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# Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs

• * Corresponding author

The second author is supported by NSERC

• In this work, we introduce a method based on piecewise polynomial interpolation to enclose rigorously solutions of nonlinear ODEs. Using a technique which we call a priori bootstrap, we transform the problem of solving the ODE into one of looking for a fixed point of a high order smoothing Picard-like operator. We then develop a rigorous computational method based on a Newton-Kantorovich type argument (the radii polynomial approach) to prove existence of a fixed point of the Picard-like operator. We present all necessary estimates in full generality and for any nonlinearities. With our approach, we study two systems of nonlinear equations: the Lorenz system and the ABC flow. For the Lorenz system, we solve Cauchy problems and prove existence of periodic and connecting orbits at the classical parameters, and for ABC flows, we prove existence of ballistic spiral orbits.

Mathematics Subject Classification: Primary: 65G20, 65P99, 65D30, 37M99, 37C27.

 Citation: • • Figure 1.  The longest orbits we are able to validate, with a total number of coefficient of approximately 14000. In blue for $p = 1$, in green for $p = 2$ and in red for $p = 3$. The initial value is given by (22)

Figure 2.  A validated periodic orbit of the Lorenz system, whose period $\tau$ is approximately $11.9973$. We used two iterations of a priori bootstrap, that is $p = 3$, for the validation. If we want to minimize the total number of coefficients to do the validation, we can take $k = 3$ and $m = 602$ (which makes $7225$ coefficients in total), and we then get a validation radius of $1.5627\times 10^{-4}$. It is possible to get a significantly lower validation radius, at the expense of a slight increase in the total number of coefficients: for instance with $k = 5$ and $m = 495$ (which makes $8911$ coefficients in total), we get a validation radius of $4.7936\times 10^{-9}$

Figure 3.  Validated connecting orbit for the Lorenz system, with parameters $(\sigma, \beta, \rho) = (10, \frac{8}{3}, 28)$. The local stable manifold of the origin is in blue, the local unstable manifold of $\left(\sqrt{\beta(\rho-1)}, \sqrt{\beta(\rho-1)}, \rho-1\right)$ in yellow, and the green connection between them (of length $\tau\simeq 17.3$) is validated using polynomial interpolation, with a priori bootstrap ($p = 3$). The proof gives a validation radius of $r = 3.1340\times 10^{-5}$

Figure 4.  These are the orbits that are described in Theorem 7.1. The color varies from blue for $A = 1$ to red for $A = 0.1$. Each proof was done with $p = 2$, $k = 2$ and $m = 50$, and gave a validation radius varying from $r = 4.8313\times 10^{-8}$ to $r = 7.4012\times 10^{-6}$

Figure 5.  This is the orbit that is described in Theorem 7.2. The proof was done with $p = 2$, $k = 2$ and $m = 300$, and gave a validation radius $r = 4.0458\times 10^{-6}$

Figure 6.  An example of the situation described just above, with $k = 11$ and $i_0 = 9$. The black squares represent the other Chebyshev points $x_{i}^k$

Table 1.  Comparisons for $p = 1$. $\tau_{max}$ is the longest integration time for which the proof succeeds, and $r$ is the associated validation radius, that is a bound of the distance (in $\mathcal{C}^0$ norm) between the numerical data used and the true solution

 $k$ $m$ $nm(k+1)$ $\tau_{max}$ $r$ 1 2333 13998 $\color{red} {0.69}$ $2.3233\times 10^{-5}$ 2 1556 14004 0.64 $1.1524\times 10^{-7}$ 3 1167 14004 0.58 $8.6805\times 10^{-9}$

Table 2.  Comparisons for $p = 2$. $\tau_{max}$ is the longest integration time for which the proof succeeds, and $r$ is the associated validation radius, that is a bound of the distance (in $\mathcal{C}^0$ norm) between the numerical data used and the true solution

 $k$ $m$ $nm(k+1)$ $\tau_{max}$ $r$ 1 2333 13998 0.97 $1.5718\times 10^{-3}$ 2 1556 14004 $\color{red} {5.6}$ $8.4373\times 10^{-5}$ 3 1167 14004 5.5 $8.4184\times 10^{-8}$ 4 933 13995 4.9 $7.9190\times 10^{-9}$

Table 3.  Comparisons for $p = 3$. $\tau_{max}$ is the longest integration time for which the proof succeeds, and $r$ is the associated validation radius, that is a bound of the distance (in $\mathcal{C}^0$ norm) between the numerical data used and the true solution

 $k$ $m$ $nm(k+1)$ $\tau_{max}$ $r$ 2 1556 14004 5.6 $7.6716\times 10^{-4}$ 3 1167 14004 $\color{red} {8.1}$ $9.3043\times 10^{-6}$ 4 933 13995 $\color{red} {8.1}$ $8.8204\times 10^{-8}$ 5 778 14004 8.0 $1.6175\times 10^{-8}$ 9 467 14010 7.9 $1.3748\times 10^{-8}$ 19 233 13980 6.9 $2.2998\times 10^{-8}$

Table 4.  Minimal number of coefficients needed to validate the orbit of length $\tau = 2$, starting from $u_0$ given in (22), and the associated validation radius provided by the proof. We repeat that in this example, we put the emphasis on getting an existence result with a minimal amount of computational power, which of course results in less sharp validation radius, especially in situations where the $Y$ bound is the limiting factor

 $k=1$ $k=2$ $k=3$ $k=4$ $p=2$ no proof $m=416$ $m=415$ $m=377$ no proof $\color{red} {nm(k+1)=3744}$ $nm(k+1)=4980$ $nm(k+1)=5655$ no proof $r=2.9\times 10^{-4}$ $r=9.4\times 10^{-7}$ $r=1.4\times 10^{-10}$ $k=2$ $k=3$ $k=4$ $k=5$ $p=3$ $m=470$ $m=125$ $m=110$ $m=99$ $nm(k+1)=4230$ $\color{red} {nm(k+1)=1500}$ $nm(k+1)=1650$ $nm(k+1)=1782$ $r=1.2\times 10^{-3}$ $r=3.4\times 10^{-4}$ $r=7.2\times 10^{-6}$ $r=1.8\times 10^{-7}$

Table 5.  The intervals $[\tau^-_A, \tau^+_A]$, for $A = 0.1, 0.2, \ldots, 1$ in which the period $\tau_A$ of the solution described in Theorem 7.1 is proved to be

 $A$ $1$ $0.9$ $0.8$ $0.7$ $0.6$ $0.5$ $0.4$ $0.3$ $0.2$ $0.1$ $\tau^-_A$ $3.23527736$ $3.41779635$ $3.62512508$ $3.86405419$ $4.14464726$ $4.48269179$ $4.90491344$ $5.46177978$ $6.26680147$ $7.67945129$ $\tau^+_A$ $3.23527746$ $3.41779647$ $3.62512521$ $3.86405436$ $4.14464749$ $4.48269213$ $4.90491401$ $5.46178092$ $6.26680442$ $7.67946552$
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