# FSI3090 Complex Systems 7.5 credits

Complex systems, a.k.a. dynamical systems, refer to mathematical models describing the time evolution of systems by means of equations of motion and initial values. It is the solutions rather than the systems, or the models of the systems, that display complex features. Examples of such features are various ordered processes and structures, such as nonlinear oscillations and waves, as well as disordered chaotic processes and fractal structures. The models are formulated in terms of coupled nonlinear differential equations or, in the discrete case, as iterated maps. Nonlinearity is essential and key concepts are bifurcations, sensitive dependence on initial values, attractors and chaos. There are applications to physics, biology, chemistry, engineering and other areas. The course deals with analytical and numerical methods for the analysis of nonlinear models based on a small number of independent variables.

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Choose semester and course offering to see information from the correct course syllabus and course offering.

Headings with content from the Course syllabus FSI3090 (Spring 2019–) are denoted with an asterisk ( )

## Content and learning outcomes

### Course contents

Coupled nonlinear differential equations. Phase space, trajectories. Iterative maps. Stability analysis of singular points. Limit cycles, strange attractors. Poincaré-Bendixson theorem. Bifurcations. Chaos. Lyapunov exponents. Feigenbaum renormalization. Fractals, fractal dimensions. Lorenz equations, logistic map, Hénon map, Rössler system. Applications to Physics, Biology, Chemistry, Engineering: Lasers. Superconducting Josephson junctions. Population dynamics. Chemical kinetics. Electronic oscillators. Nonlinear mechanical systems.

### Intended learning outcomes

After completed course, the PhD student should be able to:

• be familiar with analytical and numerical methods for the analysis of coupled nonlinear differential equations.
• interpret and characterize different solution types.
• know, and be able to develop, applications to physics, biology, chemistry, engineering, and other areas.

### Course disposition

No information inserted

## Literature and preparations

### Specific prerequisites

Basic course in differential equations.

### Recommended prerequisites

No information inserted

### Equipment

No information inserted

### Literature

Steven H. Strogatz: Nonlinear Dynamics and Chaos (Westview Press, 2000, ISBN 0-7382-0453-6).

## Examination and completion

If the course is discontinued, students may request to be examined during the following two academic years.

P, F

### Examination

• INL1 - Assignments, 5,0 hp, betygsskala: P, F
• TEN1 - Oral exam, 2,5 hp, betygsskala: P, F

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

### Other requirements for final grade

Home assignments and an oral exam.

### Opportunity to complete the requirements via supplementary examination

No information inserted

### Opportunity to raise an approved grade via renewed examination

No information inserted

### Ethical approach

• All members of a group are responsible for the group's work.
• In any assessment, every student shall honestly disclose any help received and sources used.
• In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.

## Further information

### Course web

Further information about the course can be found on the Course web at the link below. Information on the Course web will later be moved to this site.

Course web FSI3090

SCI/Physics

### Main field of study

No information inserted

Third cycle