Course Syllabus
Learning Goals:
By the end of this course, students will learn to:
Overarching Goals (not chapter specific)
- Interpret the behavior of a dynamical system in terms of a real-world application
- Convert a dynamical system to dimensionless form
Chapter 1, Section 2.0 Introduction
- Classify a dynamical system as continuous/discrete time, autonomous/nonautonomous, linear/nonlinear, and by dimension
- Explain the difference in approach between an ODEs class and a dynamical systems class (solution methods vs qualitative)
Chapter 2: 1D Flows
- Find the fixed points of a 1D (continuous time autonomous) dynamical system
- Draw a phase portrait for a 1D dynamical system
- Classify fixed points as stable/unstable/semi-stable using the phase portrait
- Give a qualitative sketch of the solution of a differential equation from the phase portrait
- Give an example of a dynamical system with given properties or a given phase portrait
- Recognize that solutions to 1D systems are monotonic
- Classify fixed points as stable/unstable using linear stability analysis
- Recognize when linear stability analysis fails
- Implement Euler’s method, improved Euler, and fourth-order Runge-Kutta
Chapter 3: Bifurcations
- Classify bifurcation points of 1D dynamical systems as saddle-node, transcritical, supercritical pitchfork, or subcritical pitchfork
- Find the bifurcation point(s) for the types listed above for a given 1D dynamical system
- Describe the qualitative changes that occur at the bifurcation point for each type of bifurcation
- Draw a bifurcation diagram
- Identify the normal forms of bifurcations
- Describe how a general bifurcation of a given type relates to the normal form
- Identify and explain hysteresis
Chapter 4: Flows on the circle
- Find and classify the fixed points of a flow on a circle
- Draw a phase portrait for a flow on a circle
- Identify and classify bifurcations for a flow on a circle
Chapter 5: 2D Linear Systems
- Convert a second-order differential equation to a system of two first-order equations
- State the definition of stable, unstable, attracting, asymptotically stable, and neutrally stable fixed points and give examples that distinguish them
- 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
- Recognize slow and fast eigendirections and stable and unstable manifolds
- Use pplane or some other computational method for drawing phase portraits
Chapter 6: Phase plane (2D nonlinear systems)
- Recognize that trajectories cannot cross in the phase portrait
- Find fixed points of 2D nonlinear systems
- Classify the fixed points using linear stability analysis
- Recognize when linear stability analysis can be trusted and when it fails
- Have a working definition of basin of attraction and separatrix
- Define conservative system
- Find a conserved quantity for a given system
- Use that a system is conservative to show that a fixed point is a center
Sections 7.0-7.3: Limit Cycles
- State the definition of limit cycle
- Construct examples of stable, unstable, and semi-stable limit cycles
- Check whether a system is a gradient system
- Find the potential function for a gradient system
- State and check the conditions of Bendixson’s Theorem
- 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
- State the conditions of the Poincaré-Bendixson Theorem
- Apply Poincaré-Bendixson to show that there is a closed trajectory by constructing a trapping region
- 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
- Define weakly nonlinear oscillator
- Explain why regular perturbation theory fails for the damped harmonic oscillator with small damping (Example 7 in class)
- Use the method of averaging to approximate limit cycles for weakly non-linear oscillators
Chapter 8: 2D Bifurcations
- Identify that saddle-node, transcritical, and pitchfork bifurcations are zero-eigenvalue bifurcations
- Describe the eigenvalue behavior of a Hopf bifurcation
- Describe the qualitative changes that occur at a Hopf bifurcation (in terms of fixed points and limit cycles)
- Classify bifurcation points of 2D dynamical systems as saddle-node, transcritical, supercritical pitchfork, subcritical pitchfork, supercritical Hopf, or subcritical Hopf
- Find the bifurcation point(s) for the types listed above for a given 2D dynamical system
Chapter 10: Discrete-time dynamical systems
- Find the fixed points of a map
- Determine the stability of the fixed points of a map
- Find periodic orbits of a map
- Determine the stability of periodic orbits of a map
- Identify transcritical or period-doubling bifurcations of a map
- Draw a cobweb diagram
- Infer properties of a dynamical system from its cobweb diagram
- Explain important properties of the Logistic map including:
- Period-doubling route to chaos
- Self-similarity of the orbit diagram
- The period-3 window
- Intermittency round to chaos
- The universality of the Feigenbaum constant
- Identify properties of chaotic dynamics (SDIC, aperiodic, transitive on a compact set)
- Calculate the Lyapunov exponent of a map
- Use the Lyapunov exponent to determine whether a system is chaotic
- Use a Poincare map to find a limit cycle of a 2D continuous system and determine its stability
Chapter 11: Fractals
- Define the terms cardinality, countable and uncountable
- Determine whether two sets have the same cardinality
- Explain the construction of the middle-third Cantor set
- State important properties of the middle-third Cantor set and derive similar properties for other Cantor sets
- Calculate the similarity dimension of a self-similar fractal
- Calculate the box-counting dimension of a fractal
Chapter 9: Lorenz Equations
- Describe a physical system modeled by the Lorenz equations
- Show that the Lorenz system is dissipative and explain what that means
- Find the fixed points of the Lorenz equations
- Argue using the Lorenz map that the Lorenz attractor is not a stable limit cycle
- State important properties of the Lorenz attractor
Chapter 12: Strange Attractors
- Define strange attractor
- List the processes involved in creating a strange attractor
- Find the invariant set of a map
Course Summary:
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