Home Work TWO

• 3.1.4 First, plot the two functions $f(x,r) = r+x/2$ and $g(x)=\frac{x}{1+x}$ for various values of $r$. The following command in Mathematica will do it for you:

  Plot[{x/(1 - x), 1 + x/2, -0.3 + x/2, -2.9 + x/2}, {x, -4, 4}]
  

  You will see that depending upon $r$ the straight line $f(x)$ either touches $g(x)$, crosses $g(x)$ or does not intersect with $g(x)$. Therefore there are two saddle node bifurcations. The fixed points are found by solving $f(x)=g(x)$, which gives
  $$x^*=\frac{1}{2}\left(1-2 r\pm \sqrt{4 r^2 -12 r +1}\right).$$
  The saddle point bifurcation occurs when square root is zero, i.e. when
  $$r=\frac{3\pm 2\sqrt{2}}{2}.$$

• 3.2.2 Taking into account that for small $x$,
  $$\log(1+x)\simeq x -\frac{x^2}{2},$$
  we obtain, for small $x$
  $$\dot x(t)\simeq x (r-1)+\frac{x^2}{2}.$$
  Changing variables by $x(t) = 2 y(t)$ we obtain the normal form for transcritical bifurcation
  $$\dot y = y (r-1) + y^2.$$
  Transcritical bifurcation occurs at $r=1$. Indeed, the only way how the $f(x) = r x$ does not cross $g(x)=\ln(1+x)$ in two points is to choose $r=1$, so that the two curves are tangential to each other at the origin. The following command in Mathematica will draw it for you:

  Plot[{Log[1 + x], x, x/2, 2 x}, {x, -1, 2}]
  

• 3.3.2 Equating $\dot P=\dot D=0$ we obtain from second and third equations:
  $$D = \frac{\lambda+l}{1+\lambda E^2}, \ \ P = E\frac{\lambda+l}{1+\lambda E^2}.$$
  We now substitute it to the first equation to obtain
  $$\dot E(t) = \kappa\lambda E(t)\frac{\lambda +l - 1-\lambda E^2(t)} {1+\lambda E^2(t)}.$$
  The fixed points of this equation are $E=0$ and
  $$E = \pm\sqrt{1+\frac{l-1}{\lambda}}.$$
  The latter fixed points exist only when $\lambda>1-l$ for positive $\lambda$ and $\lambda<1-l$ for negative $\lambda$.
• 3.4.4 The fixed points are $x=0$ and $x=\pm\sqrt{-(1+r)}.$ The latter root exists only for $r<-1$, where the pitchfork bifurcation occurs as a transition from one zero root to three roots.

  Take the value of $r$ to be equation to the critical value $r=-1$. Then all three roots are zero. We then can use Taylor series to expand the original equation to obtain

  $$\dot x = x (1+r +x^2),$$
  which is the normal form of the subcritical pitchfork bifurcation.
• 3.4.11 a If $r=0$ then there are infinitely many fixed points with even multiples of $\pi$ being stable, and odd multiples of $\pi$ being unstable.
  b If $r>1$ there is only one unstable fixed point $x=0$.
  c As $r$ decreases from $\infty$ to $0$ the first bifurcation occurs at $r=1$. This is subcritical pitchfork bifurcation, as can be seen from Taylor expanding $\sin(x)\simeq x - \frac{x^3}{6}.$ Subsequent bifurcations are saddle-node bifurcation, with the first one occuring around $r=0.1284$.
  d Bifurcations occur when two lines are “touching”, i.e. they are tangential and intersect. We need to solve therefore $r x = \sin(x)$ to find intersection and $r = \cos(x)$ for the curves to be tangential. These two equations give us $x^* = \tan(x^*)$ to find the x position and $r = \cos(x^*)$ to find the $r$ position of bifurcation.

  e To find the bifurcation diagram, we need to plot $x^*$ and $r$ solving

  $$r x^* = \sin(x^*).$$
  While we can not solve this transcendental equation, we can express $r$ as a function of $x^*$:
  $$r(x^*)=\frac{\sin(x^*)}{x^*}.$$
  The following Mathematica code will plot it for you:

  Plot[ {10 x, Sin[x]/x}, {x, -20, 20}, PlotRange -> {-0.4, 1}]
  

  Here $10 x$ is a graph of a vertical line. Hence you may find $r$ for any $x$ between $-\infty$ and $\infty$.
• 3.4.12 Consider the equation
  $$\dot x(t) = (x^2(t)-r)(x^2(t)-2 r).$$
  For this equation the fixed points are $x=\pm\sqrt{r}$ and $x=\pm \sqrt{2 r}$. These fixed points exists only for $r\ge 0$. Therefore this equation does satisfy the condition for saddle node quadfurcation. To extend it to the bifurcation of order $M$ consider
  $$\dot x(t) = \prod\limits_{k=1}^M (x^2(t)-k r).$$
  Fixed points of this equation are $x=\pm\sqrt{ k r}$ for $k=1..M$.

  Another version would be

  $$\dot x(t) = \prod\limits_{k=1}^M (x(t)-k \sqrt{r}).$$

• 3.5.4 Suppose $x(t)$ is the position of the mass. Then the length of the spring is $\sqrt{x^2+h^2}$, and the force by the spring is therefore $\vec{F} = - k (\sqrt{x^2+h^2}-L_0).$ The projection of this force on the line of motion is
  $$F\cos{\alpha} = - k (\sqrt{x^2+h^2}-L_0) \frac{x}{\sqrt{x^2+h^2}}.$$
  Equation of motion then becomes
  $$m\ddot x + b \dot x +k (\sqrt{x^2+h^2}-l) \frac{x}{\sqrt{x^2+h^2}}=0.$$
  This simplifies to
  $$m\ddot x + b \dot x +k x \left(1 -\frac{l}{\sqrt{x^2+h^2}}\right)=0.$$
  Equilibria are $x=0$ and $x = \pm \sqrt{L_0^2 - h^2}$. The latter two points exist only if $L_0>h$.

  To make equation dimensionless, take $t = \tau \frac{b}{k}$, $x=h y$ to rewrite equation as

  $$\frac{m k }{ b^2} \ddot y + \dot y + \left(1-\frac{L_0}{h\sqrt{1+y^2}}\right)y=0.$$
  One can take $m=0$ if $b^2\gg m k$, i.e. large friction.

• 3.7.6
  (a)
  $$\dot x + \dot y + \dot z = - k x y + k x y - l z + l z =0,$$
  so that $N=x+y+z$ is constant
  (b) We have
  $$\dot x = - \frac{k }{l} x \dot z,$$
  or
  $$\frac{\dot x}{x} = - \frac{k}{l}\dot z.$$
  Integrating both parts we obtain
  $$x(t)=x(t=0)e^{-frac{k}{l}z}.$$
  
  (c)
  $$\dot z = l y = l (N-z-x)=l(N-z-x_0 e^{-\frac{k z}{l}}).$$
  (d) Choose $u=\frac{k z }{l}$ and replace time variable $t$ by $t=T \tau$ with characteristic time $T=k x_0 t$. This leads to
  $$\frac{d}{d\tau} u= a - b u - e^{-u}$$
  with $a=\frac{N}{x_0}$ and $b=\frac{l}{k x_0}$.
  (e)Since $N=x_0+y_0+z_0$ with $x_0>0, y_0>0$ and $z_0>0$ we obtain $a=\frac{N}{x_0}>1$. Since $k$ and $l$ are positive, with $x_0$ also positive, the constant $b$ is therefore also positive.
  (f) To find equilibrium points, plot $a-b u$ and $e^{-u}$ functions for various values of $a$ and $b$, taking into account that $a>1$ and $b>0$. We then will see that there are two equilibria: $\bullet$ positive and stable, and $\bullet$ negative, unstable and nonphysical.
  (g) Since
  $$\dot u = \frac{k}{l}\dot z = k y,$$
  all these three functions are just scaled versions of each other, so that the maxima of these functions occur at the same time.
  (i)