A smooth periodically forced one-label Cartesian model with a transverse homoclinic horseshoe: the physical stroboscopic map has a compact invariant set conjugate to the shift, chaotic in the stated sense

Grant 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), inherited from the supplier theorem below.

(i) The model. Fix a length scale ${\mathrm{\ell }}_0>0$, a mass $m>0$, a frequency scale ${\omega }_0>0$, a dimensionless amplitude $a\ge 100$, and a pulse width $\delta >0$. Let $w$ be the normalized smooth bump and ${\beta }_{\delta }$ the smooth $P$-periodic pulse of The smooth $P$-periodic pulsed oscillator: the normalized pulse, its Hamiltonian and its stroboscopic (period) map with $P:=\pi \mathrm{/}2+\delta$, ${\beta }_{\delta }=0$ on $[0,\pi \mathrm{/}2]$ and ${\beta }_{\delta }(t)={\delta }^{-1}w((t-\pi \mathrm{/}2)\mathrm{/}\delta )$ on $[\pi \mathrm{/}2,P]$. Use full Cartesian position $\mathbf{r}=(Q,Y,Z)\in Q_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}:={\mathbb{R}}^3$ and velocity $\mathbf{v}=\dot{\mathbf{r}}\in {\mathbb{R}}^3$ for one label of mass $m$. The adopted force map on $\mathbb{R}\times Q_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}\times {\mathbb{R}}^3$ is $\mathbf{F}(T,\mathbf{r},\mathbf{v}):=-m{\omega }_0^2(Q+{\mathrm{\ell }}_0{\beta }_{\delta }({\omega }_{0}T)(a-\frac{Q^2}{{\mathrm{\ell }}_0^2}),-Y,-Z),T_p:=\frac{P}{{\omega }_0}\mathrm{.}$ It is smooth, defined on the full product domain and $T_p$-periodic, so it is a force model of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance with fixed mass $m$; its motions are the $C^2$ curves of The associated first-order system, motions, maximal motions and the Newtonian flow of a force model. The stroboscopic map and Euclidean phase chart below use the Newtonian state coordinates $(\mathbf{r},\mathbf{v})$ of that definition. The phase plane $\mathcal{P}:=\{(\mathbf{r},\mathbf{v}):Y=Z=v_Y=v_Z=0\}$ is preserved for as long as each solution exists: zero transverse position and velocity remain zero by uniqueness of their homogeneous transverse equations. This does not assert completeness of every axial solution. Define canonical momenta $\mathbf{\Pi }:=m\mathbf{v}=({\mathrm{\Pi }}_Q,{\mathrm{\Pi }}_Y,{\mathrm{\Pi }}_Z)$, with axial component $\mathrm{\Pi }:={\mathrm{\Pi }}_Q$. The restricted equation is $m\ddot{Q}=-m{\omega }_0^2[Q+{\mathrm{\ell }}_0{\beta }_{\delta }({\omega }_{0}T)(a-\frac{Q^2}{{\mathrm{\ell }}_0^2}\left)\right]\mathrm{.}$ With $q=Q\mathrm{/}{\mathrm{\ell }}_0$, $s={\omega }_{0}T$ and $p=\dot{Q}\mathrm{/}({\mathrm{\ell }}_0{\omega }_0)$ this is exactly the dimensionless equation of the mathematical supplier. On the axis $\mathrm{\Pi }=m\dot{Q}$. In the full dimensionless position-velocity chart write $\tilde{\mathbf{r}}:=\mathbf{r}\mathrm{/}{\mathrm{\ell }}_0$ and $\tilde{\mathbf{v}}:=\mathbf{v}\mathrm{/}({\mathrm{\ell }}_0{\omega }_0)$; the state scaling $(\tilde{\mathbf{r}},\tilde{\mathbf{v}})\mapsto ({\mathrm{\ell }}_0\tilde{\mathbf{r}},{\mathrm{\ell }}_0{\omega }_0\tilde{\mathbf{v}})$ identifies this chart with the physical Newtonian state chart, and ${\iota }_{\delta }$ is its restriction to the invariant axis. The full stroboscopic map at period $T_p$ is ${\mathrm{\Psi }}_{\delta }$ of The stroboscopic map of a periodic model and the Poincare return map of a transversal section: domains, iterates, fixed points and the local C1 diffeomorphism property clause (i).

(ii) The horseshoe. There is ${\delta }_0>0$, depending only on $a$ and the bump $w$, such that for every $0<\delta <{\delta }_0$ there is a homeomorphism $H_{\delta }$ of the two-sign shift space $(X_{\pm },D_{\pm })$ of The two-sign bilateral shift space, its coding map, and the ball of bounded bilateral real sequences onto a compact subset ${\mathrm{\Lambda }}_{\delta }^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\subset \mathcal{P}$ such that ${\mathrm{\Psi }}_{\delta }\circ H_{\delta }=H_{\delta }\circ S_{\pm }\mathrm{.}$ Thus ${\mathrm{\Lambda }}_{\delta }^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ is compact and ${\mathrm{\Psi }}_{\delta }$-invariant, and the restriction ${\mathrm{\Psi }}_{\delta }{\mathrm{\mid }}_{{\mathrm{\Lambda }}_{\delta }^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}$ is conjugate to the shift; with the Euclidean metric inherited from the full phase chart it is chaotic in the stated sense of Chaos in the stated sense for maps of the model: topological mixing, dense periodic points and sensitive dependence, as one precise list among several: continuous, topologically mixing, with dense periodic points and sensitive dependence with a positive constant.

