Parametric resonance

Parametric resonance is growth caused by periodic variation of a system coefficient, rather than an additive forcing term. In the Mathieu equationthe stiffness varies near twice the natural angular frequency. Put $T=ϵt$ and $u_0=a(T)\cos t+b(T)\sin t$. The order-$ϵ$ correction has resonant coefficients $-2b_T-a$ in the cosine component and $2a_T+b$ in the sine component. Eliminating them givesThe slow eigenvalues are $\pm 1/2$, so an initial component in the growing direction is amplified as $e^{ϵt/2}$ at leading order. The exactly zero solution remains zero. Detuning competes with this coupling and produces an instability tongue, while Floquet theory gives the exact periodic-coefficient stability criterion. A weak-parameter growth approximation needs a stated time interval and can fail once neglected nonlinearities become important.^[ref]

An instability tongue is a tongue-shaped region in parameter space where a periodically forced linear system has an exponentially growing mode. In the Mathieu equationFloquet theory detects instability through a Floquet multiplier of modulus greater than one. The exactly zero solution remains zero, and initial data on a decaying mode need not grow.

Near the principal resonance $a=1$ with small $q$, multiple-scale analysis gives the leading growth rateIt is real and nonzero inside the leading region $|a-1|<|q|$. The exact boundary has corrections beyond this approximation. The width therefore opens from zero at the resonance point, giving the tongue shape.

Unlike additive resonance, the drive here changes a coefficient of the equation. Detuning $a-1$ competes with that parametric coupling; damping and nonlinear saturation can further change the instability region and long-time behavior.^[ref]

For $y^{\prime \prime }+({\omega }^2+ϵ\cos t)y=0$, use the method of multiple scales with $T=ϵt$ and $y_0=a(T)\cos (t/2)+b(T)\sin (t/2)$. Removing secular terms gives $a_T=(k-1/2)b$ and $b_T=-(k+1/2)a$. The slow eigenvalues satisfy ${\lambda }^2=1/4-k^2$, so an exponentially growing mode exists inside the displayed leading instability interval. At its leading endpoints the slow matrix is nilpotent; higher-order corrections are needed to determine the exact edges. Floquet theory distinguishes this exponential instability from algebraic growth at a repeated multiplier.

For weak quadratic coupling between frequencies one and two, the first oscillator's complex amplitude can obey $2i{A}_T+\alpha A+b\overline{A}=0$ with a constant real second-oscillator amplitude $b$. In real coordinates this is a linear slow system with squared growth rate $(b^2-{\alpha }^2)/4$. Exponential amplification occurs when the modulation amplitude exceeds detuning, $|b|>|\alpha |$. The method of multiple scales is valid over bounded slow time while the physical amplitudes remain in the weak-coupling regime. At equality the slow matrix can have a Jordan block and produce linear growth for some initial phases.

For $u^{\prime \prime }+[1+ϵ\sin t+{ϵ}^{2}f]u=0$, the second parametric-instability tongue is $-1/12<f<5/12$ to leading order. Outside its closure every leading slow amplitude is oscillatory.

For $x^{\prime \prime }+[1+ϵ\cos t+f{ϵ}^2]x=0$, let $T={ϵ}^{2}t$ and $x_0=a(T)\cos t+b(T)\sin t$. First-order forcing produces a constant and second harmonics, not a resonant harmonic. Their feedback at second order gives the displayed solvability condition in the method of multiple scales. The slow squared growth rate is $(f+1/12)(5/12-f)/4$. Consequently the leading instability interval is $-1/12<f<5/12$; outside its closure the slow amplitudes oscillate. The endpoints are degenerate and require the fourth-order edges of the second Mathieu instability tongue for exact small-parameter stability.

For $u^{\prime \prime }+[1+k{ϵ}^2+ϵ\cos t]u=0$, set $t=2s$. The Mathieu equation has $a=4+4k{ϵ}^2$ and $q=-2ϵ$. Near the value $4$, the orthonormal even Fourier series basis $1,\cos 2s,\cos 4s,\cos 6s$ gives diagonal entries $0,4,16,36$ and adjacent couplings $\sqrt{2}q,q,q$. The odd basis $\sin 2s,\sin 4s,\sin 6s$ gives diagonal entries $4,16,36$ and adjacent couplings $q,q$. Higher modes require at least six coupling steps to affect the root near $4$, so these truncations determine its coefficients through $q^4$.

Substituting $a=4+c_2{q}^2+c_4{q}^4$ into the two characteristic polynomials and equating coefficients yieldsConverting back gives the displayed gap edges with remainders $O({ϵ}^4)$ in $k$. In a fixed bounded neighborhood of this parametric resonance, the true unstable gap is $k_{-}<k<k_{+}$. The leading slow-amplitude calculation cannot decide exact stability at $k=-1/12$ or $5/12$; the fourth-order shifts put those fixed values just outside the gap for sufficiently small positive $ϵ$. This does not provide a bound uniform in $ϵ$ for their transient amplification.