Students learn the mathematical structures common to complex phenomena found with non-linearity, and acquire systematic thinking skills for getting an overview of these phenomena from a systematic perspective. Specifically, the course concentrates on interaction, which is a feature of nonlinear systems, and spontaneous self-organization based on it, while students also systematically learn mathematical methodology related to modeling and analysis. Students also apply this methodology to biological, human, and social self-organization to promote a universal understanding of these as systems.
The course focuses on interaction, a feature of nonlinear systems, and self-organization based on it, as well as systematically studying mathematical methodologies related to the modeling of complex natural phenomena and their analysis. Students thereby acquire the thinking skills to have an overview of phenomena from a systematic perspective.
Non-linear systems, linear stability analysis, singularity and stability, phase plane analysis, Null-cline, bifurcation, limit cycle, phase description, phase oscillator, entrainment of the nonlinear oscillator, chaos, Lyapunov exponent, Poincare mapping, self-organization, synergetics, slaving principle, order parameter equation, contraction theory, bifurcation theory, temporal pattern, spatial pattern, simulation
|Intercultural skills||Communication skills||✔ Specialist skills||✔ Critical thinking skills||✔ Practical and/or problem-solving skills|
Starting with basic analysis methods for nonlinear systems, students will gain an understanding of various analysis methods. Then students will study examples of self-organization arising from the interaction of nonlinear systems, as well as modeling and analysis methods for them.
|Course schedule||Required learning|
|Class 1||Introduction: Nonlinear systems and self-organization||Instructions will be given in each class|
|Class 2||Analysis of nonlinear systems (1): Linear stability analysis, Singularity and stability||Instructions will be given in each class|
|Class 3||Analysis of nonlinear systems (2): Phase-plane analysis, Bifurcation theory (1-dimensional)||Instructions will be given in each class|
|Class 4||Analysis of nonlinear systems (3): Bifurcation theory (2-dimensional)||Instructions will be given in each class|
|Class 5||Analysis of nonlinear systems (4): Limit cycle, Phase description, Phase oscillator||Instructions will be given in each class|
|Class 6||Analysis of nonlinear systems (5): Entrainment of nonlinear oscillators, Return map||Instructions will be given in each class|
|Class 7||Analysis of nonlinear systems (6): Chaos, Lyapunov exponent, Poincare map||Instructions will be given in each class|
|Class 8||Self-organized systems (1): Self-organization and synergetics||Instructions will be given in each class|
|Class 9||Self-organized systems (2): Slaving principle and order parameter equations||Instructions will be given in each class|
|Class 10||Self-organized systems (3): Reduction of system dimension based on contraction theory||Instructions will be given in each class|
|Class 11||Self-organized systems (4): Analysis of self-organized systems based on bifurcation theory||Instructions will be given in each class|
|Class 12||Self-organized systems (5): Self-organization of temporal patterns||Instructions will be given in each class|
|Class 13||Self-organized systems (6): Self-organization of spatial patterns||Instructions will be given in each class|
|Class 14||Practice (1): Theoretical analysis and numerical simulations of nonlinear systems||Instructions will be given in each class|
|Class 15||Practice (2): Theoretical analysis and numerical simulations of self-organized systems||Instructions will be given in each class|
Course materials are provided during class.
General introductory book
(1) Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering (Steven H. Strogatz)
(2) Advanced Synergetics: Instability Hierarchies of Self-Organizing Systems and Devices (Hermann Haken)
Students are evaluated based on report assignments.