Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems.

*(English)*Zbl 1086.37031The paper proposes generalizations in the scale of the Gevrey spaces \(G^\alpha\), \(\alpha \geq 1\) of the celebrated Nekhoroshev Theorem (the effective stability) for analytic near integrable Hamiltonian systems [cf. N. N. Nekhoroshev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Usp. Mat. Nauk 32, No. 6, 5–66 (1977; Zbl 0383.70023)], by means of a novel functional analytic approach.

Although the statement of the problem is easy to announce, namely, to replace the real-analytic perturbation (corresponding to the Gevrey index \(\alpha =1\)) with a nonanalytic Gevrey \(G^\alpha\), \(\alpha >1\), perturbation of the integrable initial Hamiltonian system, the resolution of this problem requires subtle functional-analytic techniques. In fact, the paper contributes not only to the theory of Hamiltonian systems, but it has a high value “per se” in the context of the abstract functional analysis in the framework of Gevrey spaces.

The starting point is a near-integrable Hamiltonian system with Hamiltonian functions of the type \(H (\theta, r) =h(r) + f(\theta, r)\), where \((r,\theta)\) are the action-angle variables, \(\theta\in \mathbb T^n= \mathbb R^n/(\mathbb Z^n)\), \(r\in \overline{B_R}\), \(\overline{B_R}\) being the closed ball of radius \(R>0\) centered at the origin, \(h\in G^\alpha (\mathbb T^n)\), \(f\in G^\alpha (\mathbb T^n \times \overline{B_R})\), for some \(\alpha \geq 1\). Here \(G^\alpha (K)\) stands for the Gevrey space of index \(\alpha\). These spaces have been introduced by M. Gevrey in 1918 in order to obtain precise estimates of the regularity of solutions of PDEs. There are many equivalent definitions of the space \(G^\alpha (K)\) [e.g., cf. L. Rodino, Linear partial differential operators in Gevrey spaces (World Scientific) (1993; Zbl 0869.35005)].

The authors smart choice is the following one: a function \(f\in G^\alpha (K)\) iff there exists \(L>0\) such that \(f\) belongs to the Banach space \(G^{\alpha, L} (K)\), namely \[ \| u\|_{\alpha, L} = \sum_{k\in \mathbb N_+^{2n} } \frac{L^{\alpha |k| }}{k!^\alpha} \| \varphi \|_{C^0 (K) } < +\infty. \] Clearly, \(G^{\alpha, L} (K) \hookrightarrow G^{\alpha, L'} (K)\) if \(0<L'<L\), and in the case \(\alpha = 1\) one recaptures the space of the real-analytic functions on \(K\) The spaces \(G^{\alpha, L}(K)\) are Banach algebras and they are preserved by Gevrey \(G^\alpha\) maps under suitable conditions, as a consequence from classical and nontrivial results on analysis in spaces of ultradifferentiable functions (the Gevrey classes being a particular case) [cf. C. Roumieu, J. Anal. Math. 10, 153–192 (1962; Zbl 0122.34802)].

The choice of such norms, although a purely technical point, turns out to be one of the crucial ingredients in the proofs.

The function \(h\) is required to be quasi-convex, namely, there exists \(m>0\) such that \( < \nabla^2 h(r) v, v> \geq m \| v \|^2 \) for all \(r\in B_R\) and all vectors \(v\in \mathbb R^n\) orthogonal to \(\nabla h(r)\). The quasi-convexity is an important particular case satisfying the Nekhoroshev hypothesis of steepness of \(h\).

The first main result could be summarized as follows:

Theorem 1. Let \(\alpha \geq 1\), \(n\geq 2\). Set \[ a = \frac{1}{2n\alpha}, \qquad b = \frac{1}{2n}. \] Then for every \(R_0\in ]0,R[\), \(L>0\) one can find positive constants \( \varepsilon_0\),\(C_1,C_2\) depending on \(\alpha, R_0, L, \| f\|_{\alpha, L}\) such that for every Hamiltonian \(H\in G^\alpha \) satisfying \[ \varepsilon:= \|H - h \|_{\alpha, L}= \| f \|_{\alpha,L} \leq \varepsilon_0 \] any initial condition \((\theta_0, r_0)\in \mathbb T^n \times \overline{ B_{R_0} }\), gives rise to an integral curve \((\theta(t), r(t))\) of the Hamiltonian vector field \(X_H\) defined at least for \(t\in J_\varepsilon\), \(J_\varepsilon\) being defined by \(|t|\leq C_1\exp ( \alpha (\frac{\varepsilon_0}{\varepsilon})^a ) \) and satisfies \[ \sup_{t\in J_\varepsilon} \| r(t) - r_0 \| \leq C_2 \varepsilon^b. \]

