Past Exams and fun problems

1. Analyze the equation
   $$\dot x = 4 x^2 -16$$
   graphically. Sketch the vector field on the real line, find all fixed points, classify their stabilityu and sketch the graph of $x(t)$ for different initial conditions. Obtain the solution in the closed form.

2. Analyze the equation
   $$\dot x = 1+ \frac{1}{2}\cos(x)$$
   graphically. Sketch the vector field on the real line, find all fixed points, classify their stabilityu and sketch the graph of $x(t)$ for different initial conditions.

   Extra Credit: Obtain the solution in the closed form.

3. Consider the equation
   $$\dot x (t) = a x(t) - x^3(t).$$
   Here $a$ is a parameter that can be positive, negative or zero. Discuss all three cases.

   Find all the fixed points of this equation and classify their stability by using the linear stability analyses. If the linear stability analyses fails, use the graphical method.

4. 2.5.3 Consider the equation
   $$\dot x(t) = r x(t) + x^3(t),$$

   where $r>0$ is a fixed parameter. Show that $x(t)$ goes to infinity in finite time, for any $x(t=0)\ne 0$.

5. Consider the equation
   $$\dot x(t) = 1+ r x + x ^2.$$
   Herr $r$ is a parameter that can be positive, negative or zero. Plot all qualitatively different vector fields that occur as $r$ varies. Show that a biffurcation occurs as $r$ varies, and find the type of the biffurcation. Finally, plot the bifurcation diagram of $x^*$ as a function of $r$.

6. Consider the system

   $$\dot x(t) = r x + a x^2 - x^3.$$
   Here both $r$ and $a$ are external parameters that can be positive, negative or zero. When $a=0$ we have a normal form of the supercritical pitchfork.

   (a) For each value of $a$ there is a bifurcation diagram of $x^*$ versus $r$. As $a$ varies, the bifurcation diagram changes and goes through qualitatively different behaviour. Please plot all qualitatively different bifurcation diagrams that can be obtained as $a$ varies.

   (b) Summarize your finding by plotting the regions in the $(a,r)$ plane that corresponds to qualitatively different classes of vector fields. Bifurcations occur on the boundaries of these regions. Identify the type of these bifurcations.

7. Analyze the equation
   $$\dot x = e^x - \cos(x),$$
   find all fixed points, and classify their stability.

   HINT You do not need to find analytical solution for the fixed points

8. For each of 1)-4) below, find an equation
   $$\dot x = f(x)$$
   with that stated properties, or if there are no examples, exlain why not. (Assume $f(x)$ is a smooth continuous function).
   1. Every real number is a fixed point
   2. There are precisely three fixed points, and all of them are stable
   3. There are no fixed points
   4. There are exactly 100 fixed points.

9. Show that the solution to
   $$\dot x = 1+x^{10}$$
   escapes to $+\infty$ in a finite time, starting any initial condition.

   HINT Do not try to find an exact solution, instead, compare the solution to those of

   $$\dot x = 1+x^2.$$

10. Consider the equation
    $$\dot x = r-x+x^3$$
    for various values of $r$. Plot the potential function $V(x)$ for $r<0$, $r=0$ and $r>0$, and identify all equilibrium points and their stability.

11. Consider the equation
    $$\dot x = r-x+x^3$$
    for various values of $r$. Plot the bifurcation diagram of this equation and classify the bifurcation.

12. 2.2.5

    Analyze the equation

    $$\dot x = 1+ \frac{1}{2}\cos(x)$$
    graphically. Sketch the vector field on the real line, find all fixed points, classify their stabilityu and sketch the graph of $x(t)$ for different initial conditions.

    Extra Credit: Obtain the solution in the closed form.

13. (The Allee effect) For certain species of organisms, the effective growth rate $\dot N(t) /N$ is highest at intermediate $N$ . This is called the Allee effect (Edelstein–Keshet 1988). For example, imagine that it is too hard to find mates when N is very small, and there is too much competition for food and other resources when $N$ is large.

    a) Show that

    $$\frac{\dot N}{N} = r - a (N-b)^2$$
    provides an example of the Allee effect, if $r$, $a$, and $b$ satisfy certain constraints, to be determined.

    b) Find all the fixed points of the system and classify their stability.

    c) Sketch the solutions $N (t)$ for different initial conditions.

    d) Compare the solutions $N (t)$ to those found for the logistic equation. What are the qualitative differences, if any?

