Introduction to nonlinear physics

Dr. Aleksandar Gjurcinovski, Professor

Course content:  

1. Introduction: History of nonlinear dynamics and chaos. Examples of linearity and non-linearity in physics and other sciences. Modeling of differential and differential equation systems.

2. General properties of dynamic systems: Systems of differential equations and examples. Control parameters. Fixed points and their stability. Linear stability analysis. Numerical methods for nonlinear systems. Limit cycles and their properties. Nonlinear oscillators and application. Poincaré-Bendixon theorem. Bifurcations, their classification and examples in physics. Spatio-temporal systems, structure formation and the Turing mechanism. Strange attractors and chaotic behavior.

3. Logistic mapping: Basics of the theory of linear differential equations. One-dimensional mappings, orbits and graphical representation. Period doubling bifurcations. Feigenbaum numbers. Notion of universality. Symbolic dynamics. Routes. Types and stability of fixed points in one-dimensional and two-dimensional mappings. Chaotic behavior and determination of Lyapunov exponents. Chaotic orbits, term and definition. Conjugation and logistic mapping. Pools of attraction. Tent mirroring. Enon's two-dimensional mapping. Other types of mappings.

4. Fractals: Fractals and their construction. Cantor sets. Sierpinski and Koch Fractal Forms. Fractal Pool Boundaries. Mandelbrot's set. Fractal and correlation dimension. Emergence of fractals when describing dynamic systems.

5. Chaotic dynamics and applications: Fractal structures in simple mappings. Mechanism of emergence of strange attractors. Phase volume evolution in chaotic and non-chaotic systems. Notion of information and entropy in nonlinear chaotic systems. Hyperchaotic systems. Hamiltonian chaotic systems. Application of chaos in physics and other sciences.

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