Home Work ONE

1. 2.2.6 Fixed points are
   $$x^*{\rm unstable}= \frac{\pi}{3} +2 \pi n,$$
   and
   $$x^*{\rm stable}=\frac{5\pi}{3}+2 \pi n,$$
2. 2.2.8 There are infinitely many possibilities, including
   $$\dot x = (x+1)^2 x (x-2).$$
3. 2.3.3 Here $a$ is a growth rate of a tumor, and $b$ is akin “carrying capacity”, or “saturation”, since the tumor, according to this equation, can not grow past $N(t)=b$.
4. 2.4.8 The fixed points are $N=0$ and $N = frac{1}{b}$. The derivative is
   $$f'(N) = -a (\ln{(b N)} +1).$$
   Then $N=0$ is unstable and $N=1/b$ is stable.

5. 2.5.4 since $x=0$ is a solution, and so is
   $x(t)=0$ if $t<T$
   and
   $x(t)=(\frac{2}{3})^{\frac{3}{2}}(t-T)^{\frac{3}{2}},$ for $t\ge T$.

6. 2.6.1 Harmonic oscillator is a second order ODE, so it is equivalent to two first order ODE's. Therefore, using the definition of a book, it is a second order system.
7. 2.7.6 The potential is equal to
   $$V(x)=-r x - \frac{x^2}{2}+\frac{x^4}{4}.$$

   For $r=0$, the fixed points are $x=0$, which is unstable, and $x=\pm 1$, which are stable.

   For small values of $\vert r\vert$ those points are slightly disturbed.

   For large positive values of $r$ left two points disappear via saddle node bifurcation.

   For large negative values of $r$ right two points disappear.