% paper 3 by Z.Parsa, M.Zeitlin, A.Fedorova
% PAC97, Vancouver Canada 1997
%********************************************
\documentclass[a4paper]{pac97}
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\begin{document}
\title{ WAVELET APPROACH TO ACCELERATOR  PROBLEMS, III.\\
MELNIKOV FUNCTIONS AND SYMPLECTIC TOPOLOGY}
\author{A.~Fedorova and  M.~Zeitlin,
Institute of Problems of Mechanical Engineering,\\
 Russian Academy of Sciences, Russia,199178, St.~Petersburg,\\
  V.O., Bolshoj pr., 61, e-mail: zeitlin@math.ipme.ru\\
Z.~Parsa, Dept. of Physics, Bldg.~901A, Brookhaven National Laboratory,\\
 Upton, NY 11973-5000, USA, e-mail: parsa@bnl.gov}
\maketitle
\begin{abstract}
This is the third part of a series of talks in which we
present applications of methods of wavelet
analysis to polynomial approximations for a number of accelerator
physics problems.
 We consider the generalization of our variational wavelet
approach to nonlinear polynomial problems to the case of
Hamiltonian systems for which we need to preserve underlying
symplectic or Poissonian or quasicomplex structures in any
type of calculations. We use our approach for the problem of
explicit calculations of Arnold--Weinstein curves via
Floer variational approach from symplectic topology.
The loop solutions are parametrized by the solutions of reduced algebraical
problem -- matrix Quadratic Mirror Filters equations.
Also we consider wavelet approach to the calculations of Melnikov
functions in the theory of homoclinic chaos in perturbed
Hamiltonian systems.
\end{abstract}
\section{Introduction}
In this paper we continue the application of powerful methods of wavelet
analysis to polynomial approximations of nolinear accelerator
physics problems. In part I we considered our main example and
general approach for constructing wavelet representation for
orbital motion in storage rings.
Now we consider two problems of nontrivial dynamics
related with complicated differential geometrical and symplectic topological
structures of
system (1) from part I. In section 2 we give some points of applications
of wavelet methods from parts I, II to Melnikov approach in the theory of
homoclinic chaos in perturbed Hamiltonian systems. In section 3
we consider another type of wavelet approach, which gives a possibility
to parametrize Arnold--Weinstein curves or closed loops in Hamiltonian systems
by  generalized refinement equations or
Quadratic Mirror Filters equations.

