Section 23.2

Supplementary Problems

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Problem 23.2.1

Consider the Lotka-Volterra model

\frac{du_{1}}{dt} = f_{1}(u_{1}, u_{2}) = u_{1} - u_{1}u_{2}, \qquad \frac{du_{2}}{dt} = f_{2}(u_{1}, u_{2}) = -u_{2} + u_{1}u_{2}

with the fixed point $(u_{1}^{*} = 1, u_{2}^{*} = 1)$ . The linearized equation (variational equation) with

\frac{\partial f_{1}}{\partial x_{1}} = 1 - u_{2}, \quad \frac{\partial f_{1}}{\partial x_{2}} = -u_{1}, \quad \frac{\partial f_{2}}{\partial x_{1}} = u_{2}, \quad \frac{\partial f_{2}}{\partial x_{2}} = -1 + u_{1}

is given by

\begin{pmatrix} dy_{1}/dt \\ dy_{2}/dt \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} y_{1} \\ y_{2} \end{pmatrix} .

Solve this system of linear differential equations. First show that the eigenvalues of the $2 \times 2$ matrix are given by $\pm i$ .

Problem 23.2.2

Consider the initial problem of the matrix differential equation

\frac{dX}{dt} = A(t)X, \qquad X(0) = I_{n}

where $A(t)$ is an $n \times n$ matrix which depends smoothly on $t$ . It is known that the solution of this matrix differential equation can locally be written as

X(t) = \exp (\Omega (t))

where $\Omega (t)$ is obtained as an infinite series

\Omega (t) = \sum _{k=1}^{\infty } \Omega _{k}(t) .

This is the so-called Magnus expansion. Implement this recursion in SymbolicC++ and apply it to

A(t) = \begin{pmatrix} \cos (\omega t) & -\sin (\omega t) \\ \sin (\omega t) & \cos (\omega t) \end{pmatrix}

where $\omega$ is a constant frequency.