We suppose that ε ⪡ 1 and \[ \delta = \varepsilon \delta_1 , \] where δ₁ = O(1) as ε → 0. We consider the case k = 2, which corresponds to the strongest instability, when y(t<, ε) satisfies \[ y'' + \left( 1 + \varepsilon \delta_1 + \varepsilon\,\cos 2t \right) y = 0 . \] The idea of the multiple scale perturbation method is to describe the evolution of the solution over long timescales of the order 1/ε by the introduction of an additional ‘slow’ time variable \[ \tau = \varepsilon t . \] We then look for a solution of the form \[ y(t, \varepsilon ) = \hat{y} (t, \varepsilon t , \varepsilon ) , \] where ŷ(t, τ, ε) is a function of two time variables (t, τ) that gives y when τ is evaluated at εt.

Applying the chain rule, we find that \begin{align*} y' &= \hat{y}_t + \varepsilon \hat{y}_{\tau} , \\ y'' &= \hat{y}_{tt} + 2 \varepsilon \hat{y}_{t\tau} + \varepsilon^2 \hat{y}_{\tau\tau} , \end{align*} where the subscripts denote partial derivatives. Using this result in the original equation, and denoting partial derivatives by subscripts, we find that ŷ(t, τ, ε) satisfies \[ \hat{y}_{tt} + 2 \varepsilon\,\hat{y}_{t\tau} + \varepsilon^2 \hat{y}_{\tau\tau} + \left( 1 + \varepsilon \delta_1 + \varepsilon\, \cos 2t \right) \hat{y} = 0 . \] In fact, ŷ(t, τ, ε) only has to satisfy this equation when τ = εt, but will require that it satisfies it for all (t, τ). This requirement implies that y satisfies the original ODE. We have therefore replaced an ODE for y by a PDE for ŷ. At first sight, this may not appear to be an improvement, but as we shall see we can use the extra flexibility provided by the dependence of ŷ on two variables to obtain an asymptotic solution for y that is valid for long times of the order 1/ε. Specifically, we will require that y(t, τ, ε) is a periodic function of the ‘fast’ variable t. Moreover, we only need to solve ODEs in t to construct this asymptotic solution.

We expand \[ \hat{y} (t, \tau , \varepsilon ) = y_0 (t, \tau ) + \varepsilon y_1 (t, \tau ) + O\left( \varepsilon^2 \right) . \] We use this expansion in the equation for ŷ, and equate coefficients of ε⁰ and ε to zero. We find that \begin{align*} y_{0tt} + y_0 &= 0 , \\ y_{1tt} + y_1 + 2 y_{0t\tau} + \left( \delta_1 + \cos 2t \right) y_0 &= 0 . \end{align*} The solution of the first equation is \[ y(t, \tau ) = A(\tau )\,e^{{\bf j}t} + c.c. \] Here, it is convenient to use complex notation. The amplitude A(τ) is an arbitrary complex valued function of the ‘slow’ time, and c.c. denotes the complex conjugate of the preceding terms.

Using this solution in the second equation, and writing the cosine in terms of exponentials, we find that y₁ satisfies \begin{align*} y_{1tt} + y_1 &= -2\mathbf{j} A_{\tau} e^{\mathbf{j}t} - A\left( \delta_1 + \cos 2t \right) e^{\mathbf{j}t} + c.c \\ &= -\frac{1}{2}\,A\,e^{3\mathbf{j}t} - \left( 2\mathbf{j} A_{\tau} + \delta_1 A + \frac{1}{2}\,A^{\ast} \right) e^{\mathbf{j}t} + c.c. \end{align*} Here, the asterisk denotes a complex conjugate. The solution for y₁ is periodic in t, and does not contain secular terms in t, if and only if the coefficient of the resonant term e^jt is zero, which implies that A(τ) satisfies the ODE \[ 2\mathbf{j} A_{\tau} + \delta_1 A + \frac{1}{2}\,A^{\ast} = 0 . \] Writing A = u + jv in terms of its real and imaginary parts, we find that \[ \begin{pmatrix} u \\ v \end{pmatrix}_{\tau} = \begin{bmatrix} 0& \delta_1 /2 - 1/4 \\ -\delta_1 /2 - 1/4 & 0 \end{bmatrix} \begin{pmatrix} u \\ v \end{pmatrix} \] The solutions of this equation are proportional to e^±λτ where \[ \lambda = \frac{1}{2} \sqrt{\frac{1}{4} - \delta_1^2} . \] Thus, in the limit ε → 0 the equilibrium y = 0 is unstable when |δ₁| < ½, or \[ |\delta | < |\varepsilon | . \]    ■

We have to determine both the period T(ε) and the amplitude 𝑎(ε) of the limit cycle. Since the ODE is autonomous, we can make a time-shift so that y′(0) = 0. Thus, we want to solve the ODE subject to the conditions that \begin{align*} y(t+T, \varepsilon ) &= y(t, \varepsilon ), \\ y(0, \varepsilon )&= a(\varepsilon ) , \\ y'(0, \varepsilon )&= 0 . \end{align*}    ■

2. Lin, C.C. and Segel, L.A., Mathematics Applied to Deterministic Problems in the Natural Sciences, SIAM, Society for Industrial and Applied Mathematics, 1988.