\section{Routes to chaos}
  Now we give some points of our program of
 understanding routes to  chaos in some Hamiltonian
systems in the wavelet approach [1]-[9]. All points are:
\begin{enumerate}
 \item A model.
 \item A computer zoo. The understanding of
the  computer zoo.
 \item  A naive Melnikov function approach.
 \item A naive wavelet description of (hetero) homoclinic orbits
(separatrix) and quasiperiodic oscillations.
 \item Symplectic Melnikov function approach.
 \item Splitting of separatrix... $\longrightarrow$stochastic web
 with magic symmetry, Arnold diffusion and all
that.
\end{enumerate}
1. As a model we have two frequencies perturbations of particular
case of system (1) from part I:
\begin{eqnarray*}
\dot x_1&=&x_2      \nonumber\\
\dot x_2&=&-ax_1-b[\cos(rx_5)+\cos(sx_6)]x_1-\\ & &dx^3_1-
    mdx_1x^2_3-
    px_2-\varphi(x_5)  \nonumber\\
 \dot x_3&=&x_4 \nonumber \\
\dot x_4&=&ex_3-f[\cos(rx_5)+\cos(sx_6)]- gx_3^3-\\
 & & kx_1^2x_3-gx_4-
   \psi(x_5) \nonumber \\
\dot x_5&=&1 \nonumber\\
\dot x_6&=&1\nonumber\
\end{eqnarray*}
or in  Hamiltonian form
\begin{eqnarray*}
\dot{x}&=&J\cdot\nabla H(x)+\varepsilon g(x,\Theta), \\
\dot{\Theta}&=&\omega,\quad
(x,\Theta)\in R^4\times T^2,\\
T^2&=&S^1\times S^1,
\end{eqnarray*}
for $\varepsilon=0 $ we have:
\begin{equation}
\dot{x}=J\cdot\nabla H(x),\quad
\dot\Theta=\omega
\end{equation}
2. For pictures and details one can see [3], [8].
The key point is the
splitting of separatrix (homoclinic orbit) and transition to
fractal sets on the Poincare sections.\\
3. For $\varepsilon=0$ we
have homoclinic orbit $\bar{x}_{0}(t)$ to the hyperbolic fixed
point $x_0$. For $\varepsilon\neq 0$ we have normally hyperbolic
invariant torus $T_{\varepsilon}$ and condition on transversally
intersection of stable and unstable ma\-ni\-folds
$W^s(T_{\varepsilon})$ and $W^u(T_{\varepsilon})$ in terms of
Melnikov functions $M(\Theta)$ for $\bar{x}_{0}(t)$.  $$
M(\Theta)=\displaystyle\int\limits_{-\infty}^{\infty}\nabla
H(\bar{x}_{0}(t)) \wedge g(\bar{x}_{0}(t),\omega t+\Theta)dt
$$
This condition has the next form:
\begin{eqnarray}
M(\Theta_0)=0 \nonumber\\
\sum\limits_{j=1}^{2}\omega_j\frac{\partial}{\partial\Theta_j}
M(\Theta_0)\neq0  \nonumber
\end{eqnarray}
According to the approach of Birkhoff-Smale-Wiggins  we
determined the region in parameter space in which we observe the
chaotic behaviour [3], [8].\\
4. If we cannot solve equations (1)
explicitly in time, then we use the wavelet approach from part I
for the computations of  homoclinic (heteroclinic) loops as
the wavelet solutions of system (1).
For computations of quasiperiodic Melnikov functions
$$
M^{m/n}(t_0)=\int^{mT}_0 DH(x_\alpha(t))\wedge g(x_\alpha(t),t+t_0)dt
$$
  we used periodization of wavelet solution from part I.\\
5. We also used symplectic Melnikov function approach
\begin{eqnarray*}
M_i(z)&=&\displaystyle\lim_{j\rightarrow\infty}\int\limits_{-T_j^*}
^{T_j}\{h_i,\hat{h}\}_{\Psi (t,z)}dt  \\
d_i(z,\varepsilon)&=&h_i(z^u_\varepsilon)-h_i(z^s_\varepsilon)=
\varepsilon M_i(z)+O(\varepsilon^2)
\end{eqnarray*}
where $\{,\}$ is the Poisson bracket,
$d_i(z,\varepsilon)$ is the Melnikov distance. So, we need symplectic
invariant wavelet expressions for Poisson brackets. The computations
 are produced  according to part II.\\
6. Some hypothesis about
 strange symmetry of stochastic web
in multi-degree-of freedom Hamiltonian systems [9].


\section{Wavelet Parametrization in Floer Approach.}
Now we consider the generalization of our wavelet variational
approach  to the symplectic invariant calculation of
Arnold--Weinstein curves (closed loops) in Hamiltonian systems
[10]. We also have the parametrization of our solution by some
reduced algebraical problem but in contrast to the general case where
the solution is parametrized by construction based on scalar
refinement equation, in symplectic case we have
parametrization of the solution
by matrix problems -- Quadratic Mirror Filters equations [11].

