The smooth $P$-periodic pulsed oscillator: the normalized pulse, its Hamiltonian and its stroboscopic (period) map

The normalized pulse. By Test function cutoffs and euclidean localization, applied to $K=\{1\mathrm{/}2\}\subset (0,1)$, choose $\chi \in C_c^{\mathrm{\infty }}((0,1))$ with $0\le \chi \le 1$ and $\chi =1$ on a neighborhood of $1\mathrm{/}2$. Thus there are $0<\alpha <\beta <1$ with $\chi =1$ on $[\alpha ,\beta ]$. Before taking any integral, extend $\chi$ by zero to $\mathbb{R}$, retaining the notation $\chi$. Its compact support inside $(0,1)$ makes this extension identically zero near both endpoints, hence smooth on $\mathbb{R}$. The continuous extended function $\chi$ is integrable on $[0,1]$; by additivity and monotonicity of the integral and the integral of a constant,

$I:={\int }_0^1\chi (s)ds\ge {\int }_{\alpha }^{\beta }\chi (s)ds=\beta -\alpha >0,$

using integrability of continuous functions (A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion), additivity and monotonicity (For $a<c<b$: $f$ is integrable on $[a,b]$ if and only if it is integrable on $[a,c]$ and on $[c,b]$, and then ${\int }_a^{b}f={\int }_a^{c}f+{\int }_c^{b}f$; with the oriented form for arbitrary $a,b,c$, If $f\le g$ on $[a,b]$ and both are integrable then ${\int }_a^{b}f\le {\int }_a^{b}g$; and $m(b-a)\le {\int }_a^{b}f\le M(b-a)$), and the integral of a constant (If $m\le f\le M$ on $[a,b]$ then $m(b-a)\le L(f,P)\le \underset{‾}{{\int }_a^b}f\le \overline{{\int }_a^b}f\le U(f,P)\le M(b-a)$ for every partition $P$; in particular every constant function is integrable, with ${\int }_a^{b}c=c(b-a)$).

Define $w_0:=\chi \mathrm{/}I$ on $(0,1)$ and let $w:\mathbb{R}\to \mathbb{R}$ be its extension by zero outside $(0,1)$. Since $\chi$ has compact support in $(0,1)$, this zero extension is smooth and compactly supported in $(0,1)$. Linearity of the integral (Integrable functions on $[a,b]$ form a set closed under sums and scalar multiples, and ${\int }_a^b(\lambda f+\mu g)=\lambda {\int }_a^{b}f+\mu {\int }_a^{b}g$) gives

$w\in C_c^{\mathrm{\infty }}((0,1)),w\ge 0,{\int }_0^{1}w(s)ds=1.$

Fix such a pulse $w$ for the constructions below. The resulting coefficient and endpoint maps may depend on this choice; the later estimates apply to every such $w$, with constants allowed to depend on it.

The periodic pulse. Fix a real parameter $a\ge 100$ and a smallness parameter $\delta >0$, and put $P:=\pi \mathrm{/}2+\delta$. Define ${\beta }_{\delta }:\mathbb{R}\to \mathbb{R}$ to be the $P$-periodic function whose restriction to $[0,P]$ is

${\beta }_{\delta }(t):=0(t\in [0,\frac{\pi }{2}\left]\right),{\beta }_{\delta }(t):=\frac{1}{\delta }w\left(\frac{t-\pi \mathrm{/}2}{\delta }\right)(t\in [\frac{\pi }{2},P\left]\right)\mathrm{.}$

Both definitions use the same zero extension $w:\mathbb{R}\to \mathbb{R}$, so the endpoint values $w(0)$ and $w(1)$ are defined and equal to $0$. Since $w$ vanishes on neighborhoods of $0$ and $1$, all its derivatives vanish there; the $P$-periodic extension of ${\beta }_{\delta }$ is therefore of class $C^{\mathrm{\infty }}$ on $\mathbb{R}$ ($C^k$ Euclidean maps and diffeomorphisms). Also ${\beta }_{\delta }\ge 0$. To verify its normalization, put

