2019 Nonlinear Dynamics

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Academic unit or major
Graduate major in Systems and Control Engineering
Nakao Hiroya 
Class Format
Media-enhanced courses
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Course description and aims

Various real-world phenomena are modeled as dynamical systems. In this course, starting with the elements of dynamical systems theory, destabilization of stationary states and emergence of spontaneous rhythmic or chaotic dynamics are explained, using mathematical models of real-world systems as examples.

Student learning outcomes

The aim of this course is to provide knowledge on the elements of dynamical systems theory such as stability and bifurcation, as well as on the dynamical systems modeling of real-world phenomena. In particular, theoretical and numerical analysis of nonlinear oscillations will be discussed.


Dynamical systems, stability, nonlinear oscillations, chaos, synchronization

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

lectures, homework

Course schedule/Required learning

  Course schedule Required learning
Class 1 Introduction / Phase space and flows Notion of phase space and flows
Class 2 Gradient, Hamiltonian, and one-dimensional dynamical systems Dynamics of gradient, Hamiltonian, and one-dimensional dynamical systems
Class 3 Stability and bifurcation Notions of stability, linear stability analysis and bifurcation of fixed points
Class 4 2-dimensional systems Dynamics on the two-dimensional phase plane
Class 5 Limit-cycle oscillations Emergence of limit-cycle oscillations and typical examples
Class 6 Reduction methods and synchronization Methods to simplify dynamical systems and analyzing synchronization phenomena of nonlinear oscillations
Class 7 Chaotic dynamics Emergence of chaotic dynamics and its characterization


None specified.

Reference books, course materials, etc.

Steven Strogatz, "Nonlinear dynamics and chaos", Westview press.
Kuramoto, "Chemical Oscillations, Waves, and Turbulence", Springer.
Hoppensteadt & Izhikevich, "Weakly Connected Neural Networks", Springer.

Assessment criteria and methods

Grading will be based on the homework scores.

Related courses

  • SCE.A501 : Networks and Coupled Dynamical Systems
  • MTH.C341 : Differential Equations I
  • MTH.C342 : Differential Equations II

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Elementary knowledge of mathematics and physics

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