1. Basic mathematical methods for continuous non-linear dynamical systems

1.1. Phase portrait methods.
Dynamical system. Conservative and dissipative systems. Hamiltonian Eqs. Phase space. Liouville theorem. Poincare mapping. Poincare index.
Phase portrait of dynamical system. Main characteristics of phase portrait. Attractors. Examples.

1.2. Strange attractors. Important example: Lorenz system. Liapunov methods.
Strange attractors. Their fractal structure. Hausdorff dimension. Renyi dimension.
Lorenz system. Its properties (homogeneous, symmetric, dissipative, finite). Stable points.
Lyapunov stability. Lyapunov exponents for Lorenz system. Lyapunov dimension.

1.3. Chaos and its characteristics.
Chaos in dynamical systems. Its qualitative and quantitative characteristics. Power spectrum. Auto-correlation functions.

2. Solitons

2.1. Basic solitonic Eqs.
Self-similar solutions of PDEs. Basic solitonic Eqs: KdV (Korteweg de Vries), Nonlinear Shrodinger, Sine-Gordon, Sinh-Gordon. Fermi – Pasta - Ulam problem for oscillator chain, continualization of the discrete model and Boussinesq equation.

2.2. Direct methods of integrations for solitonic PDEs.
Hirota’s method. Backlund transforms.

2.3. Inverse methods of integrations for solitonic PDEs.
Fourier tranform and the idea of inverse scattering method of transforms. “Lax pair” (Lax operators). Inverse scattering method. Discrete and continuous parts of the spectrum.

LITERATURE

General introduction to non-linear dynamics methods:
1. Encyclopedia of Nonlinear Science. Edited by A. Scott. NY: 2005.
2. J. M. T. Thompson and H. B. Stewart. Nonlinear Dynamics and Chaos. N.Y.: John Wiley & Sons, 2002.
3. L. Lam Nonlinear Physics for Beginners. Fractals, Chaos, Solitons, Pattern Formation, Cellular Algebra and Complex Systems. Singapore: World Scientific, 1998. [set of clear introductory articles: overview, fractals, chaos, solitons]

Dynamical systems:
1. G. L. Barker and J. P. Gollub. Chaotic Dynamics. An introduction. Cambridge University Press, 1990. [visualization of dynamical systems, Poincare sections and mapping, chaotic attractors, Lyapunov exponents]
2. D. W. Jordan, P. Smith. Nonlinear ordinary differential equations. Oxford: Clarendon Press, 1977. [phase diagram, stability]

Fractals and chaos:
1. P. S. Addison. Fractals and Chaos. Bristol: Institute of Physics Publishing, 1997.
2. M. F. Barnsley. Fractals Everywhere. Boston: Academic press Professional, 1993. [introduction of fractals and its dimensions]
3. R. M. Crownover. Introduction to Fractals and Chaos. Boston: Jones and Barlett Publishers, 1995. [introduction of fractals and their applications for dynamical systems]
4. J. L. McCauley. Chaos, Dynamics and Fractals. An algorithmic approach to deterministic chaos. Cambridge University Press, 1994. [if you want to get a deep knowledge on fractals]

Literature support for the topics on physics:
1. H. Goldstein, C. Poole, J. Safko. Classical mechanics. Pearson Education.
2. N. Kryloff and N. Bogoliubov. Introduction to Non-Linear Mechanics. Princeton University Press, 1947.
3. G. B. Whitham. Linear and Nonlinear Waves. N.Y.: John Wiley & Sons, 1974.

Further reading on chaos:
1. R. L. Devaney. Chaotic dynamical systems. 2003.
2. J. L. Willems. Stability Theory of Dynamical Systems. London: Nelson, 1970.
3. C. Robinson. Dynamical Systems. Stability, Symbolic Dynamics, and Chaos. London: CRC Press, 1999.
4. A. Katok, B. Hasselblatt. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, 1995.