Attractive Invariant Circles à la Chenciner

Abstract

In studying general perturbations of a dissipative twist map depending on two parameters, a frequency $\nu$ and a dissipation $\eta$, the existence of a Cantor set $\mathcal{C}$ of curves in the $(\nu,\eta)$ plane such that the corresponding equation possesses a Diophantine quasi-periodic invariant circle can be deduced, up to small values of the dissipation, as a direct consequence of a normal form theorem in the spirit of Rüssmann and the “elimination of parameters” technique. These circles are normally hyperbolic as soon as $\eta\not=0$, which implies that the equation still possesses a circle of this kind for values of the parameters belonging to a neighborhood $\mathcal{V}$ of this set of curves. Obviously, the dynamics on such invariant circles is no more controlled and may be generic, but the normal dynamics is controlled in the sense of their basins of attraction.

As expected, by the classical graph-transform method we are able to determine a first rough region where the normal hyperbolicity prevails and a circle persists, for a strong enough dissipation $\eta\sim O(\sqrt{\varepsilon}),$ $\varepsilon$ being the size of the perturbation. Then, through normal-form techniques, we shall enlarge such regions and determine such a (conic) neighborhood $\mathcal{V}$, up to values of dissipation of the same order as the perturbation, by using the fact that the proximity of the set $\mathcal{C}$ allows, thanks to Rüssmann’s translated curve theorem, an introduction of local coordinates of the type (dissipation, translation) similar to the ones introduced by Chenciner in [7].

APPENDIX A. ON THE ELIMINATION OF PARAMETERS

In this section we discuss in more detail the content of Remark 5 and the technique of elimination of parameters applied to the question of persistence of an invariant normally hyperbolic circle of a given Diophantine rotation. Since the perturbing terms considered here are just characterized as analytic functions with uniformly bounded derivatives, there is no reason why such persistence must hold for any choices of $\eta$ uniformly w.r.t. $\varepsilon$, as it may happen in some special cases. We shall give a brief review on this matter, for the sake of clarity and completeness.

$\bullet$ In the context of Hamiltonian real analytic perturbations, in the frame of the following $n$-parameter family of dissipative twist vector fields on $\mathbb{T}^{n}\times\mathbb{R}^{n}$

$$u_{\nu}=\big{(}\dot{\theta}=\alpha+r,\quad\dot{r}=-\eta I\cdot r-\eta(\nu-\alpha)\big{)},\quad\quad(\nu,\eta)\in\mathbb{R}^{n}\times\mathbb{R},$$

(A.1)

where ${I}$ is the identity matrix, the persistence problem has a positive, strong answer. More specifically, the vector field above leaves invariant the torus $T_{0}=\mathbb{T}^{n}\times{\left\{r=0\right\}}$, whose tangential dynamics is $\alpha$-quasi-periodic, while its normal dynamics is given by the linear term $-\eta r\partial_{r}$. If a Hamiltonian perturbation is considered $u_{\nu}+\varepsilon X_{H}$, the dissipation $-\eta r\partial_{r}$ is the unique non-Hamiltonian term and, provided $\alpha$ is Diophantine, for a unique choice of the external parameter $\nu$, the survival of the reducible torus $T_{0}$ can be proved by means of a symplectic KAM scheme, thanks to the special structure “Hamiltonian + homotecy in $r$-direction” that makes these systems conformally symplectic and the scheme smooth (with uniform bounds) in $\eta$, this allowing $\eta\to 0$ smoothly, reaching the symplectic regime. See, for instance, [28, 21] or [5] in the frame of conformally symplectic maps.

$\bullet$ In the particular case of $1+\frac{1}{2}$ degrees of freedom corresponding to a nonautonomous Hamiltonian perturbation $\varepsilon X_{H}=\big{(}0,-\varepsilon\partial_{\theta}f(\theta,t)\big{)}$ in $\mathbb{T}\times\mathbb{R}$, the system $u_{\nu}+\varepsilon X_{H}$ models the well-known dissipative spin-orbit problem evoked in the introduction. An $\alpha$-diophantine torus persists as a global attractor for the dynamics, for a unique choice of $\nu$ and any dissipation $\eta$ in an interval containing the origin, see [6].