In addition, the authors give explicit estimates of the aforementioned constants \(\varepsilon_0\), \(C_1,C_2\).

If \(\alpha =1\), Theorem 1 recaptures the result for the exponents \(a=b=\frac{1}{2n}\) in the Nekhoroshev theorem shown independently by P. Lochak and A. I. Neishtadt [Chaos 2, No. 4, 495–499 (1992; Zbl 1055.37573)], P. Lochak, A. I. Neistadt and L. Niederman [Seminar on Dynamical Systems (St. Petersburg, 1991), Prog. Nonlinear Differ. Equ. Appl. 12, 15–34 (1994; Zbl 0807.70020)], and J. Pöschel [Math. Z. 213, No. 2, 187–216 (1993; Zbl 0857.70009)].

The authors propose also an improvement of the Nekhoroshev type theorems near a \(d\)-codimensional resonant surface \(S_{\mathcal M} \), \(d\in \{ 1, \ldots, n \}\), where \({\mathcal M }\) is a submodule of codimension \(d\) of \(\mathbb Z^n\) and \[ S_{\mathcal M}= \{ r\in B_{r_0}:\, k_1\partial_{r_1}h(r) +\ldots + k_n \partial_{r_n}h(r) =0, \, \forall k=(k_1,\ldots, k_n)\in \mathbb Z^n \}. \] If \(d=n\), by convention \({\mathcal M}= \{ 0\} \) and one recovers the nonresonance case.

The generalization for integral curves starting near resonant surface could be stated as follows.

Theorem 2. Let \(d\in \{1,\ldots, n-1\}\) and set \[ a = \frac{1}{2d\alpha}, \qquad b = \frac{1}{2dn}. \] Then for every \(\sigma >0\), \(L>0\), \(R_0>0\) there exist positive constants \( \varepsilon_0\), \(C_1,C_2\) depending on \(\alpha\), \(R_0\), \(L\), \(\| f\|_{\alpha, L}\), \(\sigma\) and some constants assciated to \(S_{\mathcal M}\) such that if \[ \varepsilon:= \|H - h \|_{\alpha, L}= \| f \|_{\alpha,L} \leq \varepsilon_0 \] and in addition one assumes \(\text{ dist} ( r_0, S_{\mathcal M}) \leq \sigma \varepsilon^{1/2}\), then \((\theta_0, r_0)\) gives rise to an integral curve \((\theta(t), r(t))\) of the Hamiltonian vector field \(X_H\) defined at least for \(|t|\leq C_1\exp ( \alpha (\frac{\varepsilon_0}{\varepsilon})^a ) \) and \(\| r(t) - r_0 \| \leq C_2 \varepsilon^b\) for such \(t\).

The proofs of Theorem 1 and Theorem 2 rely on an adaptation of the Lochak’s periodic orbit method [cf. P. Lochak, Usp. Mat. Nauk 47, 59–140 (1992; Zbl 0795.58042)] in the framework of Gevrey spaces \(G^\alpha\). As a byproduct of the proofs, new results on approximate Gevrey resonant normal forms are obtained as well as a generalization of A. I. Neishtadt’s [Prikl. Mat. Mekh. 48, 197–204 (1984; Zbl 0571.70022)] in the scales of Gevrey spaces \(G^\alpha\).

