Computer Assisted Proof of Transverse Saddle-to-Saddle Connecting Orbits for First Order Vector Fields

Abstract

In this paper we introduce a computational method for proving the existence of generic saddle-to-saddle connections between equilibria of first order vector fields. The first step consists of rigorously computing high order parametrizations of the local stable and unstable manifolds. If the local manifolds intersect, the Newton–Kantorovich theorem is applied to validate the existence of a so-called short connecting orbit. If the local manifolds do not intersect, a boundary value problem with boundary values in the local manifolds is rigorously solved by a contraction mapping argument on a ball centered at the numerical solution, yielding the existence of a so-called long connecting orbit. In both cases our argument yields transversality of the corresponding intersection of the manifolds. The method is applied to the Lorenz equations, where a study of a pitchfork bifurcation with saddle-to-saddle stability is done and where several proofs of existence of short and long connections are obtained.

Appendices

Appendices

1.1 Appendix 1: Parameterization Method for Lorenz

Consider $\dot{u} = g(u)$ given by the Lorenz equations (1.5). Let $p$ denote one of the fixed points, $\lambda _1$ and $\lambda _2$ denote two eigenvalues of $Dg(p)$ with similar stability (either both stable or both unstable), and $a_1, a_2$ be two associated eigenvectors. In the above considered setting $\lambda _{1} = \bar{\lambda }_{2}$ and $a_{1,2}$ are complex eigenvectors. Let $P$ denote the parameterization of the invariant manifold (whether stable or unstable) and $\Lambda \in \mathbb {C}^{2,2}$ denote the matrix with $\lambda _1$ and $\lambda _2$ as diagonal entries. Then in this case the power series is

$$\begin{aligned} P(\theta ) = f(\theta _{1}+i\theta _{2},\theta _{1}-i\theta _{2})= \sum _{n_1=0}^{\infty } \sum _{n_2=0}^{\infty } p_{(n_1, n_2)} (\theta _1+i\theta _{2})^{n_1} (\theta _{1}-i\theta _2)^{n_2}, \end{aligned}$$

with $p_{(n_1, n_2)} \in \mathbb {C}^3$ for each $n_1, n_2 \ge 0$ and $f(z) = f(z_{1},z_{2}):\mathbb {C}^{2}\rightarrow \mathbb {C}^{3}$. The linear constraints give that $p_{(0,0)} = p$, $p_{(0,1)} = a_1$, and $p_{(1,0)} = a_2$. The coefficients for $n_1 + n_2 \ge 2$ are worked out by considering the functional equation

$$\begin{aligned} \left[ \begin{array}{c} \sigma (f_2(z)-f_1(z)) \\ \rho f_1(z) - f_1(z) f_3(z) - f_2(z) \\ f_1(z)f_2(z) - \beta f_3(z) \end{array} \right] = \left[ \begin{array}{c} z_1 \lambda _1 \partial _{z_1} f_1(z) + z_2 \lambda _2 \partial _{z_2} f_1(z) \\ z_1 \lambda _1 \partial _{z_1} f_2(z) + z_2 \lambda _2 \partial _{z_2} f_2(z) \\ z_1 \lambda _1 \partial _{z_1} f_3(z) + z_2 \lambda _2 \partial _{z_2} f_3(z) \end{array} \right] . \end{aligned}$$

The right hand side expands as

$$\begin{aligned} \left[ \begin{array}{c} z_1 \lambda _1 \partial _{z_1} f_1(z) + z_2 \lambda _2 \partial _{z_2} f_1(z) \\ z_1 \lambda _1 \partial _{z_1} f_2(z) + z_2 \lambda _2 \partial _{z_2} f_2(z) \\ z_1 \lambda _1 \partial _{z_1} f_3(z) + z_2 \lambda _2 \partial _{z_2} f_3(z) \end{array} \right] = \sum _{n_1 = 0}^{\infty } \sum _{n_2 = 0}^{\infty } \left( n_1 \lambda _1 + n_2 \lambda _2 \right) \left[ \begin{array}{c} p_{(n_1, n_2)}^1 \\ p_{(n_1, n_2)}^2 \\ p_{(n_1, n_2)}^3 \end{array} \right] z_1^{n_1} z_2^{n_2}, \end{aligned}$$