In [21], the persistence results above have been recovered by means of a technique known as “elimination of parameters”, in the spirit of Moser, Rüssmann, Herman, Chenciner and many others [23, 15, 12, 13, 17, 3]. This technique is based on a normal form approach, recovering the existence of the invariant quasi-periodic object in two steps which, in terms of the specific example above (A.1), can roughly be summarized as follows:

• 1) Prove that $u_{\nu}+\varepsilon X_{H}$ admits a unique, local, normal form as $u_{\nu}+\varepsilon X_{H}=g_{*}u+b\partial_{r}$, where $g$ is a diffeomorphism in the neighborhood of the identity in an appropriate subspace of symplectic trasformations, $u$ belongs to the affine subset of vector fields passing through $(\alpha,-\eta r)$ and directed by Hamiltonian terms $\big{(}O(r),O(r^{2})\big{)}$ (i. e., the ones that admit the desired quasi-periodic invariant torus) and $b\in\mathbb{R}^{n}$ is a counter term in the neighborhood of $\alpha$, used to compensate for the possible degeneracy of the system. According to the normal form, the torus $g(T_{0})$ is translated by $b$ in the normal direction.

• 2) Since the translation/counter term $b=b(\alpha,\nu,\eta,\varepsilon)$ is proved to depend smoothly on $\nu,\eta$, one may solve implicitly $b(\alpha,\nu,\eta,\varepsilon)=0$ in terms of $\nu$, thus proving the persistence of $g(T_{0})$ for that choice of the parameter, and any $\eta$.

This is the content of [21, Theorem 6.1] from which one can deduce the particular $2$-dimensional case and phrase it in the following form, which we rewrite for convenience of the reader. See Fig. 5 below.

Theorem 6.2 and Corollary 6.2 in [21]. Let $\varepsilon_{0}$ be the maximal value that the perturbation can attain. Every Diophantine $\alpha$ identifies a surface $(\varepsilon,\eta)\mapsto\nu(\varepsilon,\eta)$ in the space $(\varepsilon,\eta,\nu)=[0,\varepsilon_{0}]\times[-\eta_{0},\eta_{0}]\times\mathbb{R}$, which is analytic in $\varepsilon$, smooth in $\eta$, for which the following holds: for any parameters $\big{(}\varepsilon,\eta,\nu(\varepsilon,\eta)\big{)}$, the vector field $u_{\nu}+\varepsilon X_{H}$ admits an invariant $\alpha$-quasi-periodic torus. This torus is $\eta$-normally attractive (resp. repulsive) if $\eta>0$ $($resp. $\eta<0)$.

Fixing $\alpha$ Diophantine and $\varepsilon$ sufficiently small, there exists a unique curve $C_{\alpha}$ $($ analytic in $\varepsilon$ , smooth in $\eta)$ in the plane $(\eta,\nu)$ of the form $\nu=\alpha+O(\varepsilon^{2})$ , along which the translation $b=b(\nu,\alpha,\eta,\varepsilon)$ vanishes, so that the perturbed system $u_{\nu}+\big{(}0,\varepsilon\partial_{\theta}f(\theta,t)\big{)}$ possesses an invariant torus carrying quasi-periodic motion of frequency $\alpha$ . This torus is normally attractive $($ resp. repulsive $)$ if $\eta>0$ $($ resp. $\eta<0)$ .

In is important to stress that the curves $C_{\alpha}$ pass continuously through $\eta=0$ (the unique value of transition between the attractive and repulsive behavior) because of the Hamiltonian nature of the perturbations. Thanks to its special structure and intrinsic symmetries, this model, of great astronomical interest, has been widely investigated in the last decade; its conformally symplectic structure leads to atypical dynamical behaviors which cannot happen in generic hyperbolic systems. Accurate numerical computations have been tailored to that special model for studying the Poincaré spin-orbit map in order to select the parameters under which the existence/breakdown of the quasi-periodic attractor may occur, or for computing possible rotation numbers of its orbits and so on (see, for instance, [4] and references therein).

On the other hand, under generic perturbation, there is no hope in general for a theorem as above, since typically the perturbation obstructs the existence of such tori for a small value of hyperbolicity. Moreover, in high dimension, the question whether it is possible to get a similar reducibility result (i. e., up to very small values of dissipation) for a system as (A.1) in the case when the homotetic dissipative matrix is replaced by a general diagonalizable one or under generic perturbations (thus breaking the special spin-orbit like conformal symplecticity symmetries) is open so far. A first step forward is provided in [20], in the context of general nonconservative diffeomorphisms of the $2n$-dimensional cylinder, close to having a reducible Diophantine torus, and in the recent study [14], in the context of flows that leaves invariant a Diophantine torus of general normal hyperbolicity (i. e., the linear dynamics is given by a diagonalizable matrix $A$ of real eigenvalues).

