Unit:
Κατεύθυνση Εφαρμοσμένα ΜαθηματικάLibrary of the School of Science
Supervisors info:
Στρατής Ιωάννης, Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Original Title:
The Melnikov method and Shilnikov bifurcations
Translated title:
The Melnikov method and Shilnikov bifurcations
Summary:
Ιn this master's thesis we shall see that the Poincaré sequences can reveal significant features of a system and that the appearance of fixed points is closely connected with periodic solutions. Poincaré maps can be used to detect underlying structure, such as periodic solutions having the forcing or a subharmonic frequency.
The main aim of this thesis is a systematic introduction to the theory of homoclinic bifurcations in second order non-autonomous dynamical systems and autonomous dynamic systems in R^3. In particular, we refer to Melnikov's method and we will prove that, if the Melnikov function has zero points, then homoclinic chaos occurs, in the sense of transversely unstable and stable manifolds. In the calculation of the Melnikov function, integrals are obtained which are calculated by applying Cauchy's integral residue theorem. We will apply the Melnikov method to the example of the perturbed Duffing system.
Thereafter, we shall study Shilnikov's theory, which gives criteria for the existence of homoclinic bifurcations in the form of saddle-spiral. We will examine important three-dimensional systems of differential equations that meet the requirements of Shilnikov theory, such as the Rössler system, the Lorenz system, and the double scroll attractor.
Studying these systems involves reducing them to lower dimensional discrete dynamical systems, and then invoking symbolic dynamics. In these cases the discrete system is a horseshoe map, which was one of the first chaotic systems to be analyzed completely.
Furthermore, we shall see that the period doubling of the Duffing oscillator arises from bifurcations of a discrete system consisting of Poincaré first returns and we will examine the simple model of the logistic equation. Then we will extend the method of the Lyapunov exponent to nonlinear differential equations, such as the Duffing equation and
the Lorenz equations, where we will apply this procedure later.
Finally, we shall see that there are features which chaos might be expected to have, such as period doubling, the Smale horseshoe map, the sensitive dependence on initial conditions etc.
Main subject category:
Science
Keywords:
Melnikov, Shilnikov, Poincare, Lorenz, Rossler, Duffing, bifurcations, chaos, attractor, Lyapunov
File:
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Melnikov_Loula_Chalou.pdf
7 MB
File access is restricted only to the intranet of UoA.