If one lets \(\varepsilon \searrow 0\), the size of the effective stability interval \(J_\varepsilon\) is estimated from below by \(\exp (c\varepsilon^{-1/(2n \alpha)}\). The explicit dependence on the Gevrey index \(\alpha\) exhibits nicely the intuitive idea that a better regularity (i.e., \(\alpha \searrow 1\)) leads to better estimates for the length of the effective stability interval while for less regular (i.e. \(\alpha \gg 1\)) the estimates are weaker. These beautiful estimates, in reviewer’s opinion, might be viewed as a unified formula generalizing the original Nekhoroshev theorem for quasi-convex \(G^\alpha\) hamiltonians depending, at least for \(\varepsilon \searrow 0\) essentially only on the Gevrey index \(\alpha\).

Next, the authors dwell upon the possible instability phenomena beyond the time of effective stability. Given \(\alpha >1\), they propose a family of examples of \(G^\alpha\) Hamilton= ian systems for which optimal estimates of the times of drift are shown.

Theorem 3. Let \(n\geq 3\), \(\alpha>1\), \(L>0\), \(R>1\), and set \[ a^*= \frac{1}{2(n-2)\alpha }. \] Then there exist a sequence \(\{f_j \}_{j=1}^\infty \subset G^\alpha (\mathbb T^n\times \overline{B_R})\) such that \( \varepsilon_j := \| f_j\|_{\alpha,L}\rightarrow 0\) for \(j\rightarrow \infty\) and an increasing sequence of positive integers \(\tau_j\) such that and for each \(j\) the Hamiltonian system generated by \[ H_j(\theta,r)=\frac{1}{2}\sum_{k=1}^{n-1}r_k^2+r_n +f_j(\theta,r) \] admits a solution \((\theta(t),r(t))\) defined at least for \(t\in [0,\tau_j]\) with \(r_1(0)=0\), \(r_1(\tau_j)=1\). Moreover, there exists a positive integer \(J\) depending only on \(n\), and positive constants \(C_1 <C_2\) depending only on \(n\), \(\alpha\), \(L\) and \(R\), such that \[ \frac{C_1}{\varepsilon_j^2} \exp ( C_1 \left(\frac{1}{\varepsilon_j}\right)^{a^*} ) \leq\tau_j \leq \frac{C_2}{\varepsilon_j^2} \exp ( C_2 \left(\frac{1}{\varepsilon_j}\right)^{a^*} ) , \] for all \(j\geq J\).

The instability result of Theorem 3 implies that for \(\alpha>1\) and \(d =(n-2)\), the value \(a=1/(2d\alpha)\) of the stability exponent \(a\) is optimal.

The proof of Theorem 3 relies on a suspension of explicit discrete systems defined on \(\mathbb T^{n-1}\times \mathbb R^{n-1}\) and on technical but subtle constructions in the framework of Gevrey spaces \(G^\alpha\) with \(\alpha >1\) of suitable perturbations of integrable symplectic mappings taking an advantage of the existence of \(G^\alpha\) cut-off functions provided \(\alpha >1\).

Finally, the authors obtain as a byproduct from the detailed analysis of their examples in Theorem 3 a new insight in Arnold’s universal mechanism of instability [cf. V. I. Arnold, Dokl. Akad. Nauk SSSR 156, 9–12 (1964; Zbl 0135.42602)].

The paper also contains a well written appendix containing useful estimates involving Gevrey norms of the type infinite sums (similar type of Gevrey norms have been used for simultaneous Gevrey normal forms in [T. Gramchev and M. Yoshino, Math. Z. 231, No. 4, 745–770 (1999; Zbl 0931.65055)].

The authors have dedicated the article to the memory of M. R. Herman. The reviewer had the occasion to meet M. R. Hermann in October 1998 and to hear some ideas for, broadly speaking, bringing the Gevrey spaces in the realm of the dynamical systems as intermediate classes between the analytic category and the space of the smooth \(C^\infty\) functions. At that time the use of the Gevrey classes, in contrast with the theory of PDEs, was rather limited. Recently there is great interest in and a lot of research on investigating dynamical systems in the Gevrey category: e.g., cf. G. Popov, Ann. Henri Poincaré 1, 223–248 (2000; Zbl 0970.37050), II. Quantum Birkhoff normal forms, Ann. Henri Poincaré 1, 249–279 (2000; Zbl 1002.37028); D. Dickinson, T. Gramchev and M. Yoshino, Proc. Edinb. Math. Soc. (2) 45, No. 3, 731–759 (2002; Zbl 1032.37010); F. Wagener, Dyn. Syst. 18, No. 2, 159–163 (2003; Zbl 1036.37022), as well as papers of the authors of the present article.

