Efficient and Reliable Algorithms for the Computation of Non-Twist Invariant Circles

Abstract

This paper presents a methodology to study non-twist invariant circles and their bifurcations for area preserving maps, which is supported on the theoretical framework developed in Gonzalez-Enriquez et al. (Mem. Amer. Math. Soc. 227:vi+115, 2014). We recall that non-twist invariant circles are characterized not only by being invariant, but also by having some specified normal behavior. The normal behavior may endow them with extra stability properties (e.g., against external noise), and hence, they appear as design goals in some applications, e.g., in plasma physics, astrodynamics and oceanography. The methodology leads to efficient algorithms to compute and continue, with respect to parameters, non-twist invariant circles. The algorithms are quadratically convergent and, when implemented using FFT, have low storage requirement and low operations count per step. Furthermore, the algorithms are backed up by rigorous a posteriori theorems, proved and discussed in detail in Gonzalez-Enriquez et al. (Mem. Amer. Math. Soc. 227:vi+115, 2014), which give sufficient conditions guaranteeing the existence of a true non-twist invariant circle, provided an approximate invariant circle is known. Hence, one can compute confidently even very close to breakdown. With some extra effort, the calculations could be turned into computer-assisted proofs, see Figueras et al. (Found. Comput. Math. 17:1123–1193, 2017) for examples of the latter. The algorithms are also guaranteed to converge up to the breakdown of the invariant circles, and then, they are suitable to compute regions of parameters where the non-twist invariant circles exist. The calculations involved in the computation of the boundary of these regions are very robust, and they do not require symmetries and can run without continuous manual adjustments, largely improving methods based on the computation of very long period periodic orbits to approximate invariant circles. This paper contains a detailed description of our algorithms, the corresponding implementation and some numerical results, obtained by running the computer programs. In particular, we include calculations for two-dimensional parameter regions where non-twist invariant circles (with a prescribed frequency) exist. Indeed, we present systematic results in systems that do not contain symmetry lines, which seem to be unaccessible for previous methods. These numerical explorations lead to some open questions, also included here.

Appendices

Appendix

The Potential and Degeneracy Conditions

One of the features of the methodology introduced in [37] is that the translation function of a family of translated tori is the negative gradient of a potential. Another strong point is the fact that non-degenerate (degenerate) critical points of the potential correspond to twist (non-twist) rotational tori. Moreover, explicit formulas are provided for the translation function and the primitive function of F (in terms of the geometrical and analytical objects associated with the family).

In the area-preserving case, even though the fact that the translation function is the gradient of a potential becomes apparent (because it depends only on one variable), some extra properties hold. This appendix reports some of these features and extra properties for area preserving maps.

In the area preserving case, in addition to the property that the translation function is the gradient of a potential (which is obvious because it depends only on one variable), we provide a formula for the potential and its first two derivatives.

1.1 The Potential of a Family of Translated Rotational Circles

Following the notation and definitions introduced in Sect. 2, the potential of the family of F-translated rotational circles is given by:

$$\begin{aligned} V(p)= -p\,\lambda (p) - \langle {S(K(\theta , p))}\rangle . \end{aligned}$$

We have the following result.

Lemma 1

$\lambda (p)= - V'(p)$

Proof

From the definition of primitive function, we get

$$\begin{aligned} \begin{aligned} \frac{\partial }{\partial p}\langle {S(K(\theta , p))}\rangle&= \langle { \frac{\partial S}{\partial x}(K(\theta , p)) \frac{\partial K^x}{\partial p}(\theta , p) + \frac{\partial S}{\partial y}(K(\theta , p)) \frac{\partial K^y}{\partial p}(\theta , p) }\rangle \\&= \langle { \left( F^y(K(\theta , p)) \frac{\partial F^x}{\partial x}(K(\theta , p)) - K^y(\theta , p) \right) \frac{\partial K^x}{\partial p}(\theta , p)}\rangle \\&\quad + \langle {F^y(K(\theta , p)) \frac{\partial F^x}{\partial y}(K(\theta , p)) \frac{\partial K^y}{\partial p}(\theta , p) }\rangle \\&= \langle { F^y(K(\theta , p)) \frac{\partial }{\partial p}( F^x(K(\theta , p)))- K^y(\theta , p) \frac{\partial K^x}{\partial p}(\theta , p) }\rangle \\&= \langle { K^y(\theta +\omega , p)\frac{\partial }{\partial p}( K^x(\theta +\omega , p)- \lambda (p)) - K^y(\theta , p)\frac{\partial K^x}{\partial p}(\theta , p)}\rangle \\&= -p \lambda '(p) , \end{aligned} \end{aligned}$$

where in the last equality we used the following equality:

$$\begin{aligned} \langle {K^y(\theta +\omega , p)}\rangle = \langle {K^y(\theta , p)}\rangle = p, \end{aligned}$$

and

$$\begin{aligned} \langle {K^y(\theta +\omega , p)\frac{\partial }{\partial p} (K^x(\theta +\omega , p)) }\rangle = \langle {K^y(\theta , p)\frac{\partial }{\partial p} (K^x(\theta , p)) }\rangle . \end{aligned}$$

The proof follows immediately. $\square$

An important consequence of Lemma 1 is that for a F-invariant rotational circle ${{\mathscr {K}}}_{p_0}$, $p_0$ is a critical point of the potential V.