$B_{\delta }(t):=\{\begin{array}{ll}0,& 0\le t\le \pi \mathrm{/}2,\\ {\displaystyle {\int }_0^{(t-\pi \mathrm{/}2)\mathrm{/}\delta }w(s)ds,}& \pi \mathrm{/}2\le t\le P\mathrm{.}\end{array}$

The integral function of the continuous $w$ has derivative $w$ by The first fundamental theorem: if $f$ is integrable on $[a,b]$ and continuous at $c$, then $F^{\mathrm{\prime }}(c)=f(c)$; in particular a continuous $f$ has $F$ as a primitive and A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, and the chain rule (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$) gives $B_{\delta }^{\mathrm{\prime }}(t)={\beta }_{\delta }(t)$ on the pulse interval. Since $w$ vanishes near its endpoints, $B_{\delta }$ is differentiable across $t=\pi \mathrm{/}2$ as well. Since ${\beta }_{\delta }$ is continuous and therefore integrable on $[0,P]$, the second part of the fundamental theorem (The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$) gives

${\int }_0^P{\beta }_{\delta }(t)dt=B_{\delta }(P)-B_{\delta }(0)={\int }_0^{1}w(s)ds=1.$

The Hamiltonian and its equations. Use the phase domain $U={\mathbb{R}}^2$ with coordinates $(q,p)$ and standard symplectic form $\omega =dq\wedge dp$ and matrix $J=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, as in A phase domain, its canonical one-form and its standard symplectic two-form. For each fixed time $t$, define the smooth Hamiltonian function

${\mathcal{H}}_{\delta }:\mathbb{R}\times {\mathbb{R}}^2⟶\mathbb{R},{\mathcal{H}}_{\delta }(t,q,p):=\frac{p^2+q^2}{2}+{\beta }_{\delta }(t)(aq-\frac{q^3}{3})\mathrm{.}$

The Hamiltonian-vector-field convention is ${\iota }_{X_H}\omega =dH$, for which The Hamiltonian field of a $C^2$ function on a phase domain is $J\mathrm{\nabla }H$, and the Poisson bracket is ${\sum }_i({\mathrm{\partial }}_{q^i}f{\mathrm{\partial }}_{p_i}g-{\mathrm{\partial }}_{p_i}f{\mathrm{\partial }}_{q^i}g)$ gives $X_H=J\mathrm{\nabla }H=({\mathrm{\partial }}_{p}H,-{\mathrm{\partial }}_{q}H)$. Here ${\mathrm{\partial }}_p{\mathcal{H}}_{\delta }=p$ and ${\mathrm{\partial }}_q{\mathcal{H}}_{\delta }=q+{\beta }_{\delta }(t)(a-q^2)$, so the associated time-dependent first-order system is

$q^{\mathrm{\prime }}=p,p^{\mathrm{\prime }}=-q+{\beta }_{\delta }(t)(q^2-a),$

equivalently the second-order equation $q^{\mathrm{\prime }\mathrm{\prime }}=-q+{\beta }_{\delta }(t)(q^2-a)$; the right-hand side is $C^{\mathrm{\infty }}$ in $(t,q,p)$, hence continuous and locally Lipschitz in the state variable, so every initial condition has a unique maximal solution (First-order systems, initial value problems, and solutions on intervals, Extensions, maximal solutions, maximal intervals, and global solutions, Every Picard–Lindelöf initial value problem has one maximal solution on an open interval). The system is $P$-periodic in $t$.