14. 2.4.7 Consider the equation
    $$\dot x (t) = a x(t) - x^3(t).$$
    Here $a$ is a parameter that can be positive, negative or zero. Discuss all three cases.

    Find all the fixed points of this equation and classify their stability by using the linear stability analyses. If the linear stability analyses fails, use the graphical method.

15. 2.5.3 Consider the equation
    $$\dot x(t) = r x(t) + x^3(t),$$

    where $r>0$ is a fixed parameter. Show that $x(t)$ goes to infinity in finite time, for any $x(t=0)\ne 0$.

16. 3.1.1 Consider the equation
    $$\dot x(t) = 1+ r x + x ^2.$$
    Herr $r$ is a parameter that can be positive, negative or zero. Plot all qualitatively different vector fields that occur as $r$ varies. Show that a biffurcation occurs as $r$ varies, and find the type of the biffurcation. Finally, plot the bifurcation diagram of $x^*$ as a function of $r$.

17. 3.6.3

    Consider the system

    $$\dot x(t) = r x + a x^2 - x^3.$$
    Here both $r$ and $a$ are external parameters that can be positive, negative or zero. When $a=0$ we have a normal form of the supercritical pitchfork.

    (a) For each value of $a$ there is a bifurcation diagram of $x^*$ versus $r$. As $a$ varies, the bifurcation diagram changes and goes through qualitatively different behaviour. Please plot all qualitatively different bifurcation diagrams that can be obtained as $a$ varies.

    (b) Summarize your finding by plotting the regions in the $(a,r)$ plane that corresponds to qualitatively different classes of vector fields. Bifurcations occur on the boundaries of these regions. Identify the type of these bifurcations.

18. Sketch all the qualitatively different vector fields that occur as $r$ is varied. Show a saddle-node bifurcation occurs at critical value of $r$ to be determined. Finally, scetch the bifurcation diagram of fixed points $x^*$ versus $r$:
    $$\dot x(t) = 1+ r x + x ^2.$$

19. Sketch all the qualitatively different vector fields that occur as $r$ is varied. Scetch the bifurcation diagram of fixed points $x^*$ versus $r$, and explain the bifurcation diagram.
    $$\dot x (t) = r^2 - x^2.$$

20. Sketch all the qualitatively different vector fields of the equation
    $$\dot x = x - r x (1-x)$$
    that occur as $r$ varied. Show that a transcritical bifurcation occurs at a critical value of $r$ to be determined.

21. Find the values of $r$ at which tbifurcations occur for the equation
    $$\dot x = 5 - r e^{-x^2},$$
    and classify those as saddle-node, transcritical, supecritical pitchfork or subcritical pitchfork. Sketch the bifurcation diagram of fixed points $x^*$ versus $r$.

22. Find and classify all fixed points of the following equation on a circle $0\le\theta\le 2Pi$:
    $$\dot \theta = \sin^3\theta,$$

23. Draw the phase potraits of the following equation on a circle $0\le\theta\le 2Pi$:
    $$\dot \theta(t)=\frac{\sin\theta(t)}{\mu+\sin\theta(t)},$$
    as a function of control parameter $\mu$, classify the bifurcation that occurs as $\mu$ varies, and find bifurcation values of $\mu$.

24. Consider the linear system
    $$\dot x= 3 x - 4 y, \ \ \dot y = x-y.$$
    Plot the phase portrait of this system and classify the fixed point. If the eigenvectors are real, plot them on your sketch.

25. Consider the system
    $$\dot x = x(x-y), \ \ \ \dot y = y(2 x - y).$$
    Find all fixed points. Sketch the nullclines, the vector field and plausible phase portrait.

26. Consider the system
    $$\dot x = x-y, \ \ \ \dot y = x^2 -4.$$
    Find the fixed points, classify them , sketch neighboring trajectories, and try to fill in the rest of the portrait.

