Time series analysis of turbulent and non-autonomous systems
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Time series analysis is commonly applied to both chaotic and stochastic systems, which are collectively described as turbulence. However, explicitly time-dependent non-autonomous systems can also generate turbulent dynamics, which makes them useful for describing many physical phenomena. Nevertheless, many of the methods used to analyse turbulence are based around autonomous systems. In this paper, time series from the chaotic, stochastic and non-autonomous Duffing system are analysed using these methods to gauge their suitability to non-autonomous systems. It is found that time-dependent representations are vitally important in the study of this class of systems. Moreover, when time-dependence is neglected in the representation a completely deterministic non-autonomous system is often indistinguishable from a stochastic system.Keywords:
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The structure of the physical and strange attractors is inherently associated with the boundedness of fluctuations. The idea behind the boundedness is that a stable long-term evolution of any natural and engineered system is possible if and only if the fluctuations that the system exerts are bounded so that the system permanently stays within its thresholds of stability. It has been established that the asymptotic structure of the physical and strange attractors is identical. Now it is found out that though the non-asymptotic behavior is universal it can be very different, namely: on coarse-graining the physical attractors can exhibit a variety of behavior while the strange attractors always have hyperuniversal properties. Yet, under certain levels of coarse-graining both physical and strange attractors match non-asymptotically a variety of noise type behavior.
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This paper presents a new 3D quadratic autonomous chaotic system which contains five system parameters and three quadratic cross-product terms, and the system can generate a single four-wing chaotic attractor with wide parameter ranges. Through theoretical analysis, the Hopf bifurcation processes are proved to arise at certain equilibrium points. Numerical bifurcation analysis shows that the system has many interesting complex dynamical behaviours; the system trajectory can evolve to a chaotic attractor from a periodic orbit or a fixed point as the proper parameter varies. Finally, an analog electronic circuit is designed to physically realize the chaotic system; the existence of four-wing chaotic attractor is verified by the analog circuit realization.
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Based on the 3D autonomous continuous Lü chaotic system, a new 3D autonomous continuous chaotic system is proposed in this paper, and there are coexisting chaotic attractors in the 3D autonomous continuous chaotic system. Moreover, there are no overlaps between the coexisting chaotic attractors; that is, there are two isolated chaotic attractors (in this paper, named “positive attractor” and “negative attractor,” resp.). The “positive attractor” and “negative attractor” depend on the distance between the initial points (initial conditions) and the unstable equilibrium points. Furthermore, by means of topological horseshoes theory and numerical computation, the topological horseshoes in this 3D autonomous continuous system is found, and the topological entropy is obtained. These results indicate that the chaotic attractor emerges in the new 3D autonomous continuous system.
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Cyber–Physical systems, which is the class of dynamical systems where physical and computational components interact in a tight coordination, are found in many applications, from large-scale distributed systems, such as the electric power grid, to micro-robotic platforms based on legged locomotion, among many others. Due to their mixed nature between physical and computational components, Cyber–Physical systems are well modeled using hybrid dynamical models, which incorporate both continuous and discrete valued state variables. Also, thanks to the flexibility and great variety of optimal control formulations, it is natural to apply optimal control algorithms to solve complex problems in the context of Cyber–Physical systems, such as the verification of a given specification, or the robust identification of parameters under state constraints.
This thesis presents three new computational tools that bring the strength of hybrid dynamical models and optimal control to applications in Cyber–Physical systems. The first tool is an algorithm that finds the optimal control of a switched hybrid dynamical system under state constraints, the second tool is an algorithm that approximates the trajectories of autonomous hybrid dynamical systems, and the third tool is an algorithm that computes the optimal control of a nonlinear dynamical system using pseudospectral approximations.
These results achieve several goals. They extend widely used algorithms to new classes of dynamical systems. They also present novel mathematical techniques that can be applied to develop new, computationally efficient, tools in the context of hybrid dynamical systems. More importantly, they enable the use of control theory in new exciting applications, that because of their number of variables or complexity of their models, cannot be addressed using existing tools.
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Cyber-Physical systems, which is the class of dynamical systems where physical and computational components interact in a tight coordination, are found in many applications, from large-scale distributed systems, such as the electric power grid, to micro-robotic platforms based on legged locomotion, among many others. Due to their mixed nature between physical and computational components, Cyber-Physical systems are well modeled using hybrid dynamical models, which incorporate both continuous and discrete valued state variables. Also, thanks to the flexibility and great variety of optimal control formulations, it is natural to apply optimal control algorithms to solve complex problems in the context of Cyber-Physical systems, such as the verification of a given specification, or the robust identification of parameters under state constraints.This thesis presents three new computational tools that bring the strength of hybrid dynamical models and optimal control to applications in Cyber-Physical systems. The first tool is an algorithm that finds the optimal control of a switched hybrid dynamical system under state constraints, the second tool is an algorithm that approximates the trajectories of autonomous hybrid dynamical systems, and the third tool is an algorithm that computes the optimal control of a nonlinear dynamical system using pseudospectral approximations.These results achieve several goals. They extend widely used algorithms to new classes of dynamical systems. They also present novel mathematical techniques that can be applied to develop new, computationally efficient, tools in the context of hybrid dynamical systems. More importantly, they enable the use of control theory in new exciting applications, that because of their number of variables or complexity of their models, cannot be addressed using existing tools.
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In this work, we define the notions of ‘impulsive non‐autonomous dynamical systems’ and ‘impulsive cocycle attractors’. Such notions generalize (we will see that not in the most direct way) the notions of autonomous dynamical systems and impulsive global attractors in the current published literature. We also establish conditions to ensure the existence of an impulsive cocycle attractor for a given impulsive non‐autonomous dynamical system, which are analogous to the continuous case. Moreover, we prove the existence of such attractor for a non‐autonomous 2D Navier–Stokes equation with impulses, using energy estimates. Copyright © 2016 John Wiley & Sons, Ltd.
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Using a proper choice of the dynamical variables, we show that a non-autonomous dynamical system transforming to an autonomous dynamical system with a certain coordinate transformations can be obtained by solving a general nonlinear first-order partial differential equations. We examine some special cases and provide particular physical examples. Our framework constitutes general machineries in investigating other non-autonomous dynamical systems.
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