On the Bifurcation of Certain Kam Tori in the Standard Mapping

Abstract

We study the ground-state {u_i} of a one-dimensional chain of atoms with energy

$$ \phi \left( {\left\{ {{u_i}} \right\}} \right) = \sum\limits_{{l = - \infty }}^{{ + \infty }} {\left[ {\lambda V\left( {{u_i}} \right) + W\left( {{u_{{i + 1}}} - {u_1}} \right)} \right]} $$

(1)

where u. is the abscissa of the i^th atom. V(u_i) is an analytic periodic and even potential with period 2a, W(u_i+1-u_i) is an analytic convex potential which couples neighbouring atoms. A ground-state is a particular solution of the equation

$$ \frac{{\partial \phi }}{{\partial {u_i}}} = \lambda V'\left( {{u_i}} \right) + W'\left( {{u_i} - {u_{{i - 1}}}} \right) - W'\left( {{u_{{i + 1}}} - {u_i}} \right) = 0 $$

(2)

This equation can be interpreted as the evolution equation of a canonical system with a discrete time I

$$ \left( {\begin{array}{*{20}{c}} {{p_{{i + 1}}}} \\ {{u_{{i + 1}}}} \\ \end{array} } \right) = T\left( {\begin{array}{*{20}{c}} {{p_i}} \\ {{u_i}} \\ \end{array} } \right) = \left( {\begin{array}{*{20}{c}} {{p_i} + \lambda V'\left( {{u_i}} \right)} \\ {{u_i} + {{W'}^{{ - 1}}}\left( {{p_i} + \lambda V'\left( {{u_i}} \right.} \right)} \\ \end{array} } \right) $$

(3)

for which Φ is the action and p_i =W’(u_i-u_i-1) the conjugate variable of u_i. u_i is mapped modulo 2a so that this mapping T is a diffeomorphism of the cylinder $\mathbb{R}{ +^1}T$. When W(u)=1/2u² and V(u)=1/4π² cos2πu, (3) becomes the standard mapping studied by B.V. CHIRIKOV [1] or J.M. GREENE [2]. With fixed boundary condition

$$ \mathop{{\lim }}\limits_{{\mathop{{N \to + \infty }}\limits_{{N' \to - \infty }} }} \frac{1}{{N - N'}}\left( {{u_N} - {u_{{N'}}}} \right) = 2\,\,and\,\,\frac{\ell }{{2a}}\,irrational $$

(4)

it is proved that there exists a hull function f_λ(x) which monotonously increases such that the configurations

$$ {u_i} = {f_{\lambda }}\left( {i\ell + \alpha } \right) = i\ell + \alpha + {g_{\lambda }}\left( {i\ell + \alpha } \right) $$

(5)

with arbitrary phase α are the ground states of (1). g_λ(x)=f_λ(x) - x is then a periodic odd function with the same period as V.