Question

Given a nonlinear system x(t)" + cx(t)' + sin(x(t)) = 0 (3-1)

a) Consider its phase plane by assuming proper parameters of c.

Do you have singular points with the cases of CENTER FOCUS, NODE and SADDLE ? Are those stable? asymptotic stable or unstable?

b) When c = 0, plot the phase plane and find these singular points

Answer #1

following nonlinear system:
x' = 2 sin y,
y'= x^2 + 2y − 1
find all singular points in the domain x, y ∈ [−1, 1],determine
their types and stability.
Find slopes of saddle separatrices.
Use this to sketch the phase portrait in the domain x, y ∈ [−1,
1].

Consider the linear system x' = x cos a − y sin a
y'= x sin a + y cos a
where a is a parameter. Show that as a ranges over [0, π], the
equilibrium point at the origin passes through the sequence stable
node, stable spiral, center, unstable spiral, unstable node.

Consider the nonlinear second-order differential
equation 4x"+4x'+2(k^2)(x^2)− 1/2 =0, where k > 0 is a constant.
Answer to the following questions.
(a) Show that there is no periodic solution in a simply connected
region R={(x,y) ∈ R2 | x <0}. (Hint: Use the corollary to
Theorem 11.5.1>>
If symply connected region R either contains no critical points of
plane autonomous system or contains a single saddle point, then
there are no periodic solutions. )
(b) Derive a plane autonomous system...

Consider the ‘spring-mass system’ represented by an ODE x′′ (t)
+ 16x(t) = 5 sin 4t with ICs: x(0) = 2, x′ (0) = 1. Answer the
questions (a)–(c):
(a) Is there damping in the system? Why or why not?
(b) Is there resonance in the system? Why or why not?
(c) Solve the ODE.

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