A.1. A Normal form Theorem and the Case of General Perturbations

For convenience of the reader, we recall here the general setting of [20] where Rüssmann’s translated curve theorem is recovered in analytic category, as a special case of the general normal form [20, Theorem 5.4], which we recall below.

Let $V$ be the space of germs along $\mathbb{T}^{n}\times{\left\{0\right\}}$ in $\mathbb{T}^{n}\times\mathbb{R}^{m}={\left\{(\theta,r)\right\}}$ of real analytic diffeomorphisms. Fix $\alpha\in\mathbb{R}^{n}$ and $A\in\mathop{\rm GL}_{m}(\mathbb{R})$, assuming that $A$ is diagonalizable with (possibly complex) eigenvalues $a_{1},\ldots,a_{m}\in{\mathbb{C}}$. Let $U(\alpha,A)$ be the affine subspace of $V$ of diffeomorphisms of the form

$$P(\theta,r)=\big{(}\theta+\alpha+O(r),A\cdot r+O(r^{2})\big{)},$$

(A.2)

where $O(r^{k})$ are terms of order $\geqslant k$ in $r$ which may depend on $\theta$. For these diffeomorphisms $\text{T}^{n}_{0}=\mathbb{T}^{n}\times{\left\{0\right\}}$ is an invariant, reducible, $\alpha$-quasi-periodic torus whose normal dynamics at the first order is characterized by $a_{1},\ldots,a_{m}.$

Let ${\mathcal{G}}$ be the space of germs of real analytic diffeomorphisms of $\mathbb{T}^{n}\times\mathbb{R}^{m}$ of the form

$$G(\theta,r)=\big{(}h(\theta),R_{0}(\theta)+R_{1}(\theta)\cdot r\big{)},$$

(A.3)

where $h$ is a diffeomorphism of the torus fixing the origin and $R_{0},R_{1}$ are functions defined on the torus $\mathbb{T}^{n}$ with values in $\mathbb{R}^{m}$ and $\mathop{\rm GL}_{m}(\mathbb{R})$, respectively, and such that $\Pi_{\mathop{\rm Ker}(A-I)}R_{0}(0)=0$ and $\Pi_{\mathop{\rm Ker}[A,\cdot]}(R_{1}(0)-I)=0$, where $I$ denotes the identity matrix in $\mathop{\rm Mat}_{m}(\mathbb{R})$ and $\Pi$ is the projection on the subspace in subscript.

Theorem 4 ([20 , Theorem 5.4])

Let $\alpha$ satisfy

$$\left|k\cdot\alpha-2\pi l\right|\geqslant\frac{\gamma}{\left|k\right|^{\mathtt{q}}},\qquad\forall k\in\mathbb{Z}^{n}\setminus{\left\{0\right\}},\forall l\in\mathbb{Z}.\vspace{-1mm}$$

In a neighborhood of $\mathbb{T}^{n}\times{\left\{0\right\}}\subset\mathbb{T}^{n}\times\mathbb{R}^{n}$, let $P^{0}\in U(\alpha,A^{0})$ be a diffeomorphism of the form

$$P^{0}(\theta,r)=\big{(}\theta+\alpha+p_{1}(\theta)\cdot r+O(r^{2}),A^{0}\cdot r+O(r^{2})\big{)},$$

where $A^{0}$ is invertible and has simple, real eigenvalues such that

$$\mathop{\rm det}{\left(\int_{\mathbb{T}^{n}}p_{1}(\theta)d\theta\right)}\neq 0.$$

If $Q^{\prime}$ is close enoughFootnote

Recall Remark 4.

to $P^{0}$ there exists a unique $A^{\prime}$ close to $A^{0}$ and a unique $(G,P,\lambda)\in{\mathcal{G}}\times U(\alpha,A^{\prime})\times\mathbb{R}^{n}$ close to $(\mathop{\rm id}\nolimits,P^{0},0)$, such that $Q^{\prime}=T_{\lambda}\circ G\circ P\circ G^{-1},$ where $T_{\lambda}(\theta,r)=(\theta,\lambda+r)$.

Recall Remark 4.

Note that, to be consistent with the notations in the present paper, in this statement we denoted by $\lambda$ and $h$ the translation and diffeomorphism, which in [20] is denoted by $b$ and $\varphi$, respectively.

Phrasing the thesis, the graph of $\gamma=R_{0}\circ h^{-1}$ is a translated torus on which the dynamics is conjugated to $R_{\alpha}$ by $h$. By stability of the simple real spectrum, the normal dynamics is of the same nature as before, characterized by $A^{\prime}$ close to $A^{0}$. As a side remark, we note that the spectrum of $A^{0}$ is used as a set of free (real) parametersFootnote

This spectrum being real, the usual Diophantine conditions on $\alpha$ only are sufficient, with no need of Melnikov’s ones.

