Relaxation oscillation

A relaxation oscillation alternates slow motion along attracting branches of a critical manifold with fast jumps near its folds. It is characteristic of a fast-slow system with widely separated time scales.

For $f(y)=y-y^2+\gamma y^3/3$ with $0<\gamma <1$, the fold points of a critical manifold are $y_{\pm }=(1\pm \sqrt{1-\gamma })/\gamma$. When $\delta \ll ϵ\ll 1$, the fastest variable satisfies $z\simeq f(y)$ even during jumps of the intermediate variable. At fixed $x$, the outer branches of $x=f(y)$ attract the fast $y$ dynamics because $f^{\prime }(y)>0$; its middle branch repels. If $y_{-}<\alpha <y_{+}$, the slow equation $\dot{x}=\alpha -y$ drives the low branch to its fold point of a critical manifold and the high branch back to the other fold point of a critical manifold. Fast jumps at almost fixed $x$ close a relaxation cycle. This explains why a reduction justified on the slow arcs must also be checked on the jump time scale. For $\gamma >1$, the cubic is strictly increasing; the reduced energy $V=[x-f(\alpha ){]}^2/2+ϵ(y-\alpha {)}^2/2$ has $\dot{V}=-(y-\alpha )[f(y)-f(\alpha )]\le 0$, precluding this oscillation mechanism.

For $ϵ\dot{x}=x-x^3/3+y$, $\dot{y}=-x$ and $ϵ\to 0^{+}$, the attracting branches of the cubic nullcline $y=x^3/3-x$ have $|x|>1$. The limiting cycle follows $x=2$ to $x=1$, jumps to $x=-2$ at fixed $y=-2/3$, follows the other outer branch to $x=-1$, and jumps back at $y=2/3$. On a slow branch, $(x^2-1)\dot{x}=-x$, so one branch takes ${\int }_2^1(1/x-x)dx=3/2-\log 2$. Reflection supplies the other half; fast-jump and fold times vanish at leading order. This proves the displayed asymptotic period. Negative $ϵ$ reverses fast stability and does not preserve this attracting cycle.

For $ϵ\dot{v}=g(v)-w$, $\dot{w}=v$, let $g=-v$ below $\alpha$, $g=1-v$ above $\beta$, and join the endpoints by an increasing line, with $-1<\alpha <0<\beta <1$ and $\beta -\alpha <1$. The slow branches obey $\dot{w}=-w$ and $\dot{w}=1-w$ between $w_L=-\alpha$ and $w_R=1-\beta$. Their travel times are $\log [(1-\beta )/(-\alpha )]$ and $\log [(1+\alpha )/\beta ]$. Fast horizontal jumps contribute vanishing time as $ϵ\to 0$, giving the displayed relaxation oscillation period.