Free rotation and pulse endpoint maps. Over the first subinterval $[0,\pi \mathrm{/}2]$ the pulse vanishes and the system is the free oscillator; define the free rotation

$R_0:{\mathbb{R}}^2\to {\mathbb{R}}^2,R_0(q,p):=(p,-q),$

which is the time-$\pi \mathrm{/}2$ solution operator of $q^{\mathrm{\prime }}=p$, $p^{\mathrm{\prime }}=-q$: the map $t\mapsto (q_0\cos t+p_0\sin t,-q_0\sin t+p_0\cos t)$ solves the free system by the derivative identities of sine and cosine (The derivatives of sine and cosine are cosine and minus sine) and has initial data $(q_0,p_0)$. Moreover $\cos (\pi \mathrm{/}2)=0$ by the zero-set statement (The zero sets of sine and cosine and the least positive common period 2 pi), $\sin (\pi \mathrm{/}2)>0$ by positivity on $(0,\pi )$ (Pi is the first positive zero of sine), and ${\sin }^2(\pi \mathrm{/}2)+{\cos }^2(\pi \mathrm{/}2)=1$ (Parity and the Pythagorean identity for sine and cosine), so $\sin (\pi \mathrm{/}2)=1$. Uniqueness of solutions then identifies the endpoint map with $R_0$.

For $s\in [0,1]$ the rescaled pulse equation is

$q_s=\delta p,p_s=-\delta q+w(s)(q^2-a),$

obtained from the second half of the period by the substitution $s=(t-\pi \mathrm{/}2)\mathrm{/}\delta$. For each initial state $z=(q,p)\in {\mathbb{R}}^2$, let $J_z^{\delta }$ be the open maximal interval containing $0$ of its solution to the rescaled equation. Define

$D_{\delta }:=\{z\in {\mathbb{R}}^2:[0,1]\subseteq J_z^{\delta }\},K_{\delta }:D_{\delta }\to {\mathbb{R}}^2,K_{\delta }(z):=(q(1),p(1)),$

where $(q(s),p(s))$ is the unique maximal solution with $(q,p)(0)=z$ (Every Picard–Lindelöf initial value problem has one maximal solution on an open interval). For the original $P$-periodic system, let $J_z$ be the maximal existence interval of the solution through $z$ at time $0$, and set $D_P:=\{z\in {\mathbb{R}}^2:[0,P]\subseteq J_z\}$. Define the stroboscopic map by

$G_{\delta }:D_{G_{\delta }}\to {\mathbb{R}}^2,D_{G_{\delta }}:=\{z\in {\mathbb{R}}^2:R_0(z)\in D_{\delta }\},G_{\delta }(z):=K_{\delta }(R_0(z))\mathrm{.}$

The change of time variable $s=(t-\pi \mathrm{/}2)\mathrm{/}\delta$ identifies solutions on $[\pi \mathrm{/}2,P]$ with solutions of the rescaled equation on $[0,1]$. Concatenating with the unique free solution on $[0,\pi \mathrm{/}2]$ shows $D_{G_{\delta }}=D_P$ and $G_{\delta }={\theta }_P$ there, where ${\theta }_P:D_P\to {\mathbb{R}}^2$ is the stroboscopic map of The stroboscopic (time-$P$) map of a $P$-periodic first-order system. Conversely, restricting any solution through time $P$ to the pulse interval and applying the same change of variables gives a solution of the rescaled equation through $s=1$; uniqueness identifies both maps.

• The phase variables $q,p$ are dimensionless in this definition; no units, masses or measured quantities occur. The parameter $a$ and the pulse data are pure numbers, and the entire construction is a mathematical model of a periodically forced one-degree-of-freedom oscillator.

• The pulse has width $\delta$ and total strength $\int {\beta }_{\delta }=1$, so ${\beta }_{\delta }$ converges to a formal unit impulse as $\delta \to 0$; no meaning is assigned to the limit $\delta =0$ as a physical force. The $\delta =0$ endpoint map is computed on the companion page cm-math-chaos-examples by explicit integration and used only as a comparison object.

• Nothing in this definition asserts that $G_{\delta }$ is smooth, invertible, symplectic, close to $F$, or defined on any particular set; these are the contents of The rescaled pulse endpoint map converges to the kick $K(q,p)=(q,p+q^2-a)$ in $C^1$, and the period map is smooth, symplectic and $C^1$-close to $F$ and A smooth periodic one-degree-of-freedom oscillator with polynomial pulse forcing has stroboscopic chaos and a transverse homoclinic point.