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MATH239101 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 Examination for the Module MATH2391 (May 2014) Nonlinear Differential Equ...

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MATH239101

MATH239101 All calculators must carry an approval sticker issued by the School of Mathematics.

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

c UNIVERSITY OF LEEDS  Examination for the Module MATH2391 (May 2014) Nonlinear Differential Equations Time allowed: 2 hours

Answer four questions. All questions carry equal marks.

1. (a) For the dynamical system dx = x2 + x − 2 dt identify the equilibrium points and their stability. Plot x(t) as a function of t for different choices of the initial condition x(0). (b) Find the general solution of the equation. (c) Use your answer to (b) to find the solutions x1 (t) and x2 (t) of the equation satisfying the initial conditions x1 (0) = 0 and x2 (0) = −2 respectively. (d) Define what is meant by saying that a fixed point of the equation dx = f (x) dt is isolated. Give an example of a nonlinear first order equation with a non-isolated fixed point at x = 1.

2. (a) Sketch the phase portraits of the following two-dimensional dynamical systems   = x − 2y, = x − 2y,  dx  dx dt dt , (ii) (i)  dy  dy = 2x + y. = y. dt dt

1

continued . . .

MATH239101

(b) Let A be a square matrix with constant coefficients. Show that x(t) = exp(At)x0 is the solution of the initial value problem d x = A · x, dt

x(0) = x0 ,

where x0 is a constant vector and the exponential function of the matrix exp(At) is defined as ∞ X A n tn exp(At) = I + n! n=1 where I is the identity matrix. (c) Let µ be a real parameter. Linearise the system  dx  dt = µx − sin y, 

dy dt

,

= µy − sin x.

near the origin (x, y) = (0, 0). Determine the values of the parameter µ such that the origin is not a hyperbolic fixed point. (d) Give an example of nonlinear system of three equations dy dz dx = f (x, y, z), = g(x, y, z), = h(x, y, z ), dt dt dt with a non-hyperbolic fixed point at x = 1, y = 1, z = 1.

3. Consider the following system of equations dy = y(6 − y − 2x) dt

dx = x(6 − x − 2y), dt

(1)

in the domain x ≥ 0, y ≥ 0. (a) Find all fixed points. (b) Linearise the system about each fixed point and sketch the corresponding phase portrait. (c) Sketch a plausible phase portrait of the non-linear system (1) in the domain x ≥ 0, y ≥ 0 and indicate a basin of attraction of any stable fixed point. (d) Use Poincar´e index theory to prove the absence of periodic orbits as solutions of (1).

4. (a) Show that, for an appropriate choice of the integers n, m and real constant a, the function V = xn + ay m is a Lyapunov function for the system dy = −y 3 − x sin(xy). dt

dx = y 3 sin(xy) − x, dt

Determine the values of n, m and a for which V is a Lyapunov function and determine whether the solution x = 0, y = 0 is asymptotically stable. 2

continued . . .

MATH239101

(b) Prove that a gradient system ∂Φ(x, y) dx =− , dt ∂x

∂Φ(x, y) dy =− , dt ∂y

where Φ(x, y) is a smooth function, cannot have periodic solutions. (c) For the system of equations dy = x(x2 + y 2 ) dt

dx = y (x2 + y 2 ), dt determine the Poincar´e index of the origin. (d) Show that the system

dy dx = −xy 2 , = −yx2 dt dt does not have periodic solutions. (To solve this problem you can use any method of your choice).

5. (a) Show that I1 = exp(2x) − exp(2y) and I2 = exp(2y) − exp(2z) are first integrals of the dynamical system dx = exp(y + z − x), dt dy = exp(z + x − y), dt dz = exp(x + y − z). dt

(2)

(b) Verify that the first integrals I1 and I2 are functionally independent. (c) Let Φ1 = Φ1 (x1 , x2 , . . . , xN ), Φ2 = Φ2 (x1 , x2 , . . . , xN ) be first integrals of a dynamical system. Let Φ3 = f (Φ1 , Φ2 ), where f is a continuously differentiable non-constant function. Show that the gradients of Φi   ∂Φi ∂Φi ∂Φi ∇(Φi ) = ,..., , , i = 1, 2, 3 ∂xN ∂x1 ∂x2 are linearly dependent vectors at each point of the phase space. (d) Motion of a small satellite of mass m in the gravity field of a planet of mass M can be described by the Newton equation d2~r ~r = −MG 3 , 2 r dt where G is a gravitational constant. Show that the vector of angular momentum ~L = m~r × d~r dt is a first integral of the system. From this result, conclude that every component of the ~ = (Lx , Ly , Lz ) is a first integral. vector L END 3...