, the variation of which allows one to cancel an additional matrix obstruction counterterm that otherwise would be present in the translation function (see the proof of [20, Theorem 5.1] or the more delicate “théorème de conjugaison hypothétique” of [15]).

This spectrum being real, the usual Diophantine conditions on $\alpha$ only are sufficient, with no need of Melnikov’s ones.

Rüssmann’s Theorem 3 recalled in Section 3 is then a particular case of dimension two where, in order to stress the hyperbolicity character, we wrote $1+A^{0}$, $A^{0}\neq 0$, and where we directly gave the statement in terms of $h$ (where $h-\mathop{\rm id}\nolimits=\varphi$ of Theorem 4, defined in (A.3), $\mathop{\rm id}\nolimits$ being the identity map). The solutions $\lambda,\gamma$ and $h$ of the nonlinear equation $Q^{\prime}=T_{\lambda}\circ G\circ P\circ G^{-1}$ are constructed as the inverse of an appropriate nonlinear normal form operator, defined from a convenient neighborhood of $(\mathop{\rm id}\nolimits,P^{0},0)$ to a neighborhood of $P^{0}$ by means of a Newton-like scheme which converges uniformly and is smooth w.r.t. the parameters on which the given diffeomorphism $Q^{\prime}$ smoothly depends (see [20, Section 2] and [21, Appendix A]). Hence, in the case we are interested in, Rüssmann’s theorem gives

$$Q\big{(}\theta,\gamma(\theta)\big{)}=\Big{(}h\circ R_{2\pi\alpha}\circ h^{-1}(\theta),\lambda+\gamma\big{(}h\circ R_{2\pi\alpha}\circ h^{-1}(\theta)\big{)}\Big{)},$$

where $\lambda=\tau+O(\varepsilon)=2\pi\eta(\nu-\alpha)+O(\varepsilon)$, $\varphi=\mathop{\rm id}\nolimits+O(\varepsilon)$, $\gamma=O(\varepsilon)$ are analytic in $\varepsilon$ and smooth in $\eta,\nu$. In order to prove that $\lambda=0$ implicitly defines $\nu$, it suffices then to show that $\nu\mapsto\lambda(\nu,\eta,\varepsilon)$ is a local diffeomorphism; since this is an open property and $Q$ is close to $P$, it suffices to show it for $P$, which is immediate. In fact, it suffices to observe that the map $\mathbb{R}^{3}\ni(\varepsilon,\nu,\eta)\mapsto\lambda(\varepsilon,\nu,\eta)$ at $p_{0}=(0,\alpha,\eta)$ gives $\lambda(p_{0})=0$ and that $\frac{\partial\lambda}{\partial\nu}_{|_{\varepsilon=0}}=2\pi\eta>0,$ which will remain so after perturbation for $\eta$ not too close to $0$.

In fact, let now $\varepsilon_{0}$ be the maximal admissible perturbation for Rüssmann’s theorem to apply, and consider the closed ball $B_{\varepsilon_{0}}(p_{0})$ of radius $\varepsilon_{0}$ centered at $p_{0}\in\mathbb{R}^{3}$. Because of the regularity of $\lambda$ with respect to $\varepsilon,\nu$ and $\eta$, there exists a positive constant $M$ independent of $\varepsilon,\eta,\nu$ such that ${\left|\lambda\right|}_{C^{2}}<M$. Considering the ball of radius $\varepsilon<\varepsilon_{0}$, the mean value theorem applied to $\frac{\partial\lambda}{\partial\nu}$ and the triangular inequality yields $\forall p_{2},p_{1}\in B_{\varepsilon/2}(p^{0})$ that

$$\left|\frac{\partial\lambda}{\partial\nu}(p_{2})\right|\geqslant\left|\frac{\partial\lambda}{\partial\nu}(p_{1})\right|-M\varepsilon.$$

Fixing $p_{1}=p_{0}$, a sufficient condition for having $\left|\frac{\partial\lambda}{\partial\nu}(p_{2})\right|>\pi\eta$ is that $\pi\eta/4M>\varepsilon$.

Hence, for every fixed value of $\varepsilon$, we can guarantee that the derivative of $\lambda$ with respect to $\nu$ is different from $0$, for those $\eta^{\prime}$s such that $\eta\geqslant\varepsilon 4M/\pi$, hence by the implicit function theorem, there exists $\nu$ such that $\lambda(\nu,\varepsilon,\eta)=0$.

