Nonlinear Dynamical Systems (AM 114/214)
Homework 1 – Due Monday October 11th (2021)
Instructions
Please submit to CANVAS one PDF file hw1.pdf (homework) and one zip file hw1code.zip that
includes any computer code you develop for the assignment.
AM 114 students: Please submit to CANVAS your homework in a PDF format (one file named
hw1.pdf). This could be a scan of your handwritten notes, compiled Latex source, or a PDF
created using any other word processor (e.g., Microsoft Word). If you develop any computer code
to produce plots or numerical results related to the assignment please attach it to your submission
as a zip file (one file named hw1code.zip).
AM 214 students: Please submit to CANVAS your homework in a PDF format (one file named
hw1.pdf) compiled from Latex source (preferred) or any other word processor. No handwritten
work should be submitted. You should also provide quantitative numerical results for all prob-
lems/questions that are amenable to computation (questions marked by (∗)). For instance, in
Question 1 you should provide a plot of the trajectories computed with a numerical integration
scheme instead of a sketch of the solution x(t) versus t for different initial conditions x0. Similarly,
in Question 4 you should provide a plot of the numerically computed flow map X(t, x0) at differ-
ent time instants ti instead of a sketch of the flow map. Attach the computer code you develop
(MATLAB or Python preferred) to your submission as a zip file (one file named hw1code.zip).
Problem sets and distribution of points
AM 114 students AM 214 students
Question 1 45 points 30 points
Question 2 10 points 10 points
Question 3 10 points 10 points
Question 4 10 points 10 points
Question 5 15 points 10 points
Question 6 10 points 10 points
Question 7 not required 10 points
Question 8 not required 10 points
1
Question 1 (*)1
Consider the following nonlinear differential equations
dx
dt
= ln(x2 + 1)− 1, (1)
dx
dt
= 2x + x3 − x5, (2)
dx
dt
= sin(x)(x2 − 5x + 6). (3)
For each case, find all fixed points, discuss their stability by using the geometric approach (i.e., the
plot of the velocity in the (x, ẋ) plane), and sketch the corresponding flow (vector field) on the real
line. In addition, sketch the graph of the solution x(t) versus t for different initial conditions x0.
Question 2 Use linear stability analysis to classify the fixed points of equation (2). Do your
results match with the geometric approach in Question 1?
Question 3 Set an arbitrary initial condition x(0) = x0 ∈ R. Does the solution to equation (1)
blow up in a finite time? Or it exists and it is unique for any finite t ≥ 0 (global solution)? Justify
your answer.
Question 4 (*) Provide an approximate plot of forward flow map X(t, x0) generated by equation
(1) versus x0 at different times, including t = 0. What happens when t→∞?
Question 5 For each of (a)-(d) below, find an equation dx/dt = f(x), where f ∈ C1(R), satisfying
the stated properties. If there are no examples, explain why not.
(a) Every real number is a fixed point.
(b) Every integer number is a fixed point, and there are no others.
(c) There are precisely two fixed points and they are both stable.
(d) There are precisely one thousand fixed points.
Question 6 Find a potential V (x) for the vector field defined by equation (3).
Question 7 Prove that the forward flow map X(t, x0) generated by any smooth one-dimensional
dynamical system of the form
ẋ = f(x), x(0) = x0, f ∈ C∞(R),
is invertible at fixed points at any finite time. (Hint: derive the evolution equation for ∂X(t, x0)/∂x0,
and solve such equation analytically at a fixed point.)
Question 8 (*) Write a computer code (e.g., Matlab or Octave code) that computes numerically
the forward and the inverse flow maps generated by equation (1), i.e., the 2D surfaces X(t, x0) and
X0(t, x). For convenience, compute such maps for t ∈ [0, 50], and for x0 and x in [−30, 30]. Include
the surface plots of X(t, x0) and X0(t, x) in your PDF report.
1Questions with an asterisk are amenable to computational work (AM 214 students).
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