## Details of PH4104 (Autumn 2016)

 Level: 4 Type: Theory Credits: 3.0

Course CodeCourse NameInstructor(s)
PH4104 Nonlinear Dynamics Anandamohan Ghosh

Preamble
In this course we introduce the basic ideas of nonlinear dynamics and chaos in classical systems modelled by ordinary differential equations and iterated maps. The level of treatment is not too physics-specific, and students of other departments can also benefit from it. Apart from physical systems, examples are also taken from chemistry and biology.

Syllabus

• Analysis of continuous-time systems: Modeling of dynamical systems: nonlinear ordinary differential equations, flows and vector fields, motion in the phase space. Different types of system behaviour in nonlinear systems: equilibrium, limit cycle, high-periodic orbits, orbit on a torus, quasiperiodicity
and chaos (examples: Lorenz system, Roessler system, Chua's circuit, Sprott circuit); Sensitive dependence on initial condition; Characterization of chaotic orbits: Hausdorff, correlation, information and Lyapunov dimensions; Lyapunov exponent; ergodicity and mixing.

• Analysis of discrete-time maps; Poincare section and reduction to discrete-time dynamical systems or maps; types of fixed points, bifurcations: period-doubling, saddle-node, Neimark-Sacker, and border collision bifurcations; Feigenbaum's universality constants; intermittency, stable and unstable manifolds, coexisting attractors and basins of attraction,
interior crisis and boundary crisis; quasiperiodicity and mode-locking, sine-circle map; Statistics of chaotic orbits: Frobenius-Perron operator and invariant density; delay coordinate embedding; synchronization and control of chaos.

• Fractal geometry: dimension of an object, Mandelbrot
set, Julia set, iterated function systems.

• Spatio-temporal dynamics: Spatio-temporal chaos, pattern formation, turbulence, solitons.

• Examples: Will be drawn from chamical, physical, and biological
systems: Weather prediction and the butterfly effect; Chemical
oscillators and the Belousov-Zabitinsky reaction; Population dynamics:
the logistic map and the predator-prey models; Dynamics of the neuron
and the human heart, etc.

Prerequisite
Exposure to linear algebra, differential equations, and numerical methods.

References

1. S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications in
Physics, Biology, Chemistry and Engineering, Perseus Publishing,
USA (1994).

2. Brian Davies, Exploring Chaos: Theory and Experiment, Perseus
Publishing, USA (1999).

3. Robert C. Hilborn, Chaos and Nonlinear Dynamics, Oxford
University Press, UK, Second Edition (2000).

4. K. T. Alligood, T. D. Sauer and J. A. Yorke, Chaos: An
Introduction to Dynamical Systems, Springer-Verlag, New York (1996).

5. R. Devaney, M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Academic Press, 2nd
edition (2003).

6. B. B. Mandelbrot, The Fractal Geometry of Nature, W.H.Freeman &
Co., New York (1982).

7. M. F. Bernsley, Fractals Everywhere, Academic Press, USA, 2nd
Edition (1993).

8. E. A. Coddington and N. Levinson, Theory of Ordinary Differential
Equations, Krieger Pub Co, 1984.

9. P. Hartman, Ordinary Differential Equations (Classics in Applied
Mathematics), SIAM, 2002.

10. V. Arnold, Ordinary Differential Equations, MIT Press (1978).

#### Course Credit Options

Sl. No.ProgrammeSemester NoCourse Choice
1 IP 1 Elective
2 IP 3 Elective
3 IP 5 Not Allowed
4 MR 1 Not Allowed
5 MR 3 Not Allowed
6 MS 7 Elective
7 RS 1 Elective
8 RS 2 Elective