27. Consider the system
    $$\ddot x = x^3 -x.$$
    • Find all equilibrium points and classify them
    • Find a conserved quantity
    • Sketch the phase portrait

28. Analyze the equation
    $$\dot x = e^{-x}\sin x$$
    graphically. Sketch vector field on the real line, find all the fixed points, classify their stability and sketch the graph of $x(t)$ for different initial conditions.

29. Find the values of $r$ at which bifurcation(s) occur for the equation
    $$\dot x = r x - \frac{x}{1+x}$$
    and classify them as saddle-node, transcritical or super(sub)critical pitchfork. Sketch the bifurcation diagram of fixed points $x^*$ versus $r$.

30. Find and classify all fixed points of
    $$\dot \theta = \sin k \theta,$$
    where $k$ is a positive integer. Sketch the phase portrait on a circle.

31. 4.1.6 Consider the equation
    $$\dot \theta(t) = 3+ \cos(2\theta(t)).$$
    Find all the fixed points, classify their stability, and plot the phase portraits on a cirle.

32. 4.3.5 Consider the system:
    $$\dot\theta(t) = \mu+\cos(\theta(t))+\cos(2 \theta(t)).$$
    Draw the phase portraits as a function of a control parameter $\mu$. Classify the biffurcations that occur as $\mu$ varies, find the values of $\mu$ where biffurcations occur.

33. Consider a nonuniform oscillator
    $$\dot\theta(t) = \mu - \cos(\theta(x)).$$
    Draw the phase portraits as a function of a control parameter $\mu$. Classify the biffurcations that occur as $\mu$ varies, find the values of $\mu$, where biffurcation occur, and plot the biffurcation diagram.

34. Find the general Solution of
    $$\frac{d}{dt} \left(\begin{array}{rr} x(t)\\ y(t)\end{array}\right... ... -1 \end{array}\right) \left(\begin{array}{rr} x(t)\\ y(t)\end{array}\right).$$
    Plot the phase portrait and classify the stability of the fixed point(s). If the eigenvectors are real, plot them on your plot.

35. Find the general solution of
    $$\frac{d}{dt} \left(\begin{array}{rr} x(t)\\ y(t)\end{array}\right... ... -6 \end{array}\right) \left(\begin{array}{rr} x(t)\\ y(t)\end{array}\right).$$
    Plot the phase portrait and classify the stability of the fixed point(s). If the eigenvectors are real, plot them on your plot.

36. Find the general solution of
    $$\frac{d}{dt} \left(\begin{array}{rr} x(t)\\ y(t)\end{array}\right... ... -5 \end{array}\right) \left(\begin{array}{rr} x(t)\\ y(t)\end{array}\right).$$
    Plot the phase portrait and classify the stability of the fixed point(s). If the eigenvectors are real, plot them on your plot.

37. Show that any matrix of the form
    $${\bf A} = \left(\begin{array}{rr} \lambda & b\\ 0 & \lambda \end{array}\right)$$
    has only onedimensional eigenspace corresponding to the eigenvalue $\lambda$.

    Solve the system

    $$\frac{d}{dt} \left(\begin{array}{rr} x(t)\\ y(t)\end{array}\right... ...da \end{array}\right) \left(\begin{array}{rr} x(t)\\ y(t)\end{array}\right),$$
    and plot the phase portrait.

38. Consider the $R$ and $J$ interactions described by
    $$\frac{d}{dt} \left(\begin{array}{rr} R(t)\\ J(t)\end{array}\right... ...& d \end{array}\right) \left(\begin{array}{rr} R(t)\\ J(t)\end{array}\right).$$
    Now answer the question:
    Do the opposite attract? In other words, consider that $J$ is an opposite of $R$, i.e.
    $$c = -a, d = -b.$$

    Now study the course of their interactions depending on relative sizes and signs of $a$ and $b$. Draw characteristic phase portraits for all possible scenarios and describe by words interactions between $R$ and $J$.

39. Consider the system
    $$\dot x = 4 x - y, \ \ \ \dot y = 2 x + y.$$
    • Find the general solution of the system
    • Classify the fixed point at the origin
    • Solve the system subject to initial conditions
      $$x(t=0) = 3, y(t=0) =4.$$

40. Consider the
    $$\dot x = x - x^3, \ \ \ \dot y = -y.$$
    Find all fixed points. Sketch the nullclines, the vector field and plausible phase portrait.

