8.E: Nonlinear Equations (Exercises)

8.1: Linearization, critical points, and equilibria

Exercise 8.1.1: Sketch the phase plane vector field for:

a) \(x'=x^2, ~~y'=y^2\),

b) \(x'=(x-y)^2, ~~y'=-x\),

c) \(x'=e^y,~~ y'=e^x\).

Exercise 8.1.2: Match systems

1) \(x'=x^2\), \(y'=y^2\), 2) \(x'=xy\), \(y'=1+y^2\), 3) \(x'=\sin(\pi y)\), \(y'=x\), to the vector fields below. Justify.

a) b) c)

Exercise 8.1.3: Find the critical points and linearizations of the following systems.

a) \(x'=x^2-y^2\), \(y'=x^2+y^2-1\),

b) \(x'=-y\), \(y'=3x+yx^2\),

c) \(x'=x^2+y\), \(y'=y^2+x\).

Exercise 8.1.4: For the following systems, verify they have critical point at \((0,0)\), and find the linearization at \((0,0)\).

a) \(x'=x+2y+x^2-y^2\), \(y'=2y-x^2\)

b) \(x'=-y\), \(y'=x-y^3\)

c) \(x'=ax+by+f(x,y)\), \(y'=cx+dy+g(x,y)\), where \(f(0,0) = 0\), \(g(0,0) = 0\), and all first partial derivatives of \(f\) and \(g\) are also zero at \((0,0)\), that is,

\(\frac{\partial f}{\partial x}(0,0) = \frac{\partial f}{\partial y}(0,0) = \frac{\partial g}{\partial x}(0,0) = \frac{\partial g}{\partial y}(0,0) = 0\).

Exercise 8.1.5:Take \(x'=(x-y)^2\), \(y'=(x+y)^2\).

a) Find the set of critical points.

b) Sketch a phase diagram and describe the behavior near the critical point(s).

c) Find the linearization. Is it helpful in understanding the system?

Exercise 8.1.6: Take \(x'=x^2\), \(y'=x^3\).

a) Find the set of critical points.

b) Sketch a phase diagram and describe the behavior near the critical point(s).

c) Find the linearization. Is it helpful in understanding the system?

Exercise 8.1.101: Find the critical points and linearizations of the following systems.

a) \(x'=\sin(\pi y)+(x-1)^2\), \(y'=y^2-y\),

b) \(x'=x+y+y^2\), \(y'=x\),

c) \(x'=(x-1)^2+y\), \(y'=x^2+y\).

Exercise 8.1.102: Match systems

1) \(x'=y^2\), \(y'=-x^2\), 2) \(x'=y\), \(y'=(x-1)(x+1)\), 3) \(x'=y+x^2\), \(y'=-x\), to the vector fields below. Justify.

a) b) c)

Exercise 8.1.103: The idea of critical points and linearization works in higher dimensions as well. You simply make the Jacobian matrix bigger by adding more functions and more variables. For the following system of 3 equations find the critical points and their linearizations:

\(x' = x + z^2,\\ y' = z^2-y, \\ z' = z+x^2.\)

Exercise 8.1.1: Any two-dimensional non-autonomous system \(x'=f(x,y,t)\), \(y'=g(x,y,t)\) can be written as a three-dimensional autonomous system (three equations). Write down this autonomous system using the variables \(u\), \(v\), \(w\).

8.2: Stability and classification of isolated critical points

Exercise 8.2.1: For the systems below, find and classify the critical points, also indicate if the equilibria are stable, asymptotically stable, or unstable.

a) \(x'=-x+3x^2, y'=-y\) b) \(x'=x^2+y^2-1\),\(y'=x\) c) \(x'=ye^x\),\(y'=y-x+y^2\)

Exercise 8.2.2: Find the implicit equations of the trajectories of the following conservative systems. Next find their critical points (if any) and classify them.

a) \(x''+ x+x^3 = 0\) b) \(\theta''+\sin \theta = 0\) c) \(z''+ (z-1)(z+1) = 0\) d) \(x''+ x^2+1 = 0\)

Exercise 8.2.3: Find and classify the critical point(s) of \(x' = -x^2\),\(y' = -y^2\).

