## MA 605 - Introduction to Nonlinear Dynamics (3-0-0)

**Instructor:**** **Dr. P.S. Dutta

Nonlinear equations: autonomous and non-autonomous systems, phase portrait, stability of equilibrium points, Lyapunov exponents, periodic solutions, local and global bifurcations, Poincare-Bendixon theorem, Hartmann-Grobmann theorem, Center Manifold theorem.

Nonlinear oscillations: perturbations and the Kolmogorov-Arnold-Moser theorem, limit cycles. Chaos: one-dimensional and two-dimensional Poincare maps, attractors, routes to chaos, intermittency, crisis and quasi periodicity. Synchronization in coupled chaotic oscillators. Applications: Examples from Biology, Chemistry, Physics and Engineering.

**Schedule: **Lectures on Monday, Tuesday, Wednesday.

## MA 718 - Evolutionary Game Theory (3-0-0)

**Instructor****: **Dr. P.S. Dutta

Introduction: Social traps and simple games. Evolutionary stability - Normal form games - Evolutionary stable strategies (ESS) - Characterization of ESS; – The replicator equation - Nash equilibrium and evolutionary stable states - Nash equilibrium strategies - Perfect equilibrium strategies - Examples of replicator dynamics and the Lotka-Volterra equation - The rock-paper-scissors game; – Other game dynamics - Imitation dynamics - General selection dynamics - Best-response dynamics.

Adaptive Dynamics: The repeated Prisoner's Dilemma - Stochastic strategies for the Prisoner's Dilemma - Adaptive Dynamics for the Prisoner's Dilemma - Adaptive dynamics and gradients; - Asymmetric games: Bimatrix games - A differential equation for asymmetric games - The case of two players and two strategies; Dynamics for bimatrix games - Partnership games and zero-sum games - Conservation of volume - Nash-Pareto pairs - Game dynamics and Nash-Pareto pairs.

The hypercycle equation - Permanence - The permanence of the hypercycle - The competition of disjoint hypercycles; - Criteria for permanence: Permanence and persistence for replicator equations - Necessary and Sufficient conditions for permanence; - Replicator networks - Cyclic symmetry.

Discrete dynamical systems in population genetics: The Hardy-Weinberg law - The selection model - The increase in average fitness - The mutation-selection equation - The selection-recombination equation - Fitness under recombination; Continuous selection dynamics: Convergence to a rest point - The location of stable rest points - Density dependent fitness - Mixed strategists and gradient systems; -The selection-mutation model - Mutation and additive selection.

**Schedule: **Lectures on Wednesday, Thursday and Friday.