41. Consider the system
    $$\dot x = x y - 1, \ \ \ \dot y = x - y^3.$$
    Find the fixed points, classify them , sketch neighboring trajectories, and try to fill in the rest of the portrait.

42. Consider the system
    $$\dot x = -y - x^3, \dot y= x.$$
    Show that the origin is a spiral, although linearization predicts a center. HINT Think about nonlinear oscillator.

43. Consider
    $$\ddot x + \alpha\dot x( x^2+\dot x^2 -1)+x=0, \ \ \ \alpha>0.$$
    • Find and classify all fixed points
    • Show that the system has a circular limit cycle, and find its amplitude and period.
    • Determine the stability of a limit cycle.
    • Plot plausible phase portrait.

44. Find equilibrium points, classify their stability and draw representative $x(t)$ trajectories for
    $$\dot x = x (x-1)(x-2).$$

45. Find Equilibrium points, classify their stability and draw representative trajectories and biffurcation diagram for the equation
    $$\dot x(t) = r (x^2(t)-r^2) x(t).$$
    Please make sure to consider the cases $r>0$, $r<0$ and $r=0$.

46. A particle travels pn the half line $x\ge 0$with velocity given by
    $$\dot x = - x^c,$$
    where $c$ is real and constant.

    (a) Find all values of $c$ such the origin $x=0$ is stable fixed point

    (b)Now assume that $c$ is chosen so that the $x=0$ is stable. Can the particle ever reach the origin in a finite time? Specifically, how long does it take for the particle to travel from $x=1$ to $x=0$, as a function of $c$?

47. Find the values of $r$ at which biffurcation occur, and classify those as saddle point, transcritical or pitchfork biffurcation. Plot the biffurcation diagram.
    $$\dot x = x + \tanh\left( r x\right).$$

48. Extra Credit Consider the equation
    $$\dot x = h + r x - x^2.$$
    When $h=0$m this system undergoes a transcritical biffurction at $r=0$. Here we are going to study how the biffurcation diagram is affected by nonzero values of $h$.

    (a) Plot the biffurcation diagrams for $h=0$, $h>0$ and $h<0$.

    (b) Sketch the regions in the $(r,h)$ plane that correspond to qualitatively different vector fields and identify the biffurcations that occur on the boundary of these regions.

    (c) Plot the potential $V(x)$ that corresponds to all these different regions in the $(r,h)$ plane.

49. Draw the phase portraits as a function of a control parameter $\mu$. Classify biffurcation that occurs as $\mu$ varies and find all biffurcation values of $\mu$:
    $$\dot \theta(t)=\mu \sin\theta(t)-\sin 2\theta(t).$$

50. Find the conditions under which it is appropriate to approximate the equation
    $$m L^2 \ddot \theta(t)+ b \dot \theta(t) + m g L \sin(\theta(t)) = {\cal G}$$
    by its overdamped limit
    $$b \dot \theta(t) + m g L \sin(\theta(t)) = {\cal G}$$

51. Find the general solution of the system
    $$\dot x = 2 x - 2 y, \ \ \dot y = 2 y - x.$$
    Here $x\equiv x(t), \ y\equiv y(t)$.

    Sketch the vector field. Indicate the length and directions of the vectors with reasonalble accuracy. Sketch some typical trajectories.

52. Plot the phase portrait and classify the fixed point of
    $$\dot x = 4 x - 3 y, \ \dot y = 8 x - 6 y.$$
    If the eigenvectors are real, indicate them on your sketch.

53. Predict the course of the events, depending on the signs and relative values of $a$ and $b$.

    Do the opposites attract? Analyze

    $$\dot R(t)= a R(t) + b J(t), \ \ \dot J(t) = - b R(t) - a J(t).$$
    This model would describe interaction of Fire and Water.

