A smooth periodic one-degree-of-freedom oscillator with polynomial pulse forcing has stroboscopic chaos and a transverse homoclinic point

Given: The parameter $a\ge 100$, the fixed normalized pulse $w$, and for each $\delta >0$ the periodic coefficient ${\beta }_{\delta }$, the Hamiltonian $H_{\delta }$, the rescaled pulse equation, the rotation $R_0$, the endpoint map $K_{\delta }$ and the stroboscopic map $G_{\delta }$ of The smooth $P$-periodic pulsed oscillator: the normalized pulse, its Hamiltonian and its stroboscopic (period) map.

Structure of the model (The smooth $P$-periodic pulsed oscillator: the normalized pulse, its Hamiltonian and its stroboscopic (period) map): ${\beta }_{\delta }$ is $C^{\mathrm{\infty }}$ and $P$-periodic with ${\beta }_{\delta }\ge 0$ and ${\int }_0^P{\beta }_{\delta }=1$; the associated system is $q^{\mathrm{\prime }}=p$, $p^{\mathrm{\prime }}=-q+{\beta }_{\delta }(t)(q^2-a)$, smooth and locally Lipschitz in the state; and $G_{\delta }=K_{\delta }\circ R_0$ is its time-$P$ map.

Pulse approximation (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 $F$): there are ${\delta }_1>0$ and $C_1,C_2<\mathrm{\infty }$ depending only on $a$ and $w$ such that for $0<\delta \le {\delta }_1$ the maps $K_{\delta }$ and $G_{\delta }=K_{\delta }\circ R_0$ are defined and $C^{\mathrm{\infty }}$ on a neighbourhood of $S=[-R-1,R+1{]}^2$, $R=\sqrt{a}+3$, have ${\sup }_S(\mathrm{\mid }{G}_{\delta }-F\mathrm{\mid }+\mathrm{\parallel }D{G}_{\delta }-DF\mathrm{\parallel })\le C_1\delta$ with $F(q,p)=(p,p^2-a-q)$, $G_{\delta }$ is a symplectic $C^{\mathrm{\infty }}$ diffeomorphism onto its image, and the conjugate map $H_{\delta }=S_w\circ G_{\delta }^{-1}\circ S_w$, $S_w(q,p)=(p,q)$, satisfies ${\sup }_S(\mathrm{\mid }{H}_{\delta }-F\mathrm{\mid }+\mathrm{\parallel }D{H}_{\delta }-DF\mathrm{\parallel })\le C_2\delta$ and is likewise $C^{\mathrm{\infty }}$ and diffeomorphic onto its image near $S$; moreover both $G_{\delta }$ and $H_{\delta }$ satisfy each of the six componentwise closeness inequalities displayed in the statement of Explicit persistence under the stated forward and conjugated-inverse bounds: coded chaos, hyperbolic fixed points and transverse homoclinic points on $S$ with the common constant $\bar{C}\delta$, where $\bar{C}:=\max (1,C_1,C_2)$.

Under the Axioms of Countable and Dependent Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain), the perturbed horseshoe (Explicit persistence under the stated forward and conjugated-inverse bounds: coded chaos, hyperbolic fixed points and transverse homoclinic points): for every $\epsilon \le 1\mathrm{/}6$ with $C\epsilon \le 1$ and every $C^2$ diffeomorphism $G$ onto its image on a neighbourhood of $S$ satisfying the six closeness inequalities with constant $\epsilon$ on $S$, and whose conjugate $S_w\circ G^{-1}\circ S_w$ does the same, the conclusions of clauses 1-4 above hold for $G$ under these same axioms with sensitivity constant $2(\sqrt{a}-3)$ and with all codes constant outside finite sets.

A homeomorphism here means a bijection $h_{\delta }$ whose forward and inverse maps are continuous in the stated metrics (Continuity of a map between metric spaces, at a point and globally, in the $\epsilon$-$\delta$ form); conjugacy means such a homeomorphism satisfying $G_{\delta }\circ h_{\delta }=h_{\delta }\circ S_{\pm }$. Compactness means that every open cover of the metric subspace has a finite subcover (Open cover, subcover, compact metric space, and compact subset of a metric space). Chaos has the four properties in Topological mixing, dense periodic points, sensitive dependence, and the stated chaos list. Hyperbolicity means that every eigenvalue of the fixed-point differential has modulus different from $1$; a transverse homoclinic point is a point different from the fixed point lying in both invariant sets, with injective $C^1$ curves in those sets through it and nonzero linearly independent tangents (Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family, using metric convergence Convergence of a sequence in a metric space: $x_k\to x$ iff $d(x_k,x)\to 0$ in $\mathbb{R}$).