Introduction to Theory of Ordinary Differential equations

1. First order equations, logistic population model, bifurcation and constant harvesting.

2. Planar Linear systems, second-order differential equations, planar systems, eigenvalues and eigenvectors, solutions an phase portraits of planar linear systems of differential equations: Real distinct eigenvalues, Complex eigenvalues, Repeated real eigenvalues; changing coordinates.

3. Classification of planar systems, the trace-determinant plane, dynamical classifications.

4. Higher dimensional linear systems, distinct eigenvalues, harmonic oscillator, the exponential of a matrix, nonautonomous linear systems,

5. Nonlinear systems, dynamical systems, the existence and uniqueness Theorem, continuous dependence of solutions on parameter and initial conditions. linearization,

6. Equilibria in nonlinear systems; sinks, sources and saddles, stability, linearization, Hartman- Grobman Theorem, saddle-node, pitchfork and Hopf bifurcations,

7. Global nonlinear techniques, nullclines, stability of equilibria Liapunov stability Theorem, Lasalle's Invariance principle, gradient

and Hamiltonian systems

8. Applications in Biology, infectious diseases, prey-predator systems, competetive species

Textbook: Hirsch, M., Smale, S., Devaney, R., Differential equations, Dynamical systems & introduction to Chaos, Academic Press, 2004