54. Draw the phase portraits as a function of a control parameter $\mu$. Classify biffurcation that occurs as $\mu$ varies and find all biffurcation values of $\mu$:
    $$\dot \theta(t)=\mu \sin\theta(t)-\sin 2\theta(t).$$

55. Find the conditions under which it is appropriate to approximate the equation
    $$m L^2 \ddot \theta(t)+ b \dot \theta(t) + m g L \sin(\theta(t)) = {\cal G}$$
    by its overdamped limit
    $$b \dot \theta(t) + m g L \sin(\theta(t)) = {\cal G}$$

56. Find the general solution of the system
    $$\dot x = 2 x - 2 y, \ \ \dot y = 2 y - x.$$
    Here $x\equiv x(t), \ y\equiv y(t)$.

    Sketch the vector field. Indicate the length and directions of the vectors with reasonalble accuracy. Sketch some typical trajectories.

57. Plot the phase portrait and classify the fixed point of
    $$\dot x = 4 x - 3 y, \ \dot y = 8 x - 6 y.$$
    If the eigenvectors are real, indicate them on your sketch.

58. Predict the course of the events, depending on the signs and relative values of $a$ and $b$.

    Do the opposites attract? Analyze

    $$\dot R(t)= a R(t) + b J(t), \ \ \dot J(t) = - b R(t) - a J(t).$$
    This model would describe interaction of Fire and Water.

59. (The Allee effect) For certain species of organisms, the effective growth rate $\dot N(t) /N$ is highest at intermediate $N$ . This is called the Allee effect (Edelstein–Keshet 1988). For example, imagine that it is too hard to find mates when N is very small, and there is too much competition for food and other resources when $N$ is large.

    a) Show that

    $$\frac{\dot N}{N} = r - a (N-b)^2$$
    provides an example of the Allee effect, if $r$, $a$, and $b$ satisfy certain constraints, to be determined.

    b) Find all the fixed points of the system and classify their stability.

    c) Sketch the solutions $N (t)$ for different initial conditions.

    d) Compare the solutions $N (t)$ to those found for the logistic equation. What are the qualitative differences, if any?

60. Sketch the bifurcation diagram of the system

    $$\dot x(t) = x(t)\left( \mu - e^{x(t)}\right).$$
    Determine what kind of bifurcations occur in this system, and at what values.

61. Draw the phase portraits as a function of a control parameter $\mu$. Classify biffurcation that occurs as $\mu$ varies and find all biffurcation values of $\mu$:
    $$\dot \theta(t)=\frac{\sin(2\theta(t))}{1+\mu\sin(\theta(t))}.$$

62. Let $A$ be a constant $n\times n$ matrix, and consider the system

      $$x\frac{d{\bf y}}{dx} = A{\bf y}, \qquad\qquad y\in \Re.$$ (1)

    (i) Show that the substitution $x=e^t$ transforms () into a system with constant coefficients, and determine the coefficient matrix of that system.

    (ii) Deduce that Euler's equation $x^2y'' + a_1xy'+a_0y=0$ can be reduced to a second-order linear equation with constant coefficients via the substitution $x=e^t$.

    (iv) Find the general solution of the system

    $$x\frac{d{\bf y}}{dx} = \left(\begin{array}{rr} -1 & -1\\ 2 & -1 \end{array}\right){\bf y}.$$

63. Consider the system
    $$\dot x = x y - 1, \ \ \ \dot y = x - y^3.$$
    Find the fixed points, classify them , sketch neighboring trajectories, and try to fill in the rest of the portrait.

64. Consider the planar system

    $$\dot r(t) = -r(t)(1-r(t))(2-r(t)),\qquad\dot\theta(t)=r^2(t),$$
    which is expressed in terms of the polar coordinates $r(t)$ and $\theta(t)$.
    • Find the fixed point(s) of this system. What is the stability of this (these) fixed point(s)?
    • Draw the phase portrait of this system in the $x-y$ plane, $x=r\cos\theta$, $y=r\sin\theta$.
    • Does this system have periodic solutions? If yes, what are the periods of these periodic solutions?

65. Consider
    $$\ddot x + \alpha\dot x( x^2+\dot x^2 -3)+x=0, \ \ \ \alpha>0.$$
    • Find and classify all fixed points
    • Show that the system has a circular limit cycle, and find its amplitude and period.
    • Determine the stability of a limit cycle.
    • Plot plausible phase portrait.

