Dynamical Systems | Department of Mathematics

# Dynamical Systems

Major Elective for B.Sc. (Research) Mathematics.

Credits (Lec:Tut:Lab): 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 102 Calculus II or MAT 103 Mathematical Methods I. And MAT 104 Mathematical Methods II or MAT 160 Linear Algebra I.

Overview:

Detailed Syllabus:

Basic Concepts: Discrete and continuous dynamical systems. Linear and nonlinear systems and principle of superposition. Linear and nonlinear forces.  Concepts of evolution, iterations, orbits, fixed points, periodic and aperiodic (chaotic) orbits. Basics of Linear Algebra: Symmetric & Skew-symmetric matrices, matrix norm and singular value decomposition. Eigenvalues, left and right eigenvectors, and dynamical interpretation. Canonical forms; simple and non-simple canonical systems. System of Equations.

Stability Analysis:

Stability of a fixed point and classification equilibrium states (for both discrete and continuous systems). Concept of bifurcation and classification of bifurcations. Concepts of Lyapunov stability & Asymptotic stability of orbits. Phase Portraits of various Linear and Nonlinear systems. Hopf bifurcation. Concept of attractors and repellers, limit cycles and torus.

Phenomena of Bifurcation:

Definition of bifurcation. Bifurcations in one, two and higher dimensional systems. Hopf, Period doubling, Saddle node, Transcritical bifurcations. Feigenbaum’s number. Local and Global bifurcations. Homoclinic & Hetero-clinic points and orbits. Poincaré-Bendixson Theorem. Conservative and Dissipative Systems.

Investigation Tools & Chaos Theory:

Time Series and Phase Plane Analysis, Poincaé Map & Section. Lyapunov Characteristic Exponents. Hamiltonian Systems and concept of iIntegrability and non-integrability. Concept of Chaos and Chaotic evolution of a Dynamical System. Measure of Chaos. Routs to Chaos.

Applications:

Applications of Dynamical Systems, (to Physics, Biology, Economics with Examples). Mathematical Models. Population Dynamics. Investigation of Evolutionary Phenomena in Logistic Map, Lotka-Volterra System, Duffing Oscillator, Oscillation of Nonlinear Pendulum, Predator-Prey Systems etc.

Tutorial:

Drawing orbits of a system for given initial values. Clear Demonstration of Linear and Nonlinear Systems. Calculation of fixed points for given system and examine their stabilities (discrete and continuous). Drawing time series graphs, phase portraits for regular and chaotic systems. Cobweb Plots. Calculations of Eigenvalues and Eigenvectors corresponding to any fixed point. Plotting Bifurcation diagrams of 1 and 2 dimensional systems. Calculations of Lyapunov exponent.

Software such as MATHEMATICA / MATLAB will be used as needed.

References:

1. Nonlinear Systems, by P. G. Drazin, Cambridge University Press India.
2. An Introduction to Chaotic Dynamical Systems by R. L. Devaney, Addison Wesley, 1989.
3. Chaos in Dynamical Systems, by Edward Ott, Cambridge University Press, 2002
4. Chaotic Dynamics – An Introduction, by G. L. Baker and J. P. Gollub, Cambridge University Press, 1996.
5. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, by J. Guckenheimer and P. Holmes, Springer, 1983.

Past Instructors: L M Saha

Course Code:
MAT386
Course Credits:
4.00
Department:
Course Level: