Nonlinear Differential Equations

Teaching semester

Spring

Objectives and Content

The course deals with existence uniqueness, and analyses in the phase space of nonlinear differential equations. Furthermore there will be focus on asymptotic theory and asymptotic series, regular and singular perturbation methods and stability analysis. In addition, an introduction to chaotic systems will be given.

Learning Outcomes

After completed course, the students are expected to be able to:

• Give account for existence and uniqueness of the solutions of ordinary differential equations solutions.
• Make use of the phase plane to analyse two-dimensional systems with emphasis on equilibrium, existence of limit cycles and linearisation.
• Summarise theorems that related to the existence of periodical solutions, and apply them to simple systems.
• Explain important terms in asymptotic theory, such as, order symbols, asymptotic sequences and asymptotic series, and give account for truncation and convergence of asymptotic series.
• Describe asymptotic perturbation methods for approximate solutions of differential equations, and be able to discuss the characteristics of the different methods.
• Apply singular perturbation methods, coordinate stretching, multiscales and boundary layers to simple problems.
• Explain harmonic and sub-harmonic response and stability to driven oscillations, and perform simple analyses of the Duffings and van der Pol equations.
• Define Poincare and Liapunov stability.
• Give account for the Floquet theory
• Apply Liapunovs stability analysis methods of two-dimensional problems.
• Explain and provide examples of the use of Poincare-Bendixon¿s theorem
• Explain central terminology in chaos theory, such as, bifurcation, strange attractors and the Liapunov exponents.

Recommended Previous Knowledge

MAT131 Differential Equations I

Credit Reduction due to Course Overlap

MAUMAT645: 10 ECTS

Forms of Assessment

Written exam 4 hours.

Grading Scale

The grading scale used is A to F. Grade A is the highest passing grade in the grading scale, grade F is a fail.