Exercise 8.2.4: Suppose \(x'=-xy\),\(y'=x^2-1-y\). a) Show there are two spiral sinks at \((-1,0)\) and \((1,0)\). b) For any initial point of the form \((0,y_0)\),find what is the trajectory. c) Can a trajectory starting at \((x_0,y_0)\) where \(x_0 > 0\) spiral into the critical point at \((-1,0)\)? Why or why not?

Exercise 8.2.5: In the example \(x'=y\),\(y'=y^3-x\) show that for any trajectory, the distance from the origin is an increasing function. Conclude that the origin behaves like is a spiral source. Hint: Consider \(f(t) = {\bigl(x(t)\bigr)}^2 + {\bigl(y(t)\bigr)}^2\) and show it has positive derivative.

Exercise 8.2.6: Suppose \(f\) is always positive. Find the trajectories of \(x''+f(x') = 0\). Are there any critical points?

Exercise 8.2.7: Suppose that \(x' = f(x,y)\),\(y' = g(x,y)\). Suppose that \(g(x,y) > 1\) for all \(x\) and \(y\). Are there any critical points? What can we say about the trajectories at \(t\) goes to infinity?

Exercise 8.2.101: For the systems below, find and classify the critical points. a) \(x'=-x+x^2\),\(y'=y\) b) \(x'=y-y^2-x\),\(y'=-x\) c) \(x'=xy\),\(y'=x+y-1\)

Exercise 8.2.102: Find the implicit equations of the trajectories of the following conservative systems. Next find their critical points (if any) and classify them. a) \(x''+ x^2 = 4\) b) \(x''+ e^x = 0\) c) \(x''+ (x+1)e^x = 0\)

Exercise 8.2.103: The conservative system \(x''+x^3 = 0\) is not almost linear. Classify its critical point(s) nonetheless.

Exercise 8.2.104:Derive an analogous classification of critical points for equations in one dimension, such as \(x'= f(x)\) based on the derivative. A point \(x_0\) is critical when \(f(x_0) = 0\) and almost linear if in addition \(f'(x_0) \not= 0\). Figure out if the critical point is stable or unstable depending on the sign of \(f'(x_0)\). Explain. Hint: see Ch. 1.6.

8.3: Applications of nonlinear systems

Exercise 8.3.1: Take the damped nonlinear pendulum equation \(\theta '' + \mu \theta' + (\frac{g}{L}) \sin \theta = 0\) for some \(\mu > 0\) (that is, there is some friction). a) Suppose \(\mu = 1\) and \(\frac{g}{L} = 1\) for simplicity, find and classify the critical points. b) Do the same for any \(\mu > 0\) and any \(g\) and \(L\), but such that the damping is small, in particular, \(\mu^2 < 4(\frac{g}{L})\). c) Explain what your findings mean, and if it agrees with what you expect in reality.

Exercise 8.3.2: Suppose the hares do not grow exponentially, but logistically. In particular consider

\[x' = (0.4-0.01y)x - \gamma x^2, ~~~~~ y' = (0.003x-0.3)y .\]

For the following two values of \(\gamma\), find and classify all the critical points in the positive quadrant, that is, for \(x \geq 0\) and \(y \geq 0\). Then sketch the phase diagram. Discuss the implication for the long term behavior of the population. a) \(\gamma=0.001\), b) \(\gamma=0.01\).

Exercise 8.3.3: a) Suppose \(x\) and \(y\) are positive variables. Show \(\frac{y x}{e^{x+y}}\) attains a maximum at \((1,1)\). b) Suppose \(a,b,c,d\) are positive constants, and also suppose \(x\) and \(y\) are positive variables. Show \(\frac{y^a x^d}{e^{cx+by}}\) attains a maximum at \((\frac{d}{c},\frac{a}{b})\).

Exercise 8.3.4: Suppose that for the pendulum equation we take a trajectory giving the spinning-around motion, for example \(\omega = \sqrt{\frac{2g}{L} \cos \theta + \frac{2g}{L} + \omega_0^2}\). This is the trajectory where the lowest angular velocity is \(\omega_0^2\). Find an integral expression for how long it takes the pendulum to go all the way around.

