Homework 1
Homework 1: Due Friday 21 September
Problems 1-3 are from (or based very closely on) Daniel Kaplan and
Leon Glass, Understanding Nonlinear dynamics, Springer-Verlag,
1995. I suggest doing problems 1 and 4 first, before doing the
others.
- Find a function for a finite-difference equation with:
- Four fixed points, all of which are unstable.
- Eleven fixed points, three of which are stable.
- No fixed points.
Hint: just give the function's graph.
- The following equation plays a role in the analysis of nonlinear
models of gene and neural networks (Glass and Pasternack, Stable
oscillations in mathematical models of biological control systems,
J. Math. Biol. 6, 207-23:1978.):
x_{t+1} = (ax_t )/ (1 + bx_t),
where a and b are positive numbers, and x_t is non-negative.
Algebraically determine the fixed points. For each fixed point, give
the range of a and b for which it exists, indicate whether the fixed
point is stable or unstable, and state whether the dynamics in the
neighborhood of the fixed point are monotonic or oscillatory.
- The population of a species is described by the finite-difference
equation: x_{t+1} = a x_t exp(-x_t), where x_t >= 0 and a is a
positive constant.
- Determine the fixed points.
- Evaluate the stability of the fixed points.
- For what value of a is the first period-doubling bifurcation?
- For what values of a will the population go extinct starting from
any initial condition?
- Suppose you were to make a bifurcation diagram for this equation
as a function of a. What do you think the bifurcation diagram would
look like and why? Which features of the diagram could you predict,
and which couldn't you?
- (optional)Using a computer, generate a bifurcation diagram
as a function of a.
- For the logistic map
- Calculate the fixed point as a function of r.
- Algebraically determine the r value at which the fixed point
loses stability.
- Using the programs available on my website here, determine whether
or not the logistic equation is chaotic for r = 3.5699456718695445 .
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