Dynamic Behavior

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Textbook Contents

Dynamic Behavior (pdf, 21Jul07)

  • 1. Solving Differential Equations
  • 2. Qualitative Analysis
  • 3. Stability
  • 4. Lyapunov Stability
  • 5. Parametric and Non-Local Behavior
  • 6. Further Reading
  • Exercises

Lecture Materials

Supplemental Information

Chapter Summary

This chapter introduces the basic concepts and tools of dynamical systems.

  1. We say that is a solution of a differential equation on the time interval to with initial value if it satisfies

    We will usually assume . For most differential equations we will encounter, there is a unique solution for a given initial condition. Numerical tools such as MATLAB and Mathematica can be used to obtain numerical solutions for given the function .

  2. An equilibrium point for a dynamical system represents a point such that if then for all . Equilibrium points represent stationary conditions for the dynamics of a system. A limit cycle for a dynamical system is a solution which is periodic with some period , so that for all .


  4. Phase portraits provide a convenient way to understand the behavior of 2-dimensional dynamical systems. A phase portrait is a graphical representation of the dynamics obtained by plotting the state in the plane. This portrait is often augmented by plotting an arrow in the plane corresponding to , which shows the rate of change of the state. The following diagrams illustrate the basic features of a dynamical systems:

    Doscpp.png Oscpp.png Stablepp.png
    An asymptotically stable equilibrium point at . A limit cycle of radius one, with an unstable equilibrium point at . A stable equlibirum point at (nearby initial conditions stay nearby).


  5. A Lyapunov function is an energy-like function math that can be used to reason about the stability of an equilibrium point. We define the derivative of along the trajectory of the system as


    Assuming and , the following conditions hold:

    Condition on math Condition on math Stability
    math math for all stable
    math math asymptotically stable

    Stability of limit cycles can also be studied using Lyapunov functions.

  6. The Krasovskii-LaSalle Principle allows one to reason about asymptotic stability even if the time derivative of is only negative semi-definite (math rather than math). Let math be a positive definite function, for all and , such that

    math on the compact set math.



    In particular, if contains no invariant sets other than , then 0 is asymptotically stable.




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