question 1 on this file is identical to a homework question i have just been given. I cannot solve part a,c or d and am in desperate need for some help.
2.050J/12.006J/18.353J Nonlinear Dynamics I: Chaos, Fall 2012
MIDTERM (At-home portion) Problem 1: Bifurcations ? a biochemical switch
A gene G, usually inactive, is activated by a biochemical substance S to produce a pigment or other gene
product when the concentration S exceeds a certain threshold.
Let g(t) denote the concentration of the gene product, and assume that the concentration s0 of S is fixed.
The model is
g? = k1 s0 ? k2 g + k3 g 2
k42 + g 2 (1) where k?s are positive constants. The production of g is stimulated by s0 at a rate k1 , and by autocatalytic
(positive) feedback process modeled by the nonlinear term. There is also a linear degradation of g at a
rate k2 .
1. Nondimensionalize the equations and bring them to the form
= s ? rx +
1 + x2 r > 0, s ? 0. (2) 2. Show that if s = 0 there are two positive fixed points x? if r < rc , where rc is to be determined.
3. Find parametric equations for the bifurcation curves in (r, s) space.
4. (MATLAB) plot quantitatively accurate plot of the stability diagram in (r, s) space
5. Classify the bifurcations that occur.
6. Assume that initially there is no gene product, i.e. g(0) = 0, and suppose that s is slowly increased
from zero (the activating signal is turned on); What happened to g(t)? What happens if s then
goes back to zero? Does the gene turn off again? Problem 2: Nonlinear oscillator
Given an oscillator x
¨ + bx? ? kx + x3 = 0, b, k can be positive, negative or zero.
1. Interpret the terms physically for different values of b and k.
2. Find the bifurcation curves in (b, k) plane, state which bifurcation happens and what kind of fixed
points one has to each side of the bifurcation curve. 1 Problem 3: Numerical study of the displaced Van der Pol oscillator
The equations for a ?displaced? Van der Pol oscillator are given by
x? = y ? a, y? = ?x + ?(1 ? x2 )y, a > 0, ? > 0. Consider a small.
1. Show that the system has two equilibrium points, one of which is a saddle. Find approximate
formula for small a of this fixed point. Study this system numerically with ode45.
2. Submit the plots of phase plane with trajectories starting at different points (to illustrate the
dynamics) for ? = 2, a = 0.1, 0.2, 0.4, and observe that the saddle point approaches the limit cycle
of the Van der Pol equation.
3. Find numerically the value of the parameter a to two decimal points when the saddle point collides
with the limit cycle. What happens to the limit cycle after this collision? 2 MIT OpenCourseWare
http://ocw.mit.edu 18.353J / 2.050J / 12.006J Nonlinear Dynamics I: Chaos