Consider the system (1) \(\dot q=p\), \(\dot p=-\omega^ 2(1+ef(t))\sin g\), with a Hamiltonian \(H(p,q,t)=(1/2)p^ 2-\omega^ 2(1+ef(t))\cos q,\) real and analytic on \(R^ 1(p)\times S^ 1(q mod 2\pi)\times S^ 1(t mod 2\pi)\times I(-\epsilon \leq e\leq \epsilon).\) Let \(\epsilon^{- 1}=\max_{t}| f(t)|,\) if it is known that for \(e\in I\), (1) possesses a hyperbolic periodic solution \(x_{\pi}(t)=(p_{\pi},q_{\pi})_ t=(0,\pi mod 2\pi)\), and there exist two 2-dim. invariant asymptotic surfaces \(\Pi^+_ e\) and \(\Pi^-_ e\), which are filled with trajectories infinitely approaching \(x_{\pi}(t)\) when \(t\to \pm \infty\). Denote by \(\gamma_ e^{\pm}=\gamma^{\pm}_{1e}\cup \gamma^{\pm}_{2e}\), where \(\gamma^{\pm}_{ie}=\{q=0\}\times \{p=p_ i^{\pm}(t,e)=p_ i^{\pm}(t+2\pi,e)\}\), \(i=1,2\), a pair of curves contained in the intersection of the transversal cylinder \(\Gamma =\{q=0\}\times R'(p)\times S'(t mod 2\pi)\) with the surfaces \(\Pi_ e^{\pm}\), which satisfy the condition: for any solution x(t) of (1) with \(x(t_ 0)\in \gamma^+_ e (\gamma^-_ e)\) we always have x(t)\(\not\in \Gamma\) for all \(t>t_ 0 (t<t_ 0)\). Let \(\phi_ e:\Gamma \backslash(\gamma^+_ e\cup \{p=0\})\to \Gamma \backslash(\gamma^- _ e\cup \{p=0\})\) be the volume preserving diffeomorphism defined by the solution x(t) of (1) with \(x(t_ 0)=(p_ 0,t_ 0)\in \Gamma \backslash(\{p_ 0=0\}\cup \gamma_ e^{\pm})\), and let \(R_ e^{\pm}=\{q=0\}\times \{(p,t):p_ 2^{\pm}(t,e)<p<p_ 1^{\pm}(t,e)\), 0\(\leq t\leq 2\pi \}\), \(H_ e^{\pm}=\Gamma \backslash \{R_ e^{\pm}\cup \gamma_ e^{\pm}\}\). The author proves the following Theorem. Let \(e\neq 0\) be so small such that the curves \(\gamma^+_{ie}\) and \(\gamma^-_{ie} (i=1,2)\) intersect transversally at the points \((p_{i\alpha},\tau_{i\alpha})\), \(\alpha =1,2,...,n_ i\). Then 1) an integer \(N=N(e)\) can be found, such that for any sequence \(s\in S_ N\) there exists a unique corresponding motion of (1); 2) the mapping \(\phi_ e\) contains the mapping \(\sigma\) as a subsystem. Here \(S_ N\) is the space of two-sided sequences of symbols taken from \(B_ N=\{R_{i,\alpha}(L)\), \(H_{i,\alpha}(L)\), \(P_{i,\alpha}(-\infty)\), \(P_{i,\alpha}(\infty)\), \(i=1,2\), \(\alpha =1,...,n_ i\), \(L=N,N+1,...\}\), which contains the following four types of sequences: i) \(\{...,s_ 1,s_ 0,s_ 1,...\}\); ii) \(\{v^-_{- \mu},s_{-\mu +1},...,s_{-1},s_ 0,s_ 1,...\}\); iii) \(\{...,s_{- 1},s_ 0,s_ 1,...,s_{\lambda -1},v^+_{\lambda}\}\); iv) \(\{v^- _{-\mu}s_{-\mu +1},...,s_{-1},s_ 0,s_ 1,...,s_{\lambda - 1},v^+_{\lambda}\}\), with \(v^{\pm}\in P(\pm \infty)=\cup_{i,\alpha}P_{i,\alpha}(\pm;\infty)\), \(s_ n\in B_ N\backslash(P(-\infty)\cup P(\infty))\), such that if \(s_ n=R_{i,\alpha}(L)\), then \(s_{n+1}=T_{1+i mod 2,\beta}(J),\) if \(s_ n=H_{i,\alpha}(L),\) then \(s_{n+1}=T_{i,\beta}(J),\) if \(v^- =P_{i,\alpha}(-\infty),\) then \(s_{-\mu +1}=T_{i,\alpha}(J),\) and T denotes R, P or H. For all \(s\in S_ N\backslash \{s:s_ 0\in P(\infty)\}\) define the shift mapping \(\sigma\) by: \((\sigma {\mathbb{O}}s)_ m=s_{m-1}\). We say that the motion x(t) corresponds to the sequence \(s\in S_ N\), if the following conditions are satisfied: a) if \(s_ m=R_{i,\alpha}(L)\), then \((p_ m,t_ m)\in Q_{i\alpha}\cap R^+_ e, (p_{m+1},t_{m+1})\in Q_{1+i mod 2,\beta}, L=[(t_{m+1}- \tau_{1+i mod 2,\beta}-t_ m+\tau_{i\alpha}+2\gamma(\delta))/2\pi]\), where \(Q_{i\alpha}(\delta)\) is a connected component of \(\Delta^+_ i(\delta)\cap \Delta^-_ i(\delta), \Delta_ i^{\pm}(\delta)=\{x:x\in \Gamma,p(x,\gamma^{\pm}_{ie})<\delta \}\), which contains just one point \(x_{i\alpha}\) for \(\delta\) sufficiently small, and \(\gamma(\delta)=\max_{i,\alpha}\sup_{x\in Q_{i,\alpha}(\delta)}p(x,x_{i\alpha}).\) b) if \(s_ m=H_{i,\alpha}(L)\), then \((p_ m,t_ m)\in Q_{i,\alpha}\cap H^+_ e\), \((p_{m+1},t_{m+1})\in Q_{i\beta}\), \(L=[(t_{m+1}- \tau_{i,\beta}-t_ m+\tau_{i,\alpha}+2\gamma(\delta))/2\pi]\); c) if \(v^+_{\lambda}=P_{i,\alpha}(\infty) (v^-_{- \mu}=P_{i,\alpha}(-\infty))\), then \((p_{\lambda},t_{\lambda})\in \gamma^+_ e\cap Q_{i\alpha}, ((p_{-\mu +1},t_{-\mu +1})\in \gamma^-_ e\cap Q_{i,\alpha})\).