For each Of the following first-order differential equations

complete the following:

(i) Find all equilibrium points

(ii) C ossify [110 stability Of each equilibrium point (ellooso the Strongest that applies): Unstable, stable, asymptotically stilble, globally asymptotically stable

(iii) Determine if the system oiliibits finite-time escape

A function F : IR- R- is said to be homogenous of de.grce 1 if, for fill E and all A C), it holds that F(Ax) AOF (.1:).

(n) Consider the dynamical system = f (x) for locally Lipscllit.z J, and as let é(t, To) denote its solution with initial condition :ro, that is,

Show that if f is homogenous of degree then, for any IR- and any

for any i O.

Hint: Let t(i) +(t, be the left-hand side and r(t) xo) the rigillt-hnncl side of the above equation. To show that (t) = for fill f 0, it is enough to show that (O) and that and r satisfy the same differential equation: i(i) = f (C(i)) and i(i) = f (TV)) for all

(b) Show that the vector field for EIIC following system is homogenous of degree = l:

where

h(.T1, T2)

for some a O and

(c) Find n Lynpunov function [hat is homogenous of degree that. establishes global asymptotic stability of the origin for the system in part (b). Dont forget to show that your function is homogenous of degree 2.

Consider the system

2f1 1 — G — —a;2

Show that there exists periodic orbit in the region

Consider the discrete-titli0 system

.1%.41 P.

(a) Find all fixed points as function Of /1. Classify the stability Of any equilibrin by linearizing the dynamics at the equilibrium. Your answer might depend and yoil should consider bill cases. If the linearization cannot be used to determine stability, explain why.

(b) Identify the bifurcation point. for the parameter ancl classify the bifurcation (ignore any period-dotibling biftrrcations, which do exist in this system).

(c) Determine stability of any equilibria wllen p by sketching the cobweb diagram for several init.ial conditions.

(n) Show [hat the origin of the following system is asymptotically stable by finding appropriate closed-form Lyapltnov function:

—T2 — :rrc-

Be careftil to establish all required properties of your Lyapunov function. Hint: notice that [lie graph of lies iti the first third qliaclratit.

(b) Determine if your Lyapttnov function folitid in part (a) In tised [o prove global asymptotic stability of [110 origin. Explain why or why not.

complete the following:

(i) Find all equilibrium points

(ii) C ossify [110 stability Of each equilibrium point (ellooso the Strongest that applies): Unstable, stable, asymptotically stilble, globally asymptotically stable

(iii) Determine if the system oiliibits finite-time escape

A function F : IR- R- is said to be homogenous of de.grce 1 if, for fill E and all A C), it holds that F(Ax) AOF (.1:).

(n) Consider the dynamical system = f (x) for locally Lipscllit.z J, and as let é(t, To) denote its solution with initial condition :ro, that is,

Show that if f is homogenous of degree then, for any IR- and any

for any i O.

Hint: Let t(i) +(t, be the left-hand side and r(t) xo) the rigillt-hnncl side of the above equation. To show that (t) = for fill f 0, it is enough to show that (O) and that and r satisfy the same differential equation: i(i) = f (C(i)) and i(i) = f (TV)) for all

(b) Show that the vector field for EIIC following system is homogenous of degree = l:

where

h(.T1, T2)

for some a O and

(c) Find n Lynpunov function [hat is homogenous of degree that. establishes global asymptotic stability of the origin for the system in part (b). Dont forget to show that your function is homogenous of degree 2.

Consider the system

2f1 1 — G — —a;2

Show that there exists periodic orbit in the region

Consider the discrete-titli0 system

.1%.41 P.

(a) Find all fixed points as function Of /1. Classify the stability Of any equilibrin by linearizing the dynamics at the equilibrium. Your answer might depend and yoil should consider bill cases. If the linearization cannot be used to determine stability, explain why.

(b) Identify the bifurcation point. for the parameter ancl classify the bifurcation (ignore any period-dotibling biftrrcations, which do exist in this system).

(c) Determine stability of any equilibria wllen p by sketching the cobweb diagram for several init.ial conditions.

(n) Show [hat the origin of the following system is asymptotically stable by finding appropriate closed-form Lyapltnov function:

—T2 — :rrc-

Be careftil to establish all required properties of your Lyapunov function. Hint: notice that [lie graph of lies iti the first third qliaclratit.

(b) Determine if your Lyapttnov function folitid in part (a) In tised [o prove global asymptotic stability of [110 origin. Explain why or why not.

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