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- September 13, 2020
- By menge

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

dx

x2

= s ? rx +

,

dT

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

Fall 2012 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.