1.2 On the Equivalence of Degeneracy and Non-Twist Conditions

In this section, we review the fact that twist (non-twist) invariant rotational circles correspond to non-degenerate (degenerate) rotational circles, by tailoring some results in [37] to the area-preserving case. We use the notations and definitions given in Sect. 2.

Lemma 2

Let $(K(\theta , p),\tau (p))$ be a family of F-translated rotational circles, with $\tau (p)= (\lambda (p),0)$, and for each p, let ${\bar{T}}(p)$ the torsion and ${\hat{T}}(p)$ the supertorsion of the rotational circle ${{\mathscr {K}}}_p$ of momentum p. We assume the supertorsion is non-degenerate. Then:

$$\begin{aligned} \lambda '(p)= \displaystyle \frac{1}{\mathrm{det} {\hat{T}}(p)}\ {\bar{T}}(p). \end{aligned}$$

Proof

By differentiating with respect to p the equation for F-translated rotational circles (2), we find the equations for $\frac{\partial K}{\partial p}(\theta , p)$ and $\tau '(p)$:

$$\begin{aligned} \left\{ \begin{aligned} { \mathrm D}F(K(\theta , p)) \frac{\partial K}{\partial p}(\theta , p) + \tau '(p) - \frac{\partial K}{\partial p}(\theta +\omega , p)&= 0 \\ \langle { \frac{\partial K}{\partial p}(\theta , p)}\rangle&= \begin{pmatrix} 0 \\ 1 \end{pmatrix}\, . \end{aligned} \right. \end{aligned}$$

(49)

Fixed p we are in the conditions of Algorithm 2, then taking $M(\theta )= { \mathrm D}F(K(\theta , p))$ and $L(\theta )= { \mathrm D}K(\theta , p)$, we have that $M(\theta )$ can be reduced to a block triangular form $\varLambda (\theta )$, so $M_0(\theta )= M(\theta )$. Algorithm 2 provides a procedure to compute the torsion of the rotational circle ${{\mathscr {K}}}_p$, ${\bar{T}}(p)= {\hat{T}}_{11}$ (see step (8) ) and the solution of (49) for the knows $V(\theta )= 0$, $v= (0,1)^\top$, and the unknowns $U(\theta )= \frac{\partial K}{\partial p}(\theta , p)$, $u= \tau '(p)= (\lambda '(p),\sigma '(p))^\top$. That is:

$$\begin{aligned} \left( \frac{\partial K}{\partial p}(\theta , p), \tau '(p) \right) = {\hat{{\mathscr {R}}}}_{M,L,\omega }(0, (0,1))^\top . \end{aligned}$$

Following the steps of Algorithm 2, we obtain in step (12)

$$\begin{aligned} \begin{pmatrix} \xi ^N_0 \\ u^x \end{pmatrix} = \begin{pmatrix} {\hat{T}}_{11} &{} {\hat{T}}_{12} \\ {\hat{T}}_{21} &{} {\hat{T}}_{22} \end{pmatrix}^{-1} \begin{pmatrix} 0 \\ 1 \end{pmatrix}\, , \end{aligned}$$

from which the result follows. $\square$

Remark 17

In terms of the potential of the family, Lemma 2 gives:

$$\begin{aligned} V''(p)= \frac{-1}{\mathrm{det} {\hat{T}}(p)}\ {\bar{T}}(p). \end{aligned}$$

Remark 18

Notice that in step (4) of Algorithm 2, we obtain $\eta (\theta )= 0$, and, then, $\sigma '(p)= u^y= \langle {\eta ^N}\rangle = 0$. This also follows from the exactness condition on F, which implies $\sigma (p)= 0$.

Remark 19

We incidentally obtain the formula

$$\begin{aligned} \langle {{ \mathrm D}_\theta K(\theta , p)^\top \varOmega _0 { \mathrm D}_p K(\theta , p)}\rangle = -\xi ^N_0= \displaystyle \frac{1}{\mathrm{det} {\hat{T}}(p)}\ {\hat{T}}_{12}(p)\, . \end{aligned}$$

In fact, it is easy to see that

$$\begin{aligned}&{ \mathrm D}_\theta K(\theta , p)^\top \varOmega _0 { \mathrm D}_p K(\theta , p) - { \mathrm D}_\theta K(\theta +\omega , p)^\top \varOmega _0 { \mathrm D}_p K(\theta +\omega , p) \\&\quad = - { \mathrm D}_\theta K^y(\theta +\omega , p) \lambda '(p), \end{aligned}$$

from where we obtain that

$$\begin{aligned}&{ \mathrm D}_\theta K(\theta , p)^\top \varOmega _0 { \mathrm D}_p K(\theta , p) = \frac{1}{\mathrm{det} {\hat{T}}(p)} \left( {\hat{T}}_{12}(p) \nonumber \right. \\&\left. \quad - {{\mathscr {R}}}_\omega ( { \mathrm D}_\theta K^y(\theta +\omega , p)) {\hat{T}}_{11}(p) \right) . \end{aligned}$$

(50)

If the expression (50) does not vanish, the translated rotational circles are transversal with respect to their momentum p, guaranteeing the existence of a (local) diffeomorphism $(\theta ,p) \rightarrow K(\theta ;p)$ giving rise to a (local) Lagrangian foliation. In particular, on a non-twist torus of momentum p,

$$\begin{aligned} { \mathrm D}_\theta K(\theta , p)^\top \varOmega _0 { \mathrm D}_p K(\theta , p) = \frac{-1}{{\hat{T}}_{12}(p)}. \end{aligned}$$