66. Weakly nonlinear oscillations: Find the dependence of the period of oscillations of a pendulum described by the equation

      $$\ddot x(t) +x(t)+ \epsilon \dot x^3(t)=0$$ (2)
    on the amplitude of oscillations $A$, assuming that both $\epsilon$ and $A$ are small. Develop a perturbation theory and multiple time scales to remove secular terms. What can you say about this system?

    Note that
    

    $$\left[A \sin(t) + B \cos(t)\right]^3 =$$

    $$\frac{3 B}{4}\cos(t)\left(A^2 + B^2\right) -\frac{B}{4}\left( 3*A... ...}{4}\sin(t)\left(A^2 + B^2\right) -\frac{A}{4}\sin(3 t) \left(A^2 + B^2\right).$$

    Also,
    $$\sin (A) \sin (B)=\frac{1}{2} (\cos (A-B)-\cos (A+B)),$$
    $$\sin (A) \cos (B)= \frac{1}{2} (\sin (A-B)+\sin (A+B)),$$
    $$\cos(A)\cos(B)= \frac{1}{2} (\cos (A-B)+\cos (A+B)),$$

67. Find equilibrium points, classify their stability and draw representative $x(t)$ trajectories for
    $$\dot x = \sin(x) (x-1)(x-2).$$

68. Consider
    $$\frac{d}{d t} x(t) = f(x(t)).$$
    Use the existence of the potential
    $$\frac{d}{d t} x(t) = - \frac{ d V (x(t))}{d x}$$
    to show that $x(t)$ can not oscillate.

69. Find Equilibrium points, classify their stability and draw representative trajectories and biffurcation diagram for the equation
    $$\dot x(t) = r (x^2(t)-r^2) x(t).$$
    Please make sure to consider the cases $r>0$, $r<0$ and $r=0$.

70. A particle travels pn the half line $x\ge 0$with velocity given by
    $$\dot x = - x^c,$$
    where $c$ is real and constant.

    (a) Find all values of $c$ such the origin $x=0$ is stable fixed point

    (b)Now assume that $c$ is chosen so that the $x=0$ is stable. Can the particle ever reach the origin in a finite time? Specifically, how long does it take for the particle to travel from $x=1$ to $x=0$, as a function of $c$?

71. Find the values of $r$ at which biffurcation occur, and classify those as saddle point, transcritical or pitchfork biffurcation. Plot the biffurcation diagram.
    $$\dot x = r x + x^3 / (1+ x ^2).$$

72. Extra Credit Consider the equation
    $$\dot x = h + r x - x^2.$$
    When $h=0$m this system undergoes a transcritical biffurction at $r=0$. Here we are going to study how the biffurcation diagram is affected by nonzero values of $h$.

    (a) Plot the biffurcation diagrams for $h=0$, $h>0$ and $h<0$.

    (b) Sketch the regions in the $(r,h)$ plane that correspond to qualitatively different vector fields and identify the biffurcations that occur on the boundary of these regions.

    (c) Plot the potential $V(x)$ that corresponds to all these different regions in the $(r,h)$ plane.

73. Consider
    $$\dot \theta(t) = \frac{\sin\theta}{\mu+\sin\theta}.$$

    Draw the phase portrait as a function of of the control parameter $\mu$. Classify the biffurcation that occur as $\mu$ varies, and find all the biffurcation values of $\mu$. Plot the biffurcation diagram.

74. Consider the motion on a circle defined by the equation

    $$\dot\theta(t) = \frac{1}{\cos(\theta)}.$$
    Plot the (circle) phase diagram for this equation, and few characteristic $\theta(t)$ curves.

    Now consider the initial condition

    $$\theta(t=0) = \frac{3}{2}\pi-\epsilon,$$
    where $\epsilon$ is extremely small positive number, say $\epsilon=10^{-100}$. Then how much time would it take for the system to reach $\theta=\frac{\pi}{2}$?

75. Find the general solution and plot the accurate phase portrait for the system
    $$\dot {\bf x}(t) = \left[\begin{array}{c c } 2 & 1 \\ 4 & -1 \end{array}\right] {\bf x}(t).$$
    Classify the stability of the origin. If the eigenvectors are real, plot them on your phase portrait.

