Introduction to nonlinear dynamics and chaos

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Upon completion of the module, the students will be capable of deriving differential equations that properly represent classical dynamic systems. They will be able to identify the expected equilibrium points and to assess the corresponding stability. Further on, the students will be able to qualitatively assess solutions of differential equations, which are hardly accessible otherwise; they will be also capable to understand the related physical meaning and implications

This course offers an intuitive introductory approach to the extensive world of nonlinear dynamic systems. The main purpose is to enhance the understanding of students on how to look at nonlinear ordinary differential equations and corresponding solutions. This will be achieved by introducing techniques that allow qualitatively assessment of the dynamics of a given system by applying a “geometrical way of thinking’. The last three sessions of the course are devoted to the analysis of classical chaotic systems. The course is based on the fascinating book “Nonlinear dynamics and chaos” of Steven H. Strogatz. This course is aimed at engineers. Rigorous mathematical analysis under the theorem-proof methodology is not the objective of this course. Instead, the method of analysis is based on geometrical representations, where phase diagrams are a perfect example. Description of sessions 1. Brief history of nonlinear dynamics and chaos. About the importance of being nonlinear. Transformation of Partial Differential Equations (PDEs) to Ordinary Differential Equations (ODEs). About the importance of the pendulum and mass-spring-damper models. Application to thermoacoustics and combustion instabilities. 2. Qualitative assessment of solutions of first-order nonlinear ODE’s: a geometric way of thinking. Fixed points and stability. Linear stability analysis. Existence and uniqueness theorems. Potentials. Application to Population growth. 3. Bifurcations: how a solution of a nonlinear ODE changes when parameters are varied? Saddle-node bifurcation. Transcritical bifurcation. Application to a Laser Threshold. 4. Super- and subcritical pitchfork bifurcation. Application to Overdamped bead on a rotating hoop and insect Outbreak 5. Second order linear ODEs. Definition and examples. Classification of linear systems. Application to Vibrations of a mass hanging from a linear spring and ‘love’ affairs. 6. Second order nonlinear ODEs. Phase portraits. Existence, uniqueness and topological consequences. Fixed points and linearization. Application to population growth in the presence of predators: rabbits versus sheeps. 7. Conservative systems. Nonlinear centers. Application to a pendulum. 8. Limit cycles: Isolated closed trajectories. Van der Pol Oscillator. Ruling out closed orbits. Poincaré-Bendixon Theorem. Relaxation oscillations. Weakly nonlinear oscillators. Regular perturbation theory and its failure. 9. Limit Cycles. Two timing. Average equations. Application to the Van der Pol oscillator. 10. Bifurcations in second-order nonlinear ODEs. Saddle-node, transcritical and pitchfork bifurcations. Super- and subcritical Hopf bifurcations. Application to oscillating chemical reactions. 11. Global bifurcation cycles. Coupled oscillators and quasi-periodicity. Poincaré Maps 12. Introduction to Chaos. Lorenz Equations. Application to a chaotic waterwheel. 13. Simple properties of the Lorenz equations. Chaos on a strange attractor. Lorenz map. 14. Logistic map: Numerics. Logistic map: analysis. Universality and experiments.

Calculus

Almost all sessions open with a practical example of nonlinear dynamic systems, which can be found in several fields of study such as biology, chemistry, neuroscience, structural mechanics, electromagnetism, acoustics and combustion, among others. These examples will be illustrated generally with power-point slides and Youtube videos in order to captivate the attention of the students. Once the system of differential equations is derived in common participation with the students, the corresponding dynamic properties are under a `rationalistic debate’ approach discussed. The purpose of this is to not only consider an answer as the truth but to be able to construct an answer and understand why it can be considered as correct. Subsequently, methods of analysis are introduced where the solutions are qualitatively assessed. Finally, the physical implications of the solutions are discussed. In addition to power-point presentations, the blackboard will be used for mathematical derivations. Also, extracts and examples of he book “Strogatz, S. H. (2014). Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Hachette UK.” will be introduced during the lectures. The course is based on 14 lectures full of examples on classical nonlinear dynamic systems. Instead of a monologue, a dialogue with the students will be aimed at each session. Following the methodology previously exposed, it is expected that the students use analysis (instead of memory) to build intuition and, consequently, be able to `look through’ nonlinear differential equations.

At the end of the course, an individual oral exam of 20 minutes will be carried out. Students should demonstrate that they are able to derive differential equations, identify expected equilibrium points and assess the corresponding stability in the form of calculation tasks. Furthermore it is checked, if they understand the related physical meaning and implications by explaining the methodology and models in their own words. The material to study for the exam will be based on the slides used during the course and some additional information coming principally from the book “Strogatz, S. H. (2014). Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Hachette UK.” Summarizing, a successful oral examination is determined by the ability of the student to qualitatively analyze a 2D system of nonlinear differential equations by means of the graphical methods introduced during the course.