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# 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

Elemetary differential equtions, Elementary martix algebra, Principles of mathematical analysis

1. Homework, 10%

2. Quizes 10%(quiz1; 16th of Mehr, quiz2; 12th of Azar)

3. Midterm 30%(14th of Aban)

4. Final exam 50%

Classes: Sunday- Tuesday, 10-12 AM.

Office hour: Sunday-Tuesday, 9:00-10:00 Am