7.2: Problem Set

Last updated
Save as PDF

Page ID: 24173

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

EXERCISE \(\PageIndex{1}\)

Consider the following autonomous vector field on the plane:

\(\dot{x} = y\),

\(\dot{y} = x-x^3-\delta y, \delta \ge 0, (x,y) \in \mathbb{R}^2\).

Use Lyapunov’s method to show that the equilibria \((x, y) = (\pm 1, 0)\) are Lyapunov stable for \(\delta = 0\) and asymptotically stable for \(\delta > 0\).

EXERCISE \(\PageIndex{2}\)

Consider the following autonomous vector field on the plane:

\(\dot{x} = y\),

\(\dot{y} = -x-\epsilon x^{2}y, \epsilon > 0, (x, y) \in \mathbb{R}^2\).

Use the LaSalle invariance principle to show that

(x, y) = (0, 0),

is asymptotically stable.

EXERCISE \(\PageIndex{3}\)

Consider the following autonomous vector field on the plane:

\(\dot{x} = y\),

\(\dot{y} = x-x^3- \alpha x^{2}y, a > 0, (x, y) \in \mathbb{R}^2\),

use the LaSalle invariance principle to describe the fate of all trajectories as \(t \rightarrow \infty\).

EXERCISE \(\PageIndex{4}\)

Consider the following autonomous vector field on the plane:

\(\dot{x} = y\),

\(\dot{y} = x-x^3+\alpha xy, (x, y) \in \mathbb{R}^2\),

where \(\alpha\) is a real parameter. Determine the equilibria and discuss their linearized stability as a function of \(\alpha\).

EXERCISE \(\PageIndex{5}\)

Consider the following autonomous vector field on the plane:

\(\dot{x} = ax+by\),

\[\dot{y} = cx + dy, (x, y) \in \mathbb{R}^2, \label{7.34}\]

where \(a, b, c, d \in \mathbb{R}\). In the questions below you are asked to give conditions on the constants a, b, c, and d so that particular dynamical phenomena are satisfied. You do not have to give all possible conditions on the constants in order for the dynamical condition to be satisfied. One condition will be sufficient, but you must justify your answer.

Give conditions on a, b, c, d for which the vector field has no periodic orbits.
Give conditions on a, b, c, d for which all of the orbits are peri- odic.
Using
\(V(x,y) = \frac{1}{2}(x^2+y^2)\)

as a Lyapunov function, give conditions on a, b, c, d for which (x, y) = (0, 0) is asymptotically stable.
Give conditions on a, b, c, d for which x = 0 is the stable man- ifold of (x,y) = (0,0) and y = 0 is the unstable manifold of (x, y) = (0, 0).

EXERCISE \(\PageIndex{6}\)

Consider the following autonomous vector field on the plane:

\(\dot{x} = y\),

\[\dot{y} = x-x^{2}y, (x,y) \in \mathbb{R}^2. \label{7.35}\]

Determine the linearized stability of (x, y) = (0, 0).
Describe the invariant manifold structure for the linearization of (7.35) about (x, y) = (0, 0).
Using \(V(x, y) = \frac{1}{2}(x^2 + y^2)\) as a Lyapunov function, what can you conclude about the stability of the origin? Does this agree with the linearized stability result obtained above? Why or why not?
Using the LaSalle invariance principle, determine the fate of a trajectory starting at an arbitrary initial condition as \(t \rightarrow \infty\)? What does this result allow you to conclude about stability of (x, y) = (0, 0)?