76. Find the general solution and plot the accurate phase portrait for the system
    $$\dot {\bf x}(t) = \left[\begin{array}{c c } -1 & 1 \\ -9 & 5 \end{array}\right] {\bf x}(t).$$
    Classify the stability of the origin. If the eigenvectors are real, plot them on your phase portrait.

77. Romeo is the robot. Nothing coud ever change the way Romeo feels about Juliet:

    $$\dot R = 0 , \dot J = a R + b J.$$
    Does Juliet end up loving him or hating him?

    Under what conditions Juliet ends up loving Romeo?

78. Consider the system
    $$\dot x = y^3 - 4 x, \dot y = y^3 - y - 3x.$$
    1. Find all the fixed points and classify their stability
    2. Show that the line $y=x$ is invariant, i.e. any trajectory which starts on this line stays on this line.
    3. Show that
       $$\lim\limits_{t\to\infty}\vert x(t)-y(t)\vert\to 0$$
       for all other trajectories.
       HINT: Form the ODE for $x(t)-y(t)$.
    4. Sketch the phase portrait
79. Consider the system
    $$\dot x = - y - x^2, \dot y =x.$$
    Show that the origin is a spiral, although the linearization predicts a center.

80. Consider the system

    $$\ddot x = x^3 -x.$$
    1. Find all fixed points and classify their stability
    2. Find a conserved quantity
    3. Does this system has periodic trajectories?
    4. Sketch the phase portrait
81. Consider the system
      $$\dot x = f(x,y), \dot y = g (x,y).$$ (3)
    Let $f$ and $g$ be a smooth vector field defined on the phase plane.
    1. Show that if
       $$\frac{\partial f }{\partial y} =\frac{\partial g }{\partial x},$$
       then the system () is a gradient system.
    2. Is the previous condition sufficient for the () to be a gradient system?

82. Show that the system

    $$\ddot x + \mu (x^2 -1 ) x + \tanh{(x)}=0, \ \ \mu>0$$
    has exactly one periodic solution.

83. Consider the Duffin oscillator, described by the differential equation

    $$\ddot x(t) + x(t) + (x(t))^3=0.$$
    • Find the equation for the closed orbit $\dot x(t) = g(x(t)).$
    • Find the period of this oscillator, if the initial conditions are given by
      $$x(t=0)=1, \dot x(t=0)=0.$$

84. Plot the bifurcation diagram of the system

    $$\dot x(t) = \mu + x \sin(x).$$
    Determine what kind of bifurcations occur in this system, and at what value(s).

    Hint To gain intuition for this problem, plot $x \sin(x)$ and move the ruller up and down the graph.

85. Draw the phase portraits as a function of a control parameter $\mu$. Classify biffurcation that occurs as $\mu$ varies and find all biffurcation values of $\mu$:
    $$\dot \theta(t)=\mu+\cos(\theta) + \cos(2\theta).$$

86. Let $A$ be a constant $n\times n$ matrix, and consider the system

      $$x\frac{d{\bf y}}{dx} = A{\bf y}, \qquad\qquad y\in \Re.$$ (4)

    (i) Show that the substitution $x=e^t$ transforms () into a system with constant coefficients, and determine the coefficient matrix of that system.

    (ii) Find the general solution of the system

    $$x\frac{d{\bf y}}{dx} = \left(\begin{array}{rr} -1 & -1\\ 2 & -1 \end{array}\right){\bf y}.$$

87. Consider the system
    $$\dot x = \sin(y), \ \ \ \dot y = \cos(x).$$
    Find the fixed points, classify them , sketch neighboring trajectories, and fill in the rest of the portrait.

88. Vasquez and Redner (2004, p. 8489) mention a highly simplified model of political opinion dynamics consisting of a population of leftists, rightists, and centrists. The leftists and rightists never talk to each other; they are too far apart politically to even begin a dialogue. But they do talk to the centrists. This is how opinion change occurs. Whenever an extremist of either type talks with a centrist, one of them convinces the other to change his or her mind, with the winner depending on the sign of the parameter r. If $r>0$ the extremist always wins and persuades the centrist to move to that end of the spectrum. If $r<0$ the centrists always wins and pulls the extremist to the middle. The model’s governing equations are

    $$\dot x = r x z, \ \ \dot y = r y z , \ \ \dot z = - r x z - r y z,$$

    where $x$, $y$, and $z$ are the relative fractions of rightists, leftists, and centrists, respectively, in the population.

