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MATH239101 This question paper consists of 3 printed pages, each of which is identified the reference MATH239101. All calculators must carry an approval sticker issued the School of Mathematics. c University of Leeds School of Mathematics May 2015 MATH239101 Nonlinear Differential Equations Time All...

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MATH239101

This question paper consists of 3 printed pages, each of which is identified by the reference MATH239101.

All calculators must carry an approval sticker issued by the School of Mathematics.

University of Leeds c School of Mathematics May 2015 MATH239101 Nonlinear Differential Equations Time Allowed: 2 hours Answer no more than 4 questions. If you attempt 5, only the best 4 will be counted. All questions carry equal marks.

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MATH239101 1. (a) For the following one-dimensional dynamical systems dx dy = y3 − 1 = (x − 1)3 , (ii.) dt dt identify the equilibrium points and determine their stability. Determine the behavior of solutions, plotting x(t) and y(t) as a functions of t for different choices of the initial condition x(0) and y(0). (i.)

(b) Find the general solution of equation (i.). (c) Find one solution x1 (t) of equation (i.) satisfying the initial condition x1 (0) = 0. Find another solution x2 (t) of equation (i.) satisfying the initial condition x2 (0) = 1. (d) A differential equation y ′ = f (y ) with a continuously differentiable vector field f (y) has exactly three fixed points: at y1 = −1, y2 = 0 and y3 = 1. It is known that the fixed point y1 = −1 is stable and that the fixed point y3 = 1 is unstable. What can you say about the stability of the fixed point y2 = 0? 2. (a) Sketch the phase portrait for the following two-dimensional dynamical system dx = 2x − 3y, dt dy = 4x − 5y. dt and classify the equilibrium point (0, 0). On the phase portrait indicate the direction of the vector field at the points (1, 0) and (1, −1). Find the general solution of the system. (b) Let   −1 −1 . B= 1 1

  1 0 , A= 0 −1

Find exp(At), exp(Bt) and exp((A + B)t). Show that exp(At) exp(Bt) 6= exp(Bt) exp(At). (c) For a given t

exp(Ct) = e



1 + t −t t 1−t



.

Find C. (d) Define what is meant by saying that a fixed point of a system dx = f (x, y, z) , dt

dy = g(x, y, z) , dt

dz = h(x, y, z) dt

is hyperbolic? In the case f (x, y, z ) = g(x, y, z ) show that any fixed point (if it exists) is nonhyperbolic.

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MATH239101 3. Consider the following system of equation dy dx = y(−1 − y + 2x) = x(−1 − x + 2y), dt dt in the domain U = {(x, y) ∈ R2 | − 3 ≤ x ≤ 3, −3 ≤ y ≤ 3}.

(1)

(a) Find all fixed points in the domain U . (b) Linearise the system about each fixed point, classify them and sketch the corresponding phase portraits. (c) Sketch a plausible phase portrait of the non-linear system (1) in the domain U . For a better accuracy use the horizontal and vertical null-clines. (d) Explain why system (1) cannot have periodic orbits. 4. (a) Give a definition of a Liapunov function. Consider the system of equations dx dy = −x − y 5 . = y 3 − x3 , dt dt Show that it has a Liapunov function of the form V = axn +y m and find appropriate values for n, m and a. Explain why this system cannot have any periodic solution. (b) Show that the system dy dx = −2xy = −y 2 , dt dt does not have periodic solutions. (To solve this problem you can use any method of your choice). (c) Give a definition of a gradient system. Prove that a gradient system cannot have periodic orbits. (d) Consider a smooth vector field x˙ = f (x, y), y˙ = g(x, y) on the plane and a simple closed curve Γ that does not pass through any critical point of the vector field. Let f (x, y) ≥ g(x, y) for all (x, y) ∈ Γ. Show that the Poincar´e index IndΓ (f (x, y), g(x, y)) = 0. 5. (a) Give a definition of a first integral for a system y˙ = F(y), y ∈Rn . Prove the statement: If the first integrals I1 = I1 (y, t), . . . , Ik = Ik (y, t) are func∂Ii ∂Ii ∂Ii ), i = 1, . . . , k tionally dependent, then their gradients ∇Ii = ( ∂y , , . . . , ∂y n 1 ∂y 2 are linearly dependent. (b) Show that H = x2 y is a first integral of the system dx = x4 y exp(x2 y), (2) dt dy = −2x3 y 2 exp(x2 y ) dt (c) Using this first integral reduce the order of system (2) and find solution of the initial value problem with initial data x(0) = x0 ,

y (0) = y0 .

(d) Sketch a phase portrait of the system indicating all fixed points. Are these fixed points isolated? Are they hyperbolic? 3

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