For each such $\delta$, the full stroboscopic map has open neighborhoods ${\mathcal{U}}_{\delta },{\mathcal{V}}_{\delta }\subseteq {\mathbb{R}}^6$ with ${\mathrm{\Lambda }}_{\delta }^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\subseteq {\mathcal{U}}_{\delta }\cap {\mathcal{V}}_{\delta }$ such that ${\mathrm{\Psi }}_{\delta }{\mathrm{\mid }}_{{\mathcal{U}}_{\delta }}:{\mathcal{U}}_{\delta }\to {\mathcal{V}}_{\delta }$ is a $C^1$ diffeomorphism. Use this branch and its inverse in clause (iii).

(iii) Hyperbolic points and transverse homoclinic points. The two constant codes give hyperbolic fixed points of the full map ${\mathrm{\Psi }}_{\delta }$, one of each sign, with hyperbolicity as defined in Stability vocabulary of the model: Lyapunov, asymptotic and orbital stability, linear stability of equilibria and map fixed points, and hyperbolicity of equilibria and fixed points clause (v); write $z_{\bar{u}}:=H_{\delta }(\bar{u})$ for the fixed point coded by the constant sequence $\bar{u}$. Define the stable and unstable sets for the stated branch by $W^s(z_{\ast }):=\{z\in {\mathcal{U}}_{\delta }:{\mathrm{\Psi }}_{\delta }^n(z)\in {\mathcal{U}}_{\delta }\text{ for every }n\ge 0\text{ and }{\mathrm{\Psi }}_{\delta }^n(z)\to z_{\ast }\},$ $W^u(z_{\ast }):=\{z\in {\mathcal{V}}_{\delta }:{\mathrm{\Psi }}_{\delta }^{-n}(z)\in {\mathcal{V}}_{\delta }\text{ for every }n\ge 0\text{ and }{\mathrm{\Psi }}_{\delta }^{-n}(z)\to z_{\ast }\},$ where forward powers are recursively defined using the restriction ${\mathrm{\Psi }}_{\delta }{\mathrm{\mid }}_{{\mathcal{U}}_{\delta }}$, backward powers use its inverse ${\mathrm{\Psi }}_{\delta }^{-1}:{\mathcal{V}}_{\delta }\to {\mathcal{U}}_{\delta }$, and convergence uses the Euclidean metric of the phase chart. ODE uniqueness ensures that inverse values from overlapping local flow neighborhoods agree. A transverse homoclinic point here means a point $z_{\ast }\mathrm{\ne }{z}_{\bar{u}}$ in both sets for which there are injective $C^1$ maps of open neighborhoods of $0$ in ${\mathbb{R}}^3$ into $W^s(z_{\bar{u}})$ and $W^u(z_{\bar{u}})$, respectively, each with rank-$3$ differential and a continuous inverse on its image. Their images pass through $z_{\ast }$, and their three-dimensional tangent spaces sum to ${\mathbb{R}}^6$. The planar stable and unstable curves used in this product construction are supplied by Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family.

For every nonconstant code $\sigma \in X_{\pm }$ for which there are a sign $u$ and $N\ge 0$ with ${\sigma }_n=u$ whenever $\mathrm{\mid }n\mathrm{\mid }\ge N$, the point $H_{\delta }(\sigma )$ is a transverse homoclinic point of the corresponding fixed point for the full map; distinct such codes give distinct points.

(iv) Explicit bounds. Write $R:=\sqrt{a}+3$. Every state in ${\mathrm{\Lambda }}_{\delta }^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ has $\mathbf{r}=(Q,0,0)$ and $\mathbf{\Pi }=(\mathrm{\Pi },0,0)$, with $\mid Q\mid \le R{\mathrm{\ell }}_0,\mid \mathrm{\Pi }\mid \le m{\mathrm{\ell }}_0{\omega }_{0}R\mathrm{.}$ Every coded trajectory exists for all times and remains in these bounds at the stroboscopic times. For $a=100$ the bounds are $\mid Q\mid \le 13{\mathrm{\ell }}_0$ and $\mid \mathrm{\Pi }\mid \le 13m{\mathrm{\ell }}_0{\omega }_0$.

(v) Scope. The pulsed axial and transverse inverted-oscillator forces are adopted model assumptions, not a reported drive or apparatus; no measurement, calibration or uncertainty is involved. The result is about this one full three-dimensional model: it does not assert chaos for every driven pendulum, for separatrix splitting in gravitational or vortex models, for Hénon-Heiles energies, or for any numerical portrait, and it does not validate a first-order Melnikov calculation by analogy (Chaos in the stated sense for maps of the model: topological mixing, dense periodic points and sensitive dependence, as one precise list among several clause (d)).

• Why this is a chaos proof and not a picture. The conjugacy to the shift, the mixing, the density of periodic points, the sensitivity and the axial homoclinic curves come from the supplier theorem's explicit constructions (a contraction on a complete sequence space, invariant cones with expansion, and weighted graph-transform curves), not from a plotted tangle. The full result then uses the explicit invariant-axis embedding and the product of the two transverse hyperbolic modes. See Chaos in the stated sense for maps of the model: topological mixing, dense periodic points and sensitive dependence, as one precise list among several clause (d).

• Choice assumptions. The countable and dependent choice assumptions are inherited from the supplier's compactness and homeomorphism clauses and are declared in the Statement; no other choice is used, since all constructions are explicit.

• Units. $[F]=\mu \mathrm{\ell }{\tau }^{-2}=$ N, $[\mathrm{\Pi }]=\mu \mathrm{\ell }{\tau }^{-1}$, $[{\mathrm{\ell }}_0]=[R{\mathrm{\ell }}_0]=\mathrm{\ell }$ and $[m{\mathrm{\ell }}_0{\omega }_{0}R]=\mu \mathrm{\ell }{\tau }^{-1}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework); the pulse duration is $\delta \mathrm{/}{\omega }_0$ and the forcing period $T_p=P\mathrm{/}{\omega }_0$, both times.