APPENDIX B. CLASSICAL TOOLS

B.1. Graph Transform Lemmata

Proof (of Lemma 1 )

Since $f$ and $g$ are real analytic on $\mathbb{T}\times[-1,1]$, they are Lipschitz.

• 1) If $u$ is a contraction, $\mathop{\rm id}\nolimits+u$ is invertible with $\mathop{\rm Lip}(\mathop{\rm id}\nolimits+u)^{-1}\leqslant\frac{1}{1-\mathop{\rm Lip}u}$.

  Letting $u=\mathop{\rm id}\nolimits+\Theta(\mathop{\rm id}\nolimits,\varpi)$ and $f$ be analytic, we have

  $$\displaystyle\left|u(\theta_{1})-u(\theta_{2})\right|\leqslant\mathop{\rm Lip}\varphi\frac{(1-e^{-2\pi\eta})}{\eta}\left|\theta_{1}-\theta_{2}\right|+\varepsilon A_{f}\left|\big{(}\theta_{1},\varphi(\theta_{1})\big{)}-\big{(}\theta_{2},\varphi(\theta_{2})\big{)}\right|$$
  $$\displaystyle\leqslant{\left(\frac{(1-e^{-2\pi\eta})}{\eta}k+\varepsilon A_{f}(1+k)\right)}\left|\theta_{1}-\theta_{2}\right|,$$

  where $A_{f}=\max{\left(\sup_{\mathbb{T}\times[-1,1]}\left|D_{\theta}f\right|,\sup_{\mathbb{T}\times[-1,1]}\left|D_{r}f\right|\right)}$.

  Since $\varepsilon,k\ll 1$, $\mathop{\rm Lip}u<1$.

• 2) It is immediate from the expression of $Q$ that for $z_{1},z_{2}$ in $\mathbb{T}\times[-1,1]$

  $$\left|R(z_{1})-R(z_{2})\right|\leqslant(e^{-2\pi\eta}+\varepsilon A_{g})\left|z_{1}-z_{2}\right|,$$

  where $A_{g}=\max{\left(\sup_{\mathbb{T}\times[-1,1]}\left|D_{\theta}g\right|,\sup_{\mathbb{T}\times[-1,1]}\left|D_{r}g\right|\right)}$, and

  $$\left|\Theta(z_{1})-\Theta(z_{2})\right|\leqslant(1+\frac{(1-e^{-2\pi\eta})}{\eta}+\varepsilon A_{f})\left|z_{1}-z_{2}\right|.$$

B.2. Difference Equation on the Torus

Consider the complex extension $\mathbb{T}_{{\mathbb{C}}}={\mathbb{C}}/{2\pi\mathbb{Z}}$ of the torus $\mathbb{T}=\mathbb{R}/{2\pi\mathbb{Z}}$ and the corresponding $s$-neighborhood defined using $\ell^{\infty}$-balls (in the real normal bundle of the torus):

$$\mathbb{T}_{s}={\left\{\theta\in\mathbb{T}_{{\mathbb{C}}}:\left|\mathop{\rm Im}\theta\right|\leqslant s\right\}}.$$

Given a real analytic function, we shall consider its unique complex extension holomorphic on some torus of a certain complex width $s$. Let $\mathcal{A}(\mathbb{T}_{s})$ be the space of holomorphic functions $f:\mathbb{T}_{s}\to{\mathbb{C}}$ with finite Banach norm

$$\left|f\right|_{s}=\sup_{\mathbb{T}_{s}}\left|f(\theta)\right|.$$

Lemma 3

Let $\alpha$ be Diophantine in the sense of (3.1) , $g\in\mathcal{A}(\mathbb{T}_{s+\sigma})$ and let some constants $a,b\in\mathbb{R}\setminus{\left\{0\right\}}$ be given. There exist a unique $f\in\mathcal{A}(\mathbb{T}_{s})$ of zero average and a unique $\mu\in\mathbb{R}$ such that the following is satisfied:

$$\mu+af(\theta+2\pi\alpha)-bf(\theta)=g(\theta),\quad\mu=\frac{1}{2\pi}\int_{\mathbb{T}}g(\theta)d\theta.$$

(B.1)

In particular, $f$ satisfies

$$\left|f\right|_{s}\leqslant\frac{C}{\gamma\sigma^{\tau+1}}\left|g\right|_{s+\sigma},$$

$C$ being a constant depending on $\tau$ .

The proof is classical and we omit it. We refer the reader interested in optimal estimates to [26].