while the left hand side is

$$\begin{aligned}&\left[ \begin{array}{c} \sigma (f_2(z)-f_1(z)) \\ \rho f_1(z) - f_1(z) f_3(z) - f_2(z) \\ f_1(z)f_2(z) - \beta f_3(z) \end{array} \right] \\&\quad =\sum _{n_1 = 0}^{\infty } \sum _{n_2 = 0}^{\infty } \left[ \begin{array}{c} \sigma \left( p_{(n_1, n_2)}^2-p_{(n_1, n_2)}^1 \right) \\ \rho \, p_{(n_1, n_2)}^1 - p_{(n_1, n_2)}^2 - \sum _{k=0}^{n_2} \sum _{j=0}^{n_1} p_{(n_1-j, n_2-k)}^1 p_{(j,k)}^3 \\ - \beta \, p_{(n_1, n_2)}^3 + \sum _{k=0}^{n_2} \sum _{j=0}^{n_1} p_{(n_1-j, n_2-k)}^1 p_{(j,k)}^2 \end{array} \right] z_1^{n_1} z_2^{n_2}. \end{aligned}$$

Matching like powers of $z$ and solving for the higher order terms in terms of the lower order terms gives the homological equation

$$\begin{aligned}&\left( \begin{array}{c@{\quad }c@{\quad }c} \sigma - (n_1 \lambda _1 + n_2 \lambda _2) &{} \sigma &{} 0 \\ \rho - p_{(0,0)}^3 &{} -1 - (n_1 \lambda _1 + n_2 \lambda _2) &{} -p_{(0,0)}^1 \\ p_{(0,0)}^2 &{} p_{(0,0)}^1 &{}-\beta - (n_1 \lambda _1 + n_2 \lambda _2) \end{array} \right) \left[ \begin{array}{c} p_{(n_1, n_2)}^1 \\ p_{(n_1, n_2)}^2 \\ p_{(n_1, n_2)}^3 \end{array} \right] \\&\quad = \sum _{k=0}^{n_2} \sum _{j=0}^{n_1} \left[ \begin{array}{c} 0 \\ \bar{p}_{(n_1-j, n_2-k)}^1 \bar{p}_{(j,k)}^3 \\ -\bar{p}_{(n_1-j, n_2-k)}^1 \bar{p}_{(j,k)}^2 \end{array} \right] , \end{aligned}$$

where

$$\begin{aligned} \bar{p}_{(j,k)} = \left\{ \begin{array}{lll} 0 \,\,\,\,\,\,\,\,&{}\text{ if } \text{ either }&{} i=j=0 \,\,\,\,\, \text{ or } \,\,\,\,\, i=n_1, j= n_2 \\ p_{(i,j)} \,\,\,\,\,\,\,\,\,\, &{}\text{ otherwise }.&{} \end{array} \right. \end{aligned}$$

The homological equation has the form

$$\begin{aligned} \left[ \, Dg(p) - (n_1 \lambda _1 + n_2 \lambda _2) I \, \right] p_{(n_1, n_2)} = s_{(n_1, n_2)}, \end{aligned}$$

with $s$ depending only on lower order terms. Moreover the matrix is a characteristic matrix for $Dg(p)$ and is invertible as long as $n_2 \lambda _1 + n_2 \lambda _2 \ne \lambda _{\ell }$ for any $n_1 + n_2 \ge 2$ and either of $\ell = 1,2$. When $\lambda _{1,2}$ are a complex conjugate pair this non-resonance condition holds for all $n_1 + n_2 \ge 2$. If $\lambda _{1,2}$ were real distinct and $\lambda _1 < \lambda _2$ then if $n_2 \lambda _2 < \lambda _1,$ it follows that $n_1 \lambda _1 + n_2 \lambda _2 < \lambda _{\ell }$, $\ell = 1,2$. So there are no resonances for any multi-index $(n_1, n_2)$ with $n_1+n_2 \ge \lambda _1/\lambda _2$. Once we check that there are no resonances for multi-indices smaller than this then we rule out resonances to all orders.

