Analytical description of the structure of chaos

We consider analytical formulae that describe the chaotic regions around the main periodic orbit $(x=y=0)$ of the H{e}non map. Following our previous paper (Efthymiopoulos, Contopoulos, Katsanikas $2014$) we introduce new variables $(xi, eta)$ in which the product $xieta=c$ (constant) gives hyperbolic invariant curves. These hyperbolae are mapped by a canonical transformation $Phi$ to the plane $(x,y)$, giving Moser invariant curves. We find that the series $Phi$ are convergent up to a maximum value of $c=c_{max}$. We give estimates of the errors due to the finite truncation of the series and discuss how these errors affect the applicability of analytical computations. For values of the basic parameter $kappa$ of the H{e}non map smaller than a critical value, there is an island of stability, around a stable periodic orbit $S$, containing KAM invariant curves. The Moser curves for $c leq 0.32$ are completely outside the last KAM curve around $S$, the curves with $0.32<c<0.41$ intersect the last KAM curve and the curves with $0.41leq c< c_{max} simeq 0.49$ are completely inside the last KAM curve. All orbits in the chaotic region around the periodic orbit $(x=y=0)$, although they seem random, belong to Moser invariant curves, which, therefore define a structure of chaos. Orbits starting close and outside the last KAM curve remain close to it for a stickiness time that is estimated analytically using the series $Phi$. We finally calculate the periodic orbits that accumulate close to the homoclinic points, i.e. the points of intersection of the asymptotic curves from $x=y=0$, exploiting a method based on the self-intersections of the invariant Moser curves. We find that all the computed periodic orbits are generated from the stable orbit $S$ for smaller values of the H{e}non parameter $kappa$, i.e. they are all regular periodic orbits.

Resonant normal form and asymptotic normal form behavior in magnetic bottle Hamiltonians

We consider normal forms in `magnetic bottle type Hamiltonians of the form $H=frac{1}{2}(rho^2_rho+omega^2_1rho^2) +frac{1}{2}p^2_z+hot$ (second frequency $omega_2$ equal to zero in the lowest order). Our main results are: i) a novel method to construct the normal form in cases of resonance, and ii) a study of the asymptotic behavior of both the non-resonant and the resonant series. We find that, if we truncate the normal form series at order $r$, the series remainder in both constructions decreases with increasing $r$ down to a minimum, and then it increases with $r$. The computed minimum remainder turns to be exponentially small in $frac{1}{Delta E}$, where $Delta E$ is the mirror oscillation energy, while the optimal order scales as an inverse power of $Delta E$. We estimate numerically the exponents associated with the optimal order and the remainders exponential asymptotic behavior. In the resonant case, our novel method allows to compute a `quasi-integral (i.e. truncated formal integral) valid both for each particular resonance as well as away from all resonances. We applied these results to a specific magnetic bottle Hamiltonian. The non resonant normal form yields theorerical invariant curves on a surface of section which fit well the empirical curves away from resonances. On the other hand the resonant normal form fits very well both the invariant curves inside the islands of a particular resonance as well as the non-resonant invariant curves. Finally, we discuss how normal forms allow to compute a critical threshold for the onset of global chaos in the magnetic bottle.