This course is suitable for senior undergraduate students or graduate students in the field of science and engineering. This goal is intended to study the basic concepts, methods, and theories of the nonlinear dynamical systems. This course contains the following contents: fixed point, stability, center manifold, norm formal, local and global bifurcations, continuous and discrete-time dynamical system, chaos, and some application in practical problems. After studying this course, students are able to deal with nonlinear dynamical behavior in different fields of natural science, social science,and engineering.
Overview
Syllabus
- Chapter I Dynamical system
- 1.1 Dynamical system
- 1.2 Linearization
- 1.3 Phase space
- Chapter II Stability
- 2.1 Lyapunov stability
- 2.2 Lyapunov function 1
- 2.3 Lyapunov function 2
- Chapter III Local bifurcation
- 3.1 Saddle-node bifurcation 1
- 3.2 Saddle-node bifurcation 2
- 3.3 Trancritical bifurcation 1
- 3.4 Transcritical bifurcation 2
- 3.5 Pitchfork bifurcation 1
- 3.6 Pitchfork bifurcation 2
- 3.7 Hopf bifurcation 1
- 3.8 Hopf bifurcation 2
- Chapter IV Center manifold
- 4.1 Center manifold theory 1
- 4.2 Center manifold theory 2
- 4.3 Center manifold theory 3
- 4.4 Center manifold theory 4
- 4.5 Center manifold theory 5
- 4.6 Center manifold theory 6
- Chapter V Normal form
- 5.1 Normal form 1
- 5.2 Normal form 2
- 5.3 Normal form 3
- 5.4 Normal form 4
- Chapter VI Melnikov method
- 6.1 Melnikov method 1
- 6.2 Melnikov method 2
- 6.3 Melnikov method 3
- 6.4 Melnikov method 4
- 6.5 Melnikov method 5
- 6.6 Melnikov method 6
- Chapter VII Li-Yorke theorem
- 7.1 Li-Yorke theorem 1
- 7.2 Li-Yorke theorem 2
- 7.3 Li-Yorke theorem 3
- 7.4 Li-Yorke theorem 4
- 7.5 Li-Yorke theorem 5
- 7.6 Li-Yorke theorem 6
- Chapter VIII Marotto theorem
- 8.1 Marotto theorem 1
- 8.2 Marotto theorem 2
- Chapter VIIII Smale horseshoe and symbolic dynamics
- 9.1 Smale Horseshoe 1
- 9.2 Smale Horseshoe 2
- 9.3 Symbolic dynamics
- Chapter X Henon map
- 10.1 Henon map 1
- 10.2 Henon map 2
- Chapter X! Application
- 11.1 Bursting of a neuron model
- 11.2 Bifurcation of a railway wheelset model
- Chapter 0 Introduction
Taught by
HONGJUN CAO