1.2 Appendix 2: Radii Polynomial Estimates and Formulas for Lorenz

First notice that

$$\begin{aligned} Dg(x,y,z) = \begin{pmatrix} -\sigma &{}\quad \sigma &{}\quad 0\\ \rho -z &{}\quad -1 &{}\quad -x\\ y &{}\quad x &{}\quad -\beta \end{pmatrix}. \end{aligned}$$

(7.1)

Let us start with the computation of the vector functions $v_{d}(u^*_{h},\tilde{u}_{1},\tilde{u}_{2})$ $(d=1,\ldots , D)$ defined in (5.8). Recall that we seek an expression of the form

$$\begin{aligned} Dg(u^*_{h}(s)+r\tilde{u}_{1}(s))\tilde{u}_{2}(s)-Dg(u^*_{h}(s))(\Pi _{h})^{n}\tilde{u}_{2}(s) = \sum _{d = 1}^{D}v_{d}r^{d-1}. \end{aligned}$$

Let $x_{i} = (\theta _{i},\alpha _{i},u_{i}) = r\tilde{x}_{i}$ ,$\tilde{x}_{i}\in B(1,\omega )$, $i=1,2$ as defined in (3.5). This in particular implies that

$$\begin{aligned} \begin{aligned} \Vert \tilde{u}_{i}\Vert _{C_{0}}&\le 1+\omega \\ \Vert \tilde{u}_{i}-(\Pi _{h})^{3}\tilde{u}_{i}\Vert _{C_{0}}&\le \omega \end{aligned} \end{aligned}$$

(7.2)

Denoting $u^*_{h} = ([u^*_{h}]_{1},[u^*_{h}]_{2},[u^*_{h}]_{3})$, $\tilde{u}_{i} = ([\tilde{u}_{i}]_{1},[\tilde{u}_{i}]_{2},[\tilde{u}_{i}]_{3})$ $(i=1,2)$ and applying (7.1) we can write (5.8) as follows.

$$\begin{aligned} \begin{aligned}&Dg(u^*_{h}(s)+r\tilde{u}_{1}(s))\tilde{u}_{2}(s)-Dg(u^*_{h}(s))(\Pi _{h})^{3}\tilde{u}_{2}(s)\\&\quad = \begin{pmatrix} -\sigma ([\tilde{u}_{2}]_{1}-\Pi _{h}[\tilde{u}_{2}]_{1})+\sigma ([\tilde{u}_{2}]_{2}-\Pi _{h}[\tilde{u}_{2}]_{2})\\ \rho ([\tilde{u}_{2}]_{1}-\Pi _{h}[\tilde{u}_{2}]_{1})-[u^*_{h}]_{3}([\tilde{u}_{2}]_{1}-\Pi _{h}[\tilde{u}_{2}]_{1})-([\tilde{u}_{2}]_{2}\\ -\Pi _{h}[\tilde{u}_{2}]_{2})-[u^*_{h}]_{1}([\tilde{u}_{2}]_{3}-\Pi _{h}[\tilde{x}_{2}]_{3})\\ [u^*_{h}]_{2}([\tilde{u}_{2}]_{1}-\Pi _{h}[\tilde{u}_{2}]_{1})+[u^*_{h}]_{1}([\tilde{u}_{2}]_{2}-\Pi _{h}[\tilde{u}_{2}]_{2})-\beta ([\tilde{u}_{2}]_{3}-\Pi _{h}[\tilde{u}_{2}]_{3}) \end{pmatrix}\\&\qquad +r\begin{pmatrix} 0\\ [\tilde{u}_{1}]_{3}[\tilde{u}_{2}]_{1}-[\tilde{u}_{1}]_{1}[\tilde{u}_{2}]_{3}\\ 2[\tilde{u}_{1}]_{2}[\tilde{u}_{2}]_{1} \end{pmatrix}\\&\quad :=v_{1}(u^*_{h},\tilde{u}_{1},\tilde{u}_{2})+rv_{2}(u^*_{h},\tilde{u}_{1},\tilde{u}_{2}). \end{aligned} \end{aligned}$$