Exercise 8.3.5:[challenging] Take the pendulum, suppose the initial position is \(\theta = 0\). a) Find the expression for \(\omega\) giving the trajectory with initial condition \((0,\omega_0)\). Hint: Figure out what \(C\) should be in terms of \(\omega_0\). b) Find the crucial angular velocity \(\omega_1\), such that for any higher initial angular velocity, the pendulum will keep going around its axis, and for any lower initial angular velocity, the pendulum will simply swing back and forth. Hint: When the pendulum doesn't go over the top the expression for \(\omega\) will be undefined for some \(\theta\)s. c) What do you think happens if the initial condition is \((0,\omega_1)\), that is, the initial angle is 0, and the initial angular velocity is exactly \(\omega_1\).

Exercise 8.3.101: Take the damped nonlinear pendulum equation \(\theta '' + \mu \theta' + (\frac{g}{L}) \sin \theta = 0\) for some \(\mu > 0\) (that is, there is friction). Suppose the friction is large, in particular \(\mu^2 > 4 (\frac{g}{L})\). a) Find and classify the critical points. b) Explain what your findings mean, and if it agrees with what you expect in reality.

Exercise 8.3.102: Suppose we have the system predator-prey system where the foxes are also killed at a constant rate \(h\) (\(h\) foxes killed per unit time): \(x' = (a-by)x,\) \(y' = (cx-d)y - h\). a) Find the critical points and the Jacobin matrices of the system. b) Put in the constants \(a=0.4\), \(b=0.01\), \(c=0.003\), \(d=0.3\), \(h=10\). Analyze the critical points. What do you think it says about the forest?

Exercise 8.3.103:[challenging] Suppose the foxes never die. That is, we have the system \(x' = (a-by)x,\) \(y' = cxy\). Find the critical points and notice they are not isolated. What will happen to the population in the forest if it starts at some positive numbers. Hint: Think of the constant of motion.

8.4: Limit cycles

Exercise 8.4.1: Show that the following systems have no closed trajectories. a) \(x'=x^3+y,y'=y^3+x^2\), b) \(x'=e^{x-y},y'=e^{x+y}\), c) \(x'=x+3y^2-y^3,y'=y^3+x^2\).

Exercise 8.4.2: Formulate a condition for a 2-by-2 linear system \({\vec{x}\,}' = A \vec{x}\) to not be a center using the Bendixson-Dulac theorem. That is, the theorem says something about certain elements of \(A\).

Exercise 8.4.3: Explain why the Bendixson-Dulac Theorem does not apply for any conservative system \(x''+h(x) = 0\).

Exercise 8.4.4: A system such as \(x'=x, y'=y\) has solutions that exist for all time \(t\), yet there are no closed trajectories or other limit cycles. Explain why the Poincare-Bendixson Theorem does not apply.

Exercise 8.4.5: Differential equations can also be given in different coordinate systems. Suppose we have the system \(r' = 1-r^2\), \(\theta' = 1\) given in polar coordinates. Find all the closed trajectories and check if they are limit cycles and if so, if they are asymptotically stable or not.

Exercise 8.4.101: Show that the following systems have no closed trajectories. a) \(x'=x+y^2\), \(y'=y+x^2\), b) \(x'=-x\sin^2(y)\), \(y'=e^x\), c) \(x'=xy\), \(y'=x+x^2\).

Exercise 8.4.102: Suppose an autonomous system in the plane has a solution \(x=\cos(t)+e^{-t}\), \(y=\sin(t)+e^{-t}\). What can you say about the system (in particular about limit cycles and periodic solutions)?

Exercise 8.4.103: Show that the limit cycle of the Van der Pol oscillator (for \(\mu > 0\)) must not lie completely in the set where \(- \sqrt{\frac{1+\mu}{\mu}} < x < \sqrt{\frac{1+\mu}{\mu}}\).

Exercise 8.4.104: Suppose we have the system \(r' = \sin(r)\), \(\theta' = 1\) given in polar coordinates. Find all the closed trajectories.

8.5: Chaos