9.2: Problem Set

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Exercise \(\PageIndex{1}\)

Consider the following autonomous vector field on the plane:

\(\dot{x} = \mu x-3y-x(x^2+y^2)^3\),

\(\dot{y} = 3x+\mu y-y(x^2+y^2)^3\),

where \(\mu\) is a parameter. Analyze possible bifurcations at (x,y) = (0, 0) for \(\mu\) in a neighborhood of zero. (Hint: use polar coordinates.)

Exercise \(\PageIndex{2}\)

These exercises are from the book of Marsden and McCracken8. Consider the following vector fields expressed in polar coordinates, i.e. \((r, \theta) \in \mathbb{R}^{+} \times \mathcal{S}^1\), depending on a parameter \(\mu\). Analyze the stability of the origin and the stability of all bifurcating periodic orbits as a function of \(\mu\).

(a)

\(\dot{r} = -r(r-\mu)^2\),

\(\dot{\theta} = 1\).

(b)

\(\dot{r} = r(\mu-r^2)(2\mu-r^2)^2\),

\(\theta{\theta} = 1\).

(c)

\(\dot{r} = r(r+\mu)(r-\mu)\),

\(\dot{\theta} = 1\).

(d)

\(\dot{r} = \mu r(r^2-μ)\),

\(\dot{\theta} = 1\).

(e)

\(\dot{r} = -\mu^{2}r(r+\mu)^2(r-\mu)^2\),

\(\dot{\theta} = 1\).

Exercise \(\PageIndex{3}\)

Consider the following vector field:

\(\dot{x} = \mu x-\frac{x^3}{2}+\frac{x^5}{4}, x \in \mathbb{R}\)

where \(\mu\) is a parameter. Classify all bifurcations of equilibria and, in the process of doing this, determine all equilibria and their stability type.