Abstract
We study the effect of time-dependent, non-conservative perturbations on the dynamics along homoclinic orbits to a normally hyperbolic invariant manifold. We assume that the unperturbed system is Hamiltonian, and the normally hyperbolic invariant manifold is parametrized via action-angle coordinates. The homoclinic excursions can be described via the scattering map, which gives the future asymptotic of an orbit as a function of its past asymptotic. We provide explicit formulas, in terms of convergent integrals, for the perturbed scattering map expressed in action-angle coordinates. We illustrate these formulas for perturbations of both uncoupled and coupled rotator-pendulum systems.
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Notes
\(g(u,v)=\omega (u,Jv)\).
An example is given by \(h_0(I)=I^n\) with \(n\geqslant 3\) odd, or \(h_0(I)=I_1^2 - I_2^2\).
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To the memory of Florin Diacu.
M. Gidea: Research of M.G. was partially supported by NSF Grant DMS-1814543. R. de la Llave: Research of R.L. was partially supported by NSF Grant DMS-1800241, and H2020-MCA-RISE #734577. M. Musser: Research of M.M. was partially supported by NSF Grant DMS-1814543.
Appendix A. Gronwall’s Inequality
Appendix A. Gronwall’s Inequality
In this section we apply Gronwall’s Inequality to estimate the error between the solution of an unperturbed system and the solution of the perturbed system, over a time of logarithmic order with respect to the size of the perturbation.
Theorem A.1
(Gronwall’s Inequality) Given a continuous real valued function \(\phi \geqslant 0\), and constants \(\delta _0,\delta _1\geqslant 0\), \(\delta _2>0\), if
then
For a reference, see, e.g., [35].
Lemma A.2
Consider the following differential equations:
Assume that \({\mathcal {X}}^0\) is Lipschitz continuous in the variable z, \(C_0\) is the Lipschitz constant of \({\mathcal {X}}^0\), and \({\mathcal {X}}^1\) is bounded with \(\Vert {\mathcal {X}}^1\Vert _{{\mathscr {C}}^0}\leqslant C_1\), for some \(C_0,C_1>0\). Let \(z_0\) be a solution of the equation (A.3) and \(z_\varepsilon \) be a solution of the equation (A.4) such that
Then, for \(0<\varrho _0<1\), \(0<k\leqslant \frac{1-{\varrho _0}}{C_0}\), and \(K=c+\frac{C_1}{C_0}\), we have
Proof
For \(z_0\) and \(z_\varepsilon \) solutions of (A.3) and (A.4), respectively, we have
Subtracting, we obtain
Using (A.5) for the first term on the right-hand side, the Lipschitz condition on \({\mathcal {X}}^0\) for the second, and the boundedness of \({\mathcal {X}}^1\) for the third we obtain:
Applying the Gronwall inequality for \(\delta _0=c\), \(\delta _1=\varepsilon C_1\), and \(\delta _2=C_0\), and recalling that \(K=c+\frac{C_1}{C_0}\) we obtain
If we let \(0\leqslant t-t_0\leqslant k\ln (1/\varepsilon )\) we obtain
Since \(k\leqslant \frac{1-\varrho }{C_0}\) we conclude
\(\square \)
We note that, with the above argument, for a time of logarithmic order with respect to the size of the perturbation, we can only obtain an error of order \(O(\varepsilon ^{\varrho })\) with \(0<\rho <1\), but we cannot obtain an error of order \(O(\varepsilon )\).
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Gidea, M., de la Llave, R. & Musser, M. Global Effect of Non-Conservative Perturbations on Homoclinic Orbits. Qual. Theory Dyn. Syst. 20, 9 (2021). https://doi.org/10.1007/s12346-020-00431-z
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DOI: https://doi.org/10.1007/s12346-020-00431-z