Qualitative theory of ordinary differential equations (ODEs). Existence and uniqueness, geometrical representation of ODEs. Construction of phase portraits. Nonlinear systems, local and global behavior, the linearization theorem. Periodic orbits and limit sets, Poincare-Bendixson theory. The stable manifold theorem, homoclinic and heteroclinic points. Bifurcation diagrams. State reconstruction from data, embedding.

### Dynamical Systems (MATH 307)

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Programs\Type | Required | Core Elective | Area Elective |

BA- Political Science | |||

BA-Cultural Studies | |||

BA-Cultural Studies | |||

BA-Economics | |||

BA-Economics | |||

BA-International Studies | |||

BA-International Studies | |||

BA-Management | |||

BA-Management | |||

BA-Political Sci.&Inter.Relat. | |||

BA-Political Sci.&Inter.Relat. | |||

BA-Social & Political Sciences | |||

BA-Visual Arts&Visual Com.Des. | |||

BA-Visual Arts&Visual Com.Des. | |||

BS-Biological Sci.&Bioeng. | |||

BS-Computer Science & Eng. | |||

BS-Computer Science & Eng. | |||

BS-Electronics Engineering | * | ||

BS-Electronics Engineering | * | ||

BS-Industrial Engineering | * | ||

BS-Manufacturing Systems Eng. | * | ||

BS-Materials Sci. & Nano Eng. | * | ||

BS-Materials Science & Eng. | * | ||

BS-Mechatronics | |||

BS-Mechatronics | |||

BS-Microelectronics | |||

BS-Molecular Bio.Gen.&Bioeng | |||

BS-Telecommunications | * | ||

Mathematics |

### CONTENT

### OBJECTIVE

To teach the fundamental theory of ODE's; the fundametals of dynamical systems and their connections with problems from a wide variety of areas.

### LEARNING OUTCOME

the students should be able to

(a) Draw phase portraits in the plane

(b) Construct proofs for existence and uniqueness for linear systems

(c) Identify fixed/equilibrium points, determine their stability, and analyze local and global behavior

(d) Rule out closed orbits for gradient systems and construct Liapunov functions

(e) Utilize theorems to establish the existence of closed orbits, such as Poincare-Bendixson theorem

(f) Study in detail Lienard systems (or those can be converted to a Lienard systems)

(g) Employ perturbation theory for weakly nonlinear oscillators

(h) Create bifurcation diagrams for saddle-node, transcritical, and pitchfork bifurcations

(i) Investigate fixed points and their stability for one-dimensional maps

### Update Date:

### ASSESSMENT METHODS and CRITERIA

Percentage (%) | |

Final | 40 |

Midterm | 30 |

Homework | 30 |

### RECOMENDED or REQUIRED READINGS

Readings |
Arrowsmith D.K., Place C.M., Dynamical Systems- Differential Equations, maps and chaotic behaviour, Chapman&Hall, 1992. |