In particular $D=2$ in this case. Now using (7.2) we can compute $\Gamma _{1,2}\in \mathbb {R}^{3}$ by applying the following estimates on the subintervals. For $i=1,\ldots ,m$

$$\begin{aligned}&\int \limits _{t_{i-1}}^{t_{i}} |v_{1}(u^*_{h}(s),\tilde{u}_{1}(s),\tilde{u}_{2}(s))|ds\nonumber \\&\quad \le \begin{pmatrix} 2|\sigma |\\ \max _{t\in [t_{i-1},t_{i}]}\{|\rho -[u^*_{h}]_{3}(t)|+1+|[u^*_{h}]_{1}(t)|\}\\ \max _{t\in [t_{i-1},t_{i}]}\{|[u^*_{h}]_{2}(t)|+|[\hat{x}_{h}]_{1}(t)|+|\beta |\} \end{pmatrix}\omega (t_{i}-t_{i-1}) \end{aligned}$$

(7.3)

where the absolute value is to be understood component-wise. Similarly

$$\begin{aligned} \begin{aligned} \int \limits _{t_{i-1}}^{t_{i}} |v_{2}(u^*_{h}(s),\tilde{u}_{1}(s),\tilde{u}_{2}(s))|ds\le \begin{pmatrix} 0\\ 2\\ 2 \end{pmatrix}(t_{i}-t_{i-1})(1+\omega )^{2} \end{aligned}. \end{aligned}$$

(7.4)

Splitting the integral from 0 to 1 into the sum of the integrals over the subintervals $[t_{i-1},t_{i}]$ $(i=1,\ldots ,m)$ we can use (7.3) and (7.4) to compute $\Gamma _{1,2}$. More explicitely

$$\begin{aligned} \Gamma _{1} = \omega \sum _{i=1}^{m}\begin{pmatrix} 2|\sigma |\\ \max _{t\in [t_{i-1},t_{i}]}\{|\rho -[u^*_{h}]_{3}(t)|+1+|[u^*_{h}]_{1}(t)|\}\\ \max _{t\in [t_{i-1},t_{i}]}\{|[u^*_{h}]_{2}(t)|+|[u^*_{h}]_{1}(t)|+|\beta |\} \end{pmatrix} \end{aligned}$$

and

$$\begin{aligned} \Gamma _{2} = (1+\omega )^{2}\begin{pmatrix} 0\\ 2\\ 2 \end{pmatrix}. \end{aligned}$$

By a similar reasoning we can compute $\Gamma _{1,2}^{i}\in \mathbb {R}^{m+1}$ for $i=1,2,3$. The bounds $\Lambda _{s,u}$ were derived in general in (5.6) and (5.7). Therefore we have all the ingredients to compute the bounds $Z_{l}(r)$ $(l=1,\ldots ,3(m+2))$.

We continue to derive explicit expressions for $Y_{\infty }$ and $Z_{\infty }(r)$. Recall that

$$\begin{aligned} (u^*_{h})'(t)|_{(t_{i-1},t_{i})} = \frac{1}{t_{i}-t_{i-1}}\Delta u^*_{i}, \end{aligned}$$

where $u^*_{i} = u^*_{h}(t_{i})$ for $i=0,\ldots ,m$ and $\Delta u^*_{i} = u^*_{i}-u^*_{i-1}$.

$\underline{Y_{\infty }}$: Let $i = 1,\ldots ,m$:

$$\begin{aligned} \begin{aligned} \!Dg(u^*_{h})(u^*_{h})'|_{(t_{i-1},t_{i})}&= \frac{1}{t_{i}-t_{i-1}}\!\begin{pmatrix} -\sigma &{} \sigma &{} 0\\ \rho -[u^*_{h}]_{3}|_{(t_{i-1},t_{i})} &{} -1 &{}\quad -[u^*_{h}]_{1}|_{(t_{i-1},t_{i})}\\ [u^*_{h}]_{2}|_{(t_{i-1},t_{i})} &{}\quad [u^*_{h}]_{1}|_{(t_{i-1},t_{i})} &{} -\beta \end{pmatrix}\! \Delta u^*_{i}. \end{aligned} \end{aligned}$$

