Global Effect of Non-Conservative Perturbations on Homoclinic Orbits

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.

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

$$\begin{aligned} \phi (t)\leqslant \delta _0+\delta _1(t-t_0)+\delta _2\int _{t_0}^{t}\phi (s)ds \end{aligned}$$

(A.1)

then

$$\begin{aligned} \phi (t)\leqslant \left( \delta _0+\frac{\delta _1}{\delta _2} \right) e^{\delta _2(t-t_0)}-\frac{\delta _1}{\delta _2}. \end{aligned}$$

(A.2)

For a reference, see, e.g., [35].

Lemma A.2

Consider the following differential equations:

$$\begin{aligned} {\dot{z}}(t)= & {} {\mathcal {X}}^0(z,t) \end{aligned}$$

(A.3)

$$\begin{aligned} {\dot{z}}(t)= & {} {\mathcal {X}}^0(z,t)+\varepsilon {\mathcal {X}}^1(z,t;\varepsilon ) \end{aligned}$$

(A.4)

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

$$\begin{aligned} \Vert z_0(t_0)-z_\varepsilon (t_0)\Vert <c\varepsilon . \end{aligned}$$

(A.5)

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

$$\begin{aligned} \Vert z_0(t)-z_\varepsilon (t)\Vert < K\varepsilon ^{\varrho _0}, \text { for } 0\leqslant t-t_0\leqslant k\ln (1/\varepsilon ). \end{aligned}$$

(A.6)

Proof

For $z_0$ and $z_\varepsilon$ solutions of (A.3) and (A.4), respectively, we have

$$\begin{aligned} z_0(t)= & {} z_0(t_0)+\int _{t_0}^{t}{\mathcal {X}}^0(z_0(s),s)ds, \end{aligned}$$

(A.7)

$$\begin{aligned} z_\varepsilon (t)= & {} z_\varepsilon (t_0)+\int _{t_0}^{t}{\mathcal {X}}^0(z_\varepsilon (s),s)ds+\varepsilon \int _{t_0}^{t_1}{\mathcal {X}}^1(z_\varepsilon (s),s;\varepsilon )ds. \end{aligned}$$

(A.8)

Subtracting, we obtain

$$\begin{aligned} \Vert z_\varepsilon (t)-z_0(t)\Vert&\leqslant \Vert z_\varepsilon (t_0)-z_0(t_0)\Vert +\int _{t_0}^{t}\Vert {\mathcal {X}}^0(z_\varepsilon (s),s)-{\mathcal {X}}^0(z_0(s),s)\Vert ds\nonumber \\&\quad +\varepsilon \int _{t_0}^{t}\Vert {\mathcal {X}}^1(z_\varepsilon (s),s;\varepsilon )\Vert ds. \end{aligned}$$

(A.9)

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:

$$\begin{aligned} \Vert z_\varepsilon (t)-z_0(t)\Vert&\leqslant c\varepsilon +C_0\int _{t_0}^{t}\Vert z_\varepsilon (s)-z_0(s)\Vert ds\nonumber \\&\quad +\varepsilon C_1(t-t_0). \end{aligned}$$

(A.10)

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

$$\begin{aligned} \Vert z_\varepsilon (t)-z_0(t)\Vert&\leqslant \varepsilon \left( c+\frac{C_1}{C_0}\right) e^{C_0(t-t_0)}-\varepsilon \frac{C_1}{C_0}\nonumber \\&\leqslant \varepsilon K e^{C_0(t-t_0)}. \end{aligned}$$

(A.11)

If we let $0\leqslant t-t_0\leqslant k\ln (1/\varepsilon )$ we obtain

$$\begin{aligned} \begin{aligned} \Vert z_\varepsilon (t)-z_0(t)\Vert&\leqslant \varepsilon \left( c+\frac{C_1}{C_0}\right) e^{C_0(t-t_0)}-\varepsilon \frac{C_1}{C_0}\\&\leqslant \varepsilon K e^{C_0k\ln (1/\varepsilon )}\\ {}&=\varepsilon K \left( \frac{1}{\varepsilon }\right) ^{C_0 k}. \end{aligned}\end{aligned}$$

(A.12)

Since $k\leqslant \frac{1-\varrho }{C_0}$ we conclude

$$\begin{aligned} \begin{aligned} \Vert z_\varepsilon (t)-z_0(t)\Vert&\leqslant \varepsilon K \left( \frac{1}{\varepsilon }\right) ^{1-\varrho }=K\varepsilon ^{\varrho }. \end{aligned}\end{aligned}$$

(A.13)

$\square$

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 $O(\varepsilon )$.