In reviewer’s opinion, the paper under review is one of the fundamental papers in this field.

Although the statement of the problem is easy to announce, namely, to replace the real-analytic perturbation (corresponding to the Gevrey index \(\alpha =1\)) with a nonanalytic Gevrey \(G^\alpha\), \(\alpha >1\), perturbation of the integrable initial Hamiltonian system, the resolution of this problem requires subtle functional-analytic techniques. In fact, the paper contributes not only to the theory of Hamiltonian systems, but it has a high value “per se” in the context of the abstract functional analysis in the framework of Gevrey spaces.

The starting point is a near-integrable Hamiltonian system with Hamiltonian functions of the type \(H (\theta, r) =h(r) + f(\theta, r)\), where \((r,\theta)\) are the action-angle variables, \(\theta\in \mathbb T^n= \mathbb R^n/(\mathbb Z^n)\), \(r\in \overline{B_R}\), \(\overline{B_R}\) being the closed ball of radius \(R>0\) centered at the origin, \(h\in G^\alpha (\mathbb T^n)\), \(f\in G^\alpha (\mathbb T^n \times \overline{B_R})\), for some \(\alpha \geq 1\). Here \(G^\alpha (K)\) stands for the Gevrey space of index \(\alpha\). These spaces have been introduced by M. Gevrey in 1918 in order to obtain precise estimates of the regularity of solutions of PDEs. There are many equivalent definitions of the space \(G^\alpha (K)\) [e.g., cf. L. Rodino, Linear partial differential operators in Gevrey spaces (World Scientific) (1993; Zbl 0869.35005)].

The authors smart choice is the following one: a function \(f\in G^\alpha (K)\) iff there exists \(L>0\) such that \(f\) belongs to the Banach space \(G^{\alpha, L} (K)\), namely \[ \| u\|_{\alpha, L} = \sum_{k\in \mathbb N_+^{2n} } \frac{L^{\alpha |k| }}{k!^\alpha} \| \varphi \|_{C^0 (K) } < +\infty. \] Clearly, \(G^{\alpha, L} (K) \hookrightarrow G^{\alpha, L'} (K)\) if \(0<L'<L\), and in the case \(\alpha = 1\) one recaptures the space of the real-analytic functions on \(K\) The spaces \(G^{\alpha, L}(K)\) are Banach algebras and they are preserved by Gevrey \(G^\alpha\) maps under suitable conditions, as a consequence from classical and nontrivial results on analysis in spaces of ultradifferentiable functions (the Gevrey classes being a particular case) [cf. C. Roumieu, J. Anal. Math. 10, 153–192 (1962; Zbl 0122.34802)].

The choice of such norms, although a purely technical point, turns out to be one of the crucial ingredients in the proofs.

The function \(h\) is required to be quasi-convex, namely, there exists \(m>0\) such that \( < \nabla^2 h(r) v, v> \geq m \| v \|^2 \) for all \(r\in B_R\) and all vectors \(v\in \mathbb R^n\) orthogonal to \(\nabla h(r)\). The quasi-convexity is an important particular case satisfying the Nekhoroshev hypothesis of steepness of \(h\).

The first main result could be summarized as follows:

Theorem 1. Let \(\alpha \geq 1\), \(n\geq 2\). Set \[ a = \frac{1}{2n\alpha}, \qquad b = \frac{1}{2n}. \] Then for every \(R_0\in ]0,R[\), \(L>0\) one can find positive constants \( \varepsilon_0\),\(C_1,C_2\) depending on \(\alpha, R_0, L, \| f\|_{\alpha, L}\) such that for every Hamiltonian \(H\in G^\alpha \) satisfying \[ \varepsilon:= \|H - h \|_{\alpha, L}= \| f \|_{\alpha,L} \leq \varepsilon_0 \] any initial condition \((\theta_0, r_0)\in \mathbb T^n \times \overline{ B_{R_0} }\), gives rise to an integral curve \((\theta(t), r(t))\) of the Hamiltonian vector field \(X_H\) defined at least for \(t\in J_\varepsilon\), \(J_\varepsilon\) being defined by \(|t|\leq C_1\exp ( \alpha (\frac{\varepsilon_0}{\varepsilon})^a ) \) and satisfies \[ \sup_{t\in J_\varepsilon} \| r(t) - r_0 \| \leq C_2 \varepsilon^b. \]

In addition, the authors give explicit estimates of the aforementioned constants \(\varepsilon_0\), \(C_1,C_2\).