Hence we obtain:

$$\begin{aligned} \begin{aligned} \left| \frac{d^{2}}{dt^{2}}h^{1}(t)|_{(t_{i-1},t_{i})}\right|&\le \frac{L}{t_{i}-t_{i-1}}\sigma \left| \Delta [u^*_{i}]_{2}-\Delta [u^*_{i}]_{1}\right| \\ \left| \frac{d^{2}}{dt^{2}}h^{2}(t)|_{(t_{i-1},t_{i})}\right|&\!\le \! \frac{L}{t_{i}\!-\!t_{i-1}}\left| (\rho -[u^*_{h}]_{3}\right| _{(t_{i-1},t_{i})})\Delta [u^*_{i}]_{1}\!-\!\Delta [u^*_{i}]_{2}-[u^*_{h}]_{1}\left| _{(t_{i-1},t_{i})}\Delta [u^*_{i}]_{3}\right| \\ \left| \frac{d^{2}}{dt^{2}}h^{3}(t)|_{(t_{i-1},t_{i})}\right|&\le \frac{L}{t_{i}-t_{i-1}}|[u^*_{h}]_{2}|_{(t_{i-1},t_{i})} \Delta [u^*_{i}]_{1}+ [u^*_{h}]_{1}\left| _{(t_{i-1},t_{i})}\Delta [u^*_{i}]_{2} -\beta \Delta [u^*_{i}]_{3}\right| . \end{aligned} \end{aligned}$$

Using interval arithmetic we are able to evaluate $[u^*_{h}]_{i}|_{(t_{i-1},t_{i})}$ for $i=1,2,3$ and finalize the computations for $Y_{\infty }$ using Lemma 5.

$\underline{Z_{\infty }}$: Using (7.1) one obtains after computing

$$\begin{aligned}&Dg(u^*_{h}+r\tilde{u}_{1})\tilde{u}_{2} \nonumber \\&\quad = \begin{pmatrix} -\sigma [\tilde{u}_{2}]_{1}+\sigma [\tilde{u}_{2}]_{1}\\ \rho [\tilde{u}_{2}]_{2}-[u^*_{h}]_{3}[\tilde{u}_{2}]_{1}-r[\tilde{u}_{1}]_{3}[\tilde{u}_{2}]_{1}-[\tilde{x}_{2}]_{2}-[u^*_{h}]_{1}[\tilde{u}_{2}]_|{3}-r[\tilde{x}_{1}]_{1}[\tilde{u}_{2}]_{3}\\ [u^*_{h}]_{2}[\tilde{x}_{2}]_{1}+r[\tilde{x}_{1}]_{2}[\tilde{u}_{2}]_{1}+[u^*_{h}]_{1}[\tilde{u}_{2}]_{2}+r[\tilde{x}_{1}]_{2}[\tilde{u}_{2}]_{2}-\beta [\tilde{u}_{2}]_{3} \end{pmatrix} \end{aligned}$$

that

$$\begin{aligned} \begin{aligned} |Dg^{1}(u^*_{h}+r\tilde{u}_{1})\tilde{u}_{2}|&\le 2\sigma (1+\omega )\\ |Dg^{2}(u^*_{h}+r\tilde{u}_{1})\tilde{u}_{2}|&\le (\rho +1+|[u^*_{h}]_{3}|+|[u^*_{h}]_{1}|+2r)(1+\omega )^2\\ |Dg^{3}(u^*_{h}+r\tilde{u}_{1})\tilde{u}_{2}|&\le (\beta +|[u^*_{h}]_{2}|+|[\hat{x}_{h}]_{1}|+2r)(1+\omega )^2. \end{aligned} \end{aligned}$$

Using Lemma 6 by evaluating the above bounds with interval arithmetic on the subintervals $(t_{i-1},t_{i})$ $(i=1,\ldots ,m)$ we can use this finalize the computation of $Z_{\infty }(r)$.