Cm Math Chaos — Examples

Take $a=100$ in The square-root recurrence $T_s(x{)}_n=s_n\sqrt{a+x_{n-1}+x_{n+1}}$ is a contraction of $B_R$ with a unique coded solution, so that

$R=\sqrt{100}+3=13,L=\sqrt{a-2R}=\sqrt{74},\sqrt{a+2R}=\sqrt{126}\mathrm{.}$

Then $126<169={13}^2$, so $\sqrt{126}<R=13$; $74>1$, so $L>1$; and the contraction constant of $T_s$ on $B_{13}$ is

$\frac{1}{L}=\frac{1}{\sqrt{74}}<1.$

The constant codes $\bar{u}$ ($u=\pm 1$) are solved by the constant sequences $x_n=r_u$ with

$11<r_{+}=1+\sqrt{101}<12,-10<r_{-}=1-\sqrt{101}<-9,$

the two displayed bounds hold because $100<101<121$; and both satisfy the exact identity $r_u^2=100+2r_u$. Every coded sequence has $\mathrm{\mid }{x}_n\mathrm{\mid }\ge \sqrt{74}$, and its sign is the corresponding symbol.

• All verifications are exact algebraic identities involving $\sqrt{101}$, $\sqrt{74}$ and $\sqrt{126}$, with rational comparisons for the inequalities; no numerical integration, simulation or measurement is involved.

Take $a=100$, so $L=\sqrt{74}$, $k=1\mathrm{/}2$, and let $r_{+}=1+\sqrt{101}$ be the constant-code value of the all-$+1$ code. The fixed slope of the forward transform ${\mathrm{\Phi }}_y(h)=1\mathrm{/}(2y-h)$ at $y=r_{+}$ is the small root of $h=1\mathrm{/}(2r_{+}-h)$, i.e. of

$h^2-2r_{+}h+1=0:h=r_{+}-\sqrt{r_{+}^2-1}=\frac{1}{r_{+}+\sqrt{r_{+}^2-1}},$

and one has $0<h<1\mathrm{/}2$. Moreover $\lambda =2L-k=2\sqrt{74}-\frac{1}{2}>1$ and ${\mathrm{\Phi }}_{r_{+}}$ maps $[-1\mathrm{/}2,1\mathrm{/}2]$ into $[-{\lambda }^{-1},{\lambda }^{-1}]\subseteq [-1\mathrm{/}2,1\mathrm{/}2]$ with Lipschitz constant ${\lambda }^{-2}<1\mathrm{/}2$.

• The fixed slope $h$ is the slope of the invariant unstable line at the fixed point in the cone $C_{1\mathrm{/}2}^u$; the computation uses only the exact fixed-point relation $r_{+}^2=100+2r_{+}$ and rational estimates for the two square roots.

Take $a=100$ and let $s\in X_{\pm }$ be the code with

$s_0=-1,s_n=+1(n\mathrm{\ne }0),$

the single-flip code. Then $s$ is constant outside a finite set (indeed outside $\{0\}$), is not the constant code $\overline{+1}$, and is not periodic: $S_{\pm }^{m}s\mathrm{\ne }s$ for every $m\ge 1$, because the unique coordinate at which $S_{\pm }^{m}s$ equals $-1$ is $-m\mathrm{\ne }0$ whereas $s$ equals $-1$ only at $0$. Consequently the coded point $z:=h(s)$ satisfies $z\mathrm{\ne }(r_{+},r_{+})$ and $F^m(z)\mathrm{\ne }z$ for every $m\ge 1$, while

$F^n(z)⟶(r_{+},r_{+})(n\to +\mathrm{\infty }\text{ and }n\to -\mathrm{\infty })\mathrm{.}$

• The verification uses only the coding dictionary and the finite-word convergence; it does not compute the coordinates of $h(s)$, which have no closed form.

• The same argument applies to every nonconstant code that is constant outside a finite set: none of these coded orbits is periodic, and each converges to the corresponding fixed point in both time directions.

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

1. the free oscillator $q^{\mathrm{\prime }}=p$, $p^{\mathrm{\prime }}=-q$ on the interval $[0,\pi \mathrm{/}2]$ carries the initial data $(q_0,p_0)$ to

$(q,p)\left(\frac{\pi }{2}\right)=(q_0\cos \frac{\pi }{2}+p_0\sin \frac{\pi }{2},-q_0\sin \frac{\pi }{2}+p_0\cos \frac{\pi }{2})=(p_0,-q_0)=R_0(q_0,p_0);$

2. the $\delta =0$ rescaled equation $q_s=0$, $p_s=w(s)(q^2-a)$ has the exact solution $q(s)=q_0$, $p(s)=p_0+(q_0^2-a){\int }_0^{s}w$, so its endpoint map is

$K(q_0,p_0)=(q_0,p_0+q_0^2-a)$

because ${\int }_0^{1}w=1$; and

$(K\circ R_0)(q_0,p_0)=K(p_0,-q_0)=(p_0,-q_0+p_0^2-a)=F(q_0,p_0)\mathrm{.}$

Moreover ${\beta }_{\delta }\ge 0$ with ${\int }_0^P{\beta }_{\delta }=1$ and the restriction of the pulse to $[0,P]$ has support contained in $(\pi \mathrm{/}2,P)$. On $\mathbb{R}$ its support is the periodic union ${\bigcup }_{j\in \mathbb{Z}}(\pi \mathrm{/}2+\delta \mathrm{supp}w+jP)$.

• The $\delta =0$ computation is a formal comparison object: the endpoint map $K$ is the endpoint map of the limiting pulse of unit total strength concentrated in zero time. The genuine model has $\delta >0$, and the comparison is quantified in 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 $F$.

Take $a=100$ in Root bounds and $C^1$ dependence for the perturbed scalar recurrence $v^2-a-u-w+e(u,v,w)=0$, so that $R=13$, $l=7$, and suppose the defect bound is $\eta =1$. Then:

1. for all $u,w\in [-13,13]$ the unperturbed quadratic $v^2-100-u-w$ satisfies

$v^2-100-u-w{\mid }_{v=7}\le 49-100+26=-25,v^2-100-u-w{\mid }_{v=13}\ge 169-100-26=43,$

so the signs at the endpoints are strictly opposite with margins $25$ and $43$; 2. the derivative bound is $d=(1+\eta )\mathrm{/}(2l-\eta )=(1+1)\mathrm{/}(14-1)=2\mathrm{/}13$, and

$2d=\frac{4}{13}<1,d(\frac{1}{2}+2)=\frac{5}{13}<1,\frac{d}{1-d\mathrm{/}2}=\frac{2\mathrm{/}13}{12\mathrm{/}13}=\frac{1}{6}<\frac{2}{3};$

3. the root margins are at least $24\mathrm{/}27>3\mathrm{/}4$ and $42\mathrm{/}27>3\mathrm{/}4$, since the slope of $P_{u,w}$ is bounded by $2R+\eta =27$ on $[7,13]$.

• Every number is evaluated symbolically at the worst case $a=100$ and $\eta =1$; larger $a$ or smaller $\eta$ only improve the margins.