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AMATH 502 A: Introduction to Dynamical Systems and Chaos

Meeting Time: 
MTWF 1:30pm - 2:20pm
LOW 216
Joint Sections: 

Syllabus Description:





Learning Goals:

By the end of this course, students will learn to:

Overarching Goals (not chapter specific)

  1. Interpret the behavior of a dynamical system in terms of a real-world application
  2. Convert a dynamical system to dimensionless form

Chapter 1, Section 2.0 Introduction

  1. Classify a dynamical system as continuous/discrete time, autonomous/nonautonomous, linear/nonlinear, and by dimension
  2. Explain the difference in approach between an ODEs class and a dynamical systems class (solution methods vs qualitative)

Chapter 2: 1D Flows

  1. Find the fixed points of a 1D (continuous time autonomous) dynamical system
  2. Draw a phase portrait for a 1D dynamical system
  3. Classify fixed points as stable/unstable/semi-stable using the phase portrait
  4. Give a qualitative sketch of the solution of a differential equation from the phase portrait
  5. Give an example of a dynamical system with given properties or a given phase portrait
  6. Recognize that solutions to 1D systems are monotonic
  7. Classify fixed points as stable/unstable using linear stability analysis
  8. Recognize when linear stability analysis fails
  9. Implement Euler’s method, improved Euler, and fourth-order Runge-Kutta

Chapter 3: Bifurcations

  1. Classify bifurcation points of 1D dynamical systems as saddle-node, transcritical, supercritical pitchfork, or subcritical pitchfork
  2. Find the bifurcation point(s) for the types listed above for a given 1D dynamical system
  3. Describe the qualitative changes that occur at the bifurcation point for each type of bifurcation
  4. Draw a bifurcation diagram
  5. Identify the normal forms of bifurcations
  6. Describe how a general bifurcation of a given type relates to the normal form
  7. Identify and explain hysteresis

Chapter 4: Flows on the circle

  1. Find and classify the fixed points of a flow on a circle
  2. Draw a phase portrait for a flow on a circle
  3. Identify and classify bifurcations for a flow on a circle

Chapter 5: 2D Linear Systems

  1. Convert a second-order differential equation to a system of two first-order equations
  2. State the definition of stable, unstable, attracting, asymptotically stable, and neutrally stable fixed points and give examples that distinguish them
  3. Use eigenvalues/eigenvectors to classify the fixed points of a 2D linear system as a stable/unstable node, saddle point, line of stable/unstable fixed points, center, stable/unstable spiral, stable/unstable star, or stable/unstable degenerate node
  4. Recognize slow and fast eigendirections and stable and unstable manifolds
  5. Use pplane or some other computational method for drawing phase portraits

Chapter 6: Phase plane (2D nonlinear systems)

  1. Recognize that trajectories cannot cross in the phase portrait
  2. Find fixed points of 2D nonlinear systems
  3. Classify the fixed points using linear stability analysis
  4. Recognize when linear stability analysis can be trusted and when it fails
  5. Have a working definition of basin of attraction and separatrix
  6. Define conservative system
  7. Find a conserved quantity for a given system
  8. Use that a system is conservative to show that a fixed point is a center

Sections 7.0-7.3: Limit Cycles

  1. State the definition of limit cycle
  2. Construct examples of stable, unstable, and semi-stable limit cycles
  3. Check whether a system is a gradient system
  4. Find the potential function for a gradient system
  5. State and check the conditions of Bendixson’s Theorem
  6. Show that a given dynamical system has no closed trajectories (in some region) using Bendixson’s theorem or the fact that it is a gradient system
  7. State the conditions of the Poincaré-Bendixson Theorem
  8. Apply Poincaré-Bendixson to show that there is a closed trajectory by constructing a trapping region
  9. Explain the consequences of the Poincaré-Bendixson Theorem in terms of the types of behavior that are possible for 2D systems

End of midterm material

Section 7.6: Limit Cycles

  1. Define weakly nonlinear oscillator
  2. Explain why regular perturbation theory fails for the damped harmonic oscillator with small damping (Example 7 in class)
  3. Use the method of averaging to approximate limit cycles for weakly non-linear oscillators

Chapter 8: 2D Bifurcations

  1. Identify that saddle-node, transcritical, and pitchfork bifurcations are zero-eigenvalue bifurcations
  2. Describe the eigenvalue behavior of a Hopf bifurcation
  3. Describe the qualitative changes that occur at a Hopf bifurcation (in terms of fixed points and limit cycles)
  4. Classify bifurcation points of 2D dynamical systems as saddle-node, transcritical, supercritical pitchfork, subcritical pitchfork, supercritical Hopf, or subcritical Hopf
  5. Find the bifurcation point(s) for the types listed above for a given 2D dynamical system

Chapter 10: Discrete-time dynamical systems

  1. Find the fixed points of a map
  2. Determine the stability of the fixed points of a map
  3. Find periodic orbits of a map
  4. Determine the stability of periodic orbits of a map
  5. Identify transcritical or period-doubling bifurcations of a map
  6. Draw a cobweb diagram
  7. Infer properties of a dynamical system from its cobweb diagram
  8. Explain important properties of the Logistic map including:
    1. Period-doubling route to chaos
    2. Self-similarity of the orbit diagram
    3. The period-3 window
    4. Intermittency round to chaos
    5. The universality of the Feigenbaum constant
  9. Identify properties of chaotic dynamics (SDIC, aperiodic, transitive on a compact set)
  10. Calculate the Lyapunov exponent of a map
  11. Use the Lyapunov exponent to determine whether a system is chaotic
  12. Use a Poincare map to find a limit cycle of a 2D continuous system and determine its stability

Chapter 11: Fractals

  1. Define the terms cardinality, countable and uncountable
  2. Determine whether two sets have the same cardinality
  3. Explain the construction of the middle-third Cantor set
  4. State important properties of the middle-third Cantor set and derive similar properties for other Cantor sets
  5. Calculate the similarity dimension of a self-similar fractal
  6. Calculate the box-counting dimension of a fractal

Chapter 9: Lorenz Equations

  1. Describe a physical system modeled by the Lorenz equations
  2. Show that the Lorenz system is dissipative and explain what that means
  3. Find the fixed points of the Lorenz equations
  4. Argue using the Lorenz map that the Lorenz attractor is not a stable limit cycle
  5. State important properties of the Lorenz attractor

Chapter 12: Strange Attractors

  1. Define strange attractor
  2. List the processes involved in creating a strange attractor
  3. Find the invariant set of a map
Catalog Description: 
Overview methods describing qualitative behavior of solutions on nonlinear differential equations. Phase space analysis of fixed pointed and periodic orbits. Bifurcation methods. Description of strange attractors and chaos. Introductions to maps. Applications: engineering, physics, chemistry, and biology. Prerequisite: either a course in differential equations or permission of instructor.
Last updated: 
August 2, 2019 - 9:00pm