    (a) Show that the set $x + y + z = 1$ is invariant. What does this invariant represent in the context of the model?

    (b) Use the invariant to reduce this to a two variable system from the three variable system.

    (c) Analyze the long term behavior predicted by the model for both positive and negative values of r.

    (d) Interpret the results in political terms

    (e) Propose a more realistic model

89. Consider the competition model

    $$\dot N_1 = r_1 N_1(1 - N_1/K_1) - b_1 N_1 N_2, \dot N_2 = r_2 N_2(1 - N_2/K_22) - b_2N_1N_2.$$
    Useing the Dulac's criterion with the weighting function
    $$g =\frac{1}{N_1 N_2},$$
    show that the system has no periodic orbits in the first quadrant $N_1, N_2 > 0$.

90. Consider the planar system

    $$\dot r(t) = -r(t)(1-r(t))(2-r(t)),\qquad\dot\theta(t)=r^2(t),$$
    which is expressed in terms of the polar coordinates $r(t)$ and $\theta(t)$.
    • Find the fixed point(s) of this system. What is the stability of this (these) fixed point(s)?
    • Draw the phase portrait of this system in the $x-y$ plane, $x=r\cos\theta$, $y=r\sin\theta$.
    • Does this system have periodic solutions? If yes, what are the periods of these periodic solutions?

Bonus Problem Consider

$$\ddot x + \alpha\dot x( x^2+\dot x^2 -3)+x=0, \ \ \ \alpha>0.$$

• Find and classify all fixed points
• Show that the system has a circular limit cycle, and find its amplitude and period.
• Determine the stability of a limit cycle.
• Plot plausible phase portrait.

Bonus Problem

Weakly nonlinear oscillations: Find the dependence of the period of oscillations of a pendulum described by the equation

  $$\ddot x(t) +x(t)+ \epsilon \dot x^3(t)=0$$ (5)

on the amplitude of oscillations $A$, assuming that both $\epsilon$ and $A$ are small. Develop a perturbation theory and multiple time scales to remove secular terms. What can you say about this system?

Note that

$$\left[A \sin(t) + B \cos(t)\right]^3 =$$

$$\frac{3 B}{4}\cos(t)\left(A^2 + B^2\right) -\frac{B}{4}\left( 3*A... ...}{4}\sin(t)\left(A^2 + B^2\right) -\frac{A}{4}\sin(3 t) \left(A^2 + B^2\right).$$

Also,

$$\sin (A) \sin (B)=\frac{1}{2} (\cos (A-B)-\cos (A+B)),$$

$$\sin (A) \cos (B)= \frac{1}{2} (\sin (A-B)+\sin (A+B)),$$

$$\cos(A)\cos(B)= \frac{1}{2} (\cos (A-B)+\cos (A+B)),$$

(The Allee effect) For certain species of organisms, the effective growth rate $\dot N(t) /N$ is highest at intermediate $N$ . This is called the Allee effect (Edelstein–Keshet 1988). For example, imagine that it is too hard to find mates when N is very small, and there is too much competition for food and other resources when $N$ is large.

a) Show that

$$\frac{\dot N}{N} = r - a (N-b)^2$$

provides an example of the Allee effect, if $r$, $a$, and $b$ satisfy certain constraints, to be determined.

b) Find all the fixed points of the system and classify their stability.

c) Sketch the solutions $N (t)$ for different initial conditions.

d) Compare the solutions $N (t)$ to those found for the logistic equation. What are the qualitative differences, if any?

ANSWER Equilibria are $N=0$ and $N = b \pm \sqrt{\frac{r}{a}}$. For $N\ll 1$, $\dot N = N(r-a b^2)$ so that the rate is $r - a b^2$. If $r< a b^2$ the rate is negative. Constraints are

$$a>0,r>0,b^2>r/a.$$

The difference with Logistics equation that rate may be negative, so small population decreases to zero. Zero is unstable Fixed Point