The action functional for loops in the phase space is [10]
$$
F(\gamma)=\displaystyle\int_\gamma pdq-\int_0^1H(t,\gamma(t))dt
$$
The critical points of $F$ are those loops $\gamma$, which solve
the Hamiltonian equations associated with the Hamiltonian $H$
and hence are periodic orbits. By the way, all critical points of $F$ are
the saddle points of infinite Morse index, but surprisingly this approach  is
very effective. This will be demonstrated using several
variational techniques starting from minimax due to Rabinowitz
and ending with Floer homology. So, $(M,\omega)$ is symplectic
manifolds, $H: M \to R $, $H$ is Hamiltonian, $X_H$ is
unique Hamiltonian vector field defined  by
$$
\omega(X_H(x),\upsilon)=-dH(x)(\upsilon),\quad \upsilon\in T_xM,
\quad x\in M,
$$
where $ \omega$ is the symplectic structure.
A T-periodic solution $x(t)$ of the Hamiltonian equations
$$
\dot x=X_H(x) \quad \mbox{ on $M$}
$$
is a solution, satisfying the boundary conditions $x(T)$ $=x(0), T>0$.
Let us consider the loop space $\Omega=C^\infty(S^1, R^{2n})$,
where $S^1=R/{\bf Z}$, of smooth loops in $R^{2n}$.
Let us define a function $\Phi: \Omega\to R $ by setting
$$
\Phi(x)=\displaystyle\int_0^1\frac{1}{2}<-J\dot x, x>dt-
\int_0^1 H(x(t))dt, \quad x\in\Omega
$$
The critical points of $\Phi$ are the periodic solutions of $\dot x=X_H(x)$.
Computing the derivative at $x\in\Omega$ in the direction of $y\in\Omega$,
we find
\begin{eqnarray*}
\Phi'(x)(y)&=&\frac{d}{d\epsilon}\Phi(x+\epsilon y)\vert_{\epsilon=0}
=\\
&&\displaystyle\int_0^1<-J\dot x-\bigtriangledown H(x),y>dt
\end{eqnarray*}
Consequently, $\Phi'(x)(y)=0$ for all $y\in\Omega$ iff the loop $x$ satisfies
the equation
$$
-J\dot x(t)-\bigtriangledown H(x(t))=0,
$$
i.e. $x(t)$ is a solution of the Hamiltonian equations, which also satisfies
$x(0)=x(1)$, i.e. periodic of period 1. Periodic loops may be represented by
their Fourier series:
$$
x(t)=\displaystyle\sum_{k\in{\bf Z}}e^{k2\pi Jt}x_k, \quad x_k\in R^{2k},
$$
where $J$ is quasicomplex structure. We give relations between
quasicomplex structure and wavelets in part IV.
But now we use  the  construction [11]
for loop parametrization. It is based on the theorem about
explicit bijection between the Quadratic Mirror Filters (QMF) and
the whole loop group: $LG: S^1\to G$.
In particular case we have relation  between {\bf QMF}-systems and
measurable functions
$\chi: S^1 \to U(2)$ satisfying

\begin{displaymath}
\chi(\omega+\pi)=\chi(\omega)\left [ \begin{array}{ll}
                                 0 & 1\\
                                 1 & 0
                              \end{array}\right ],
\end{displaymath}
in the next explicit form
\begin{eqnarray*}
\left [ \begin{array}{ll}
          \hat\Phi_0(\omega) & \hat\Phi_0(\omega+\pi)\\
          \hat\Phi_1(\omega) & \hat\Phi_1(\omega+\pi)
         \end{array}\right ]
      &=& \chi(\omega)\left [\begin{array}{ll}
             0 & 1\\
             1 & 0
         \end{array}\right ]\\
       &+&\chi(\omega+\pi)\left [\begin{array}{ll}
             0 & 0\\
             0 & 1
           \end{array}\right ],
\end{eqnarray*}
where
$$
\left |\hat\Phi_i(\omega)\right |^2+\left |\hat\Phi_i(\omega+\pi)\right |^2=2,
\quad i=0,1.
$$

Also, we have symplectic structure
on $LG$
$$
\omega(\xi,\eta)=\frac{1}{2\pi}\int_0^{2\pi}<\xi(\theta),\eta'(\theta)>d\theta
$$
So, we have the parametrization of periodic orbits (Arnold--Weinstein curves)
 by reduced QMF equations.

Extended version and related results may be found in [1]-[9].

This research was supported in part under "New Ideas for Particle Accelerator
Program" NSF-Grant no.~PHY94-07194.
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