic behavior is a result of sensitivity to initial conditions applied to a reiteration of the rule describing the system. However, the complex phenomena and artifacts of this world are more fittingly perceived as nonlinear CSs. Consequently, it is of value to investigate the conditions under which behavior of CSs could become chaotic. It is important to emphasize that a chaotic system is a system that exhibits, or is capable of exhibiting, a chaotic behavior—a system may or may not be behaving chaotically at a given point in time. In general, there are three stages, of phases, of the behavior of a system—ordered, critical (edge of chaos), and chaotic. The process of the changing the behavior—as the system transitions between the stages—is referred to as phase transition.
Equilibria, periodic orbits and limit cycles
Published in Christian Mazza, Michel Benaïm, Stochastic Dynamics for Systems Biology, 2016
Christian Mazza, Michel Benaïm
In dimension 1 every alpha or omega limit set (for a bounded trajectory) consists of equilibria (see exercice 10.2.3). In dimension 2 (see section 10.3), it may also be a periodic orbit or an heteroclinic cycle. In larger dimensions the structure of limit sets is much richer. It is easy to construct toral limit sets (see exercise (10.6.2)), and there are dynamics which exhibit chaotic behaviours. Deterministic chaos in small dimensions is a relatively new discovery. It goes back to 1963 with the discovery by Edward Norton Lorenz1 of what is now known as the Lorenz Attractor. Before Lorenz we believed that turbulent behaviours required a large number of degrees of freedom. Since the work of Lorenz we know that three dimensional differential equations can produce “deterministic chaos.” Lorenz wanted to explain with a simple model certain climate phenomena. With this purpose, he considered fluid in a two dimensional box heated from above and cooled down from below. Drastic simplifications of fluid mechanics equations led him to consider the following system: () {x˙=σ(y−x),y˙=ρx−y−xz,z˙=xy−βz.
Fractional optical solitons with stochastic properties of a wick-type stochastic fractional NLSE driven by the Brownian motion
Published in Waves in Random and Complex Media, 2021
Ben-Hai Wang, Yue-Yue Wang, Chao-Qing Dai
Functions and are white noise functions, which are related to the Brownian motion. As we all know, chaos theory is about bifurcation, periodic motion and aperiodic motion entanglement in nonlinear systems under certain parameters, and chaotic motion in natural sciences usually refers to a random behavior in deterministic systems. Considering the random nature of the Brownian motion, we use the Lorentz chaotic system to influence fractional stochastic soliton solutions of wick-type SFNLSE. Lorenz found that the so-called ‘butterfly effect’ is a chaotic phenomenon of climate change caused by simple convection, and chaotic systems are highly sensitive to initial values. Lorentz differential equation system is a typical chaotic system model, and it describes a singular attractor (Lorentz attractor) in motion [38]. In Lorentz equations of the following form, chaos occurs by selecting different parameters [38]. The evolution diagram of a typical singular chaotic attractor is shown in Figure 1(a) when we set the parameter and initial value of Lorentz chaos model as [39,40].
Understanding the geometry of dynamics: the stable manifold of the Lorenz system
Published in Journal of the Royal Society of New Zealand, 2018
The Lorenz system is a classical example of a dynamical system that exhibits sensitive dependence on initial conditions. This means that any two points that lie arbitrarily close to each other will move apart under the flow Φ so dramatically that it is impossible to verify, after a reasonably short period of time, whether they were ever close to each other. Lorenz (1963) derived his example as a much-simplified model of convection in the atmosphere. The standard parameter values , , and are the ones provided by Lorenz as a relatively realistic choice. For these parameter values, all trajectories will converge quickly to a strange attracting object, known as the butterfly attractor or Lorenz attractor, which occupies only a small part of the phase space . Hence, most studies in the literature have focussed on explaining the behaviour of the system on the Lorenz attractor. Geometrically, the Lorenz attractor is a rather intriguing object: it is larger than a one-dimensional curve, or even set of curves, but it is not a two-dimensional object; mathematically, we say that it has fractal dimension and such an attractor is called a strange attractor.
Virtual sensing network for statistical process monitoring
Published in IISE Transactions, 2023
Alexander Krall, Daniel Finke, Hui Yang
The Lorenz attractor is a well-known chaotic system whose signal vector is represented by The system changes according to where and are model parameters. We assume that the system is in control for the set of parameters Shifts in these model parameters will cause the system to become out of control. We vary parameters according to Figure 6 to test the performance of the proposed VS monitoring framework.