If \(\alpha =1\), Theorem 1 recaptures the result for the exponents \(a=b=\frac{1}{2n}\) in the Nekhoroshev theorem shown independently by P. Lochak and A. I. Neishtadt [Chaos 2, No. 4, 495–499 (1992; Zbl 1055.37573)], P. Lochak, A. I. Neistadt and L. Niederman [Seminar on Dynamical Systems (St. Petersburg, 1991), Prog. Nonlinear Differ. Equ. Appl. 12, 15–34 (1994; Zbl 0807.70020)], and J. Pöschel [Math. Z. 213, No. 2, 187–216 (1993; Zbl 0857.70009)].

The authors propose also an improvement of the Nekhoroshev type theorems near a \(d\)-codimensional resonant surface \(S_{\mathcal M} \), \(d\in \{ 1, \ldots, n \}\), where \({\mathcal M }\) is a submodule of codimension \(d\) of \(\mathbb Z^n\) and \[ S_{\mathcal M}= \{ r\in B_{r_0}:\, k_1\partial_{r_1}h(r) +\ldots + k_n \partial_{r_n}h(r) =0, \, \forall k=(k_1,\ldots, k_n)\in \mathbb Z^n \}. \] If \(d=n\), by convention \({\mathcal M}= \{ 0\} \) and one recovers the nonresonance case.

The generalization for integral curves starting near resonant surface could be stated as follows.

Theorem 2. Let \(d\in \{1,\ldots, n-1\}\) and set \[ a = \frac{1}{2d\alpha}, \qquad b = \frac{1}{2dn}. \] Then for every \(\sigma >0\), \(L>0\), \(R_0>0\) there exist positive constants \( \varepsilon_0\), \(C_1,C_2\) depending on \(\alpha\), \(R_0\), \(L\), \(\| f\|_{\alpha, L}\), \(\sigma\) and some constants assciated to \(S_{\mathcal M}\) such that if \[ \varepsilon:= \|H - h \|_{\alpha, L}= \| f \|_{\alpha,L} \leq \varepsilon_0 \] and in addition one assumes \(\text{ dist} ( r_0, S_{\mathcal M}) \leq \sigma \varepsilon^{1/2}\), then \((\theta_0, r_0)\) gives rise to an integral curve \((\theta(t), r(t))\) of the Hamiltonian vector field \(X_H\) defined at least for \(|t|\leq C_1\exp ( \alpha (\frac{\varepsilon_0}{\varepsilon})^a ) \) and \(\| r(t) - r_0 \| \leq C_2 \varepsilon^b\) for such \(t\).

The proofs of Theorem 1 and Theorem 2 rely on an adaptation of the Lochak’s periodic orbit method [cf. P. Lochak, Usp. Mat. Nauk 47, 59–140 (1992; Zbl 0795.58042)] in the framework of Gevrey spaces \(G^\alpha\). As a byproduct of the proofs, new results on approximate Gevrey resonant normal forms are obtained as well as a generalization of A. I. Neishtadt’s [Prikl. Mat. Mekh. 48, 197–204 (1984; Zbl 0571.70022)] in the scales of Gevrey spaces \(G^\alpha\).

If one lets \(\varepsilon \searrow 0\), the size of the effective stability interval \(J_\varepsilon\) is estimated from below by \(\exp (c\varepsilon^{-1/(2n \alpha)}\). The explicit dependence on the Gevrey index \(\alpha\) exhibits nicely the intuitive idea that a better regularity (i.e., \(\alpha \searrow 1\)) leads to better estimates for the length of the effective stability interval while for less regular (i.e. \(\alpha \gg 1\)) the estimates are weaker. These beautiful estimates, in reviewer’s opinion, might be viewed as a unified formula generalizing the original Nekhoroshev theorem for quasi-convex \(G^\alpha\) hamiltonians depending, at least for \(\varepsilon \searrow 0\) essentially only on the Gevrey index \(\alpha\).

