Lecture 2 Notes, Tuesday 30 August 2005

  • Strogatz, Chap. 2, pp. 15-35
  • Fall et al., Appendix B, pp. 410-418, which is supplemented by this XPPAUT online tutorial
  • Homework due: Thur 1 Sept 2005

    • Today's lesson focuses on one-dimensional flows, which are first-order dynamical systems. In particular, we limit our discussion to autonomous systems with no explicit time dependence. According to our general definition, the dynamical system x′ = f(x) would be autonomous, whereas x′ = f(x,t) would be non-autonomous because some terms in f(x,t) depend on time. For example, the system Y′ = Ysin(Y) would be autonomous, whereas Y′ = Ysin(t) would not.

    With one state variable, the state space of a first order system is the line. We will analyze several systems based on the vector field, which plots the state variable x versus the change equation x′. The vector field gives an overall picture of the dynamics of the system and allows us to create a phase portrait (i.e., phase diagram) illustrating all the qualitatively different trajectories that are possible in the system. The vector field and phase portrait emphasize the key role played by the fixed points (i.e., equilibrium points) of the dynamical system, which are denoted by x*. Using these new tools we will discuss important concepts including:

  • how to translate an arbitrary system (e.g., a pendulum, an electric circuit...) into differential equations
  • stable and unstable fixed points (attractors and repellors)
  • graphical analysis methods
  • Kirchoff's voltage law (this will come in handy later on...)

    • The general approach to analyzing any dynamical system x′ = f(x), which we will apply throughout the course, is as follows:

  • plot the vector field: f(x) vs x
  • find the roots f(x)=0 to determine the fixed points (x*) of the system
  • examine the slope of the vector field x′ vs x to determine the stability of the fixed points, which is f′(x*) in 1D
  • draw the phase portrait

    • We first examine the simple pendulum system described by Garfinkel (1983):

    x′ = f(x), which is our general definition of a dynamical system

    F = ma, is the law from Newtonian physics

    -mg sin(x) = mx″

    m cancels out, for small x we will say that sin(x) ~ x, and assume that g=1

    define x′ = v, and x″ = v′

    so the system becomes:

    x′ = v
    v′=-x

    We will analyze the trajectory of this simple example in the x vs v state space, drawing change vectors (x′,v′) for each state point (x,v) to get a sense of the trajectory. We will then use XPP to actually simulate this system, which confirms our sense of pendulum behavior for small x, i.e., it swings back and forth.

    It is essential that you familiarize yourself with XPPAUT, which will be used extensively throughout the class. Working through the Appendix B in Fall et al. (pp. 410 - 418) and the XPPAUT online tutorial should help you with re-running simulations done in class as well as your homework assignment due Thur 1 Sept 2005.

    Examples and models

    We will discuss these examples in class, taken from Strogatz Chap. 2 listed below with a link to *.ode files that you can run in XPPAUT.

  • x′ = sin(x), Figs. 2.1.1 and 2.1.2: S_2_01.ode
  • x′ = x2 - 1, Example 2.2.1 and Fig. 2.2.2: S_2_02_01.ode
  • Q′ = V0/R - Q/RC, Example 2.2.2, Figs. 2.2.3 and 2.2.4: S_2_02_02.ode
  • N′ = rN(1 - N/K), Section 2.3 and Figs. 2.3.1 - 2.3.4: S_2_03.ode
    • Lecture 2 Top - Cellular Biophysics and Modeling - Del Negro Homepage - Applied Science - The College of William and Mary