The emergence of chaotic behavior in many physical systems has triggered the curiosity of scientists for a long time. Their study has been concentrated in understanding which are the underlying laws that govern such dynamics and eventually aim to suppress such (often) undesired behavior. In layman terms, a system is defined chaotic when two orbits that initially are very near to each other will diverge in exponential time. Clearly, this translates to the fact that a chaotic system can hardly have regular behavior, a property that is also often required even for humanmade systems. An example is that of particle accelerators used a lot in the study of experimental physics. The main principle is that of forcing a large number of particles to move periodically in a toroidal space in order to collide with each other. Another example is that of the tokamak, a particular accelerator built to generate plasma, one of the states of the matter. In both cases, it is crucial for the sake of the accelerating process, to have regular periodic behavior of the particles instead of a chaotic one. In this dissertation, we have studied the question of chaos in mathematical models for the motion of magnetically charged particles inside the tokamak in the presence or absence of plasma. We start by a model introduced by Cambon et al., which describes in general mathematical terms, also known as the Duffing modes, the formalism of the above problem. The central core of this work reviews the necessary mathematical tools to tackle this problem, such as the theorem of the Linked Twisted maps and the variational Hamiltonian equations which describe the evolutionary dynamics of the system under consideration. Extensive analytical and numerical tools are required in this thesis work to investigate the presence of chaos, known as chaos indicator. The main ones we have used here are the Poincar ́e Map, the Maximum Lyapunov Exponent (MLE), and the SALI and GALI methods. Using the techniques mentioned above, we have studied our problem analytically and validated our results numerically for the particular case of the Duffing equation, which applies to the motion of charged particles in the tokamak. In detail, we first discuss the presence of chaotic dynamics of charged particles inside an idealized magnetic field, sug gested by a tokamak type configuration. Our model is based on a periodically perturbed Hamiltonian system in a halfplane r ¿ 0. We propose a simple mechanism producing complex dynamics, based on the theory of Linked Twist Maps jointly with the method of stretching along the paths. A key step in our argument relies on the monotonicity of the period map associated with the unperturbed planar system. In the second part of our results, we give an analytical proof of the presence of complex dynamics for a model of charged particles in a magnetic field. Our method is based on the theory of topological horseshoes and applied to a periodically perturbed Duffing equation. The existence of chaos is proved for sufficiently large, but explicitly computable, periods. In the latter part, we study the generalized forementioned Duffing equations and study the chaoticity using the Melnikov topological method and verify the results numerically for the models of Wang & You and the tokamak one.
Periodic solutions and chaotic dynamics in a Duffing equation model of charged particles / Oltiana Gjata , 2020 Jul 06. 32. ciclo, Anno Accademico 2018/2019.
Periodic solutions and chaotic dynamics in a Duffing equation model of charged particles
GJATA, OLTIANA
20200706
Abstract
The emergence of chaotic behavior in many physical systems has triggered the curiosity of scientists for a long time. Their study has been concentrated in understanding which are the underlying laws that govern such dynamics and eventually aim to suppress such (often) undesired behavior. In layman terms, a system is defined chaotic when two orbits that initially are very near to each other will diverge in exponential time. Clearly, this translates to the fact that a chaotic system can hardly have regular behavior, a property that is also often required even for humanmade systems. An example is that of particle accelerators used a lot in the study of experimental physics. The main principle is that of forcing a large number of particles to move periodically in a toroidal space in order to collide with each other. Another example is that of the tokamak, a particular accelerator built to generate plasma, one of the states of the matter. In both cases, it is crucial for the sake of the accelerating process, to have regular periodic behavior of the particles instead of a chaotic one. In this dissertation, we have studied the question of chaos in mathematical models for the motion of magnetically charged particles inside the tokamak in the presence or absence of plasma. We start by a model introduced by Cambon et al., which describes in general mathematical terms, also known as the Duffing modes, the formalism of the above problem. The central core of this work reviews the necessary mathematical tools to tackle this problem, such as the theorem of the Linked Twisted maps and the variational Hamiltonian equations which describe the evolutionary dynamics of the system under consideration. Extensive analytical and numerical tools are required in this thesis work to investigate the presence of chaos, known as chaos indicator. The main ones we have used here are the Poincar ́e Map, the Maximum Lyapunov Exponent (MLE), and the SALI and GALI methods. Using the techniques mentioned above, we have studied our problem analytically and validated our results numerically for the particular case of the Duffing equation, which applies to the motion of charged particles in the tokamak. In detail, we first discuss the presence of chaotic dynamics of charged particles inside an idealized magnetic field, sug gested by a tokamak type configuration. Our model is based on a periodically perturbed Hamiltonian system in a halfplane r ¿ 0. We propose a simple mechanism producing complex dynamics, based on the theory of Linked Twist Maps jointly with the method of stretching along the paths. A key step in our argument relies on the monotonicity of the period map associated with the unperturbed planar system. In the second part of our results, we give an analytical proof of the presence of complex dynamics for a model of charged particles in a magnetic field. Our method is based on the theory of topological horseshoes and applied to a periodically perturbed Duffing equation. The existence of chaos is proved for sufficiently large, but explicitly computable, periods. In the latter part, we study the generalized forementioned Duffing equations and study the chaoticity using the Melnikov topological method and verify the results numerically for the models of Wang & You and the tokamak one.File  Dimensione  Formato  

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