Next, the authors dwell upon the possible instability phenomena beyond the time of effective stability. Given \(\alpha >1\), they propose a family of examples of \(G^\alpha\) Hamilton= ian systems for which optimal estimates of the times of drift are shown.

Theorem 3. Let \(n\geq 3\), \(\alpha>1\), \(L>0\), \(R>1\), and set \[ a^*= \frac{1}{2(n-2)\alpha }. \] Then there exist a sequence \(\{f_j \}_{j=1}^\infty \subset G^\alpha (\mathbb T^n\times \overline{B_R})\) such that \( \varepsilon_j := \| f_j\|_{\alpha,L}\rightarrow 0\) for \(j\rightarrow \infty\) and an increasing sequence of positive integers \(\tau_j\) such that and for each \(j\) the Hamiltonian system generated by \[ H_j(\theta,r)=\frac{1}{2}\sum_{k=1}^{n-1}r_k^2+r_n +f_j(\theta,r) \] admits a solution \((\theta(t),r(t))\) defined at least for \(t\in [0,\tau_j]\) with \(r_1(0)=0\), \(r_1(\tau_j)=1\). Moreover, there exists a positive integer \(J\) depending only on \(n\), and positive constants \(C_1 <C_2\) depending only on \(n\), \(\alpha\), \(L\) and \(R\), such that \[ \frac{C_1}{\varepsilon_j^2} \exp ( C_1 \left(\frac{1}{\varepsilon_j}\right)^{a^*} ) \leq\tau_j \leq \frac{C_2}{\varepsilon_j^2} \exp ( C_2 \left(\frac{1}{\varepsilon_j}\right)^{a^*} ) , \] for all \(j\geq J\).

The instability result of Theorem 3 implies that for \(\alpha>1\) and \(d =(n-2)\), the value \(a=1/(2d\alpha)\) of the stability exponent \(a\) is optimal.

The proof of Theorem 3 relies on a suspension of explicit discrete systems defined on \(\mathbb T^{n-1}\times \mathbb R^{n-1}\) and on technical but subtle constructions in the framework of Gevrey spaces \(G^\alpha\) with \(\alpha >1\) of suitable perturbations of integrable symplectic mappings taking an advantage of the existence of \(G^\alpha\) cut-off functions provided \(\alpha >1\).

Finally, the authors obtain as a byproduct from the detailed analysis of their examples in Theorem 3 a new insight in Arnold’s universal mechanism of instability [cf. V. I. Arnold, Dokl. Akad. Nauk SSSR 156, 9–12 (1964; Zbl 0135.42602)].

The paper also contains a well written appendix containing useful estimates involving Gevrey norms of the type infinite sums (similar type of Gevrey norms have been used for simultaneous Gevrey normal forms in [T. Gramchev and M. Yoshino, Math. Z. 231, No. 4, 745–770 (1999; Zbl 0931.65055)].

The authors have dedicated the article to the memory of M. R. Herman. The reviewer had the occasion to meet M. R. Hermann in October 1998 and to hear some ideas for, broadly speaking, bringing the Gevrey spaces in the realm of the dynamical systems as intermediate classes between the analytic category and the space of the smooth \(C^\infty\) functions. At that time the use of the Gevrey classes, in contrast with the theory of PDEs, was rather limited. Recently there is great interest in and a lot of research on investigating dynamical systems in the Gevrey category: e.g., cf. G. Popov, Ann. Henri Poincaré 1, 223–248 (2000; Zbl 0970.37050), II. Quantum Birkhoff normal forms, Ann. Henri Poincaré 1, 249–279 (2000; Zbl 1002.37028); D. Dickinson, T. Gramchev and M. Yoshino, Proc. Edinb. Math. Soc. (2) 45, No. 3, 731–759 (2002; Zbl 1032.37010); F. Wagener, Dyn. Syst. 18, No. 2, 159–163 (2003; Zbl 1036.37022), as well as papers of the authors of the present article.

In reviewer’s opinion, the paper under review is one of the fundamental papers in this field.

Reviewer: Todor Gramchev