Weak chaotic orbit

The second curve corresponds to a weak chaotic orbit of initial conditions (0.4839,0) while the third one corresponds to a regular invariant torus of initial conditions (2.5,0) and the lowest one to a resonant curve of initial conditions (1.2,0). 

Figure 2. Variation of the Fast Lyapunov Indicator with time, a) Orbits of the standard map of equation (5) with e = 0.7. The upper curve is for a chaotic orbit with initial conditions x(O) = 10 4,j/(0) = 0, the second one is for a weak chaotic orbit with x(0) = 0.4839, y(0) = 0, the third one is for a non resonant orbit with x(0) = 2.5,2/(0) = 0 and the lowest one is for a resonant orbit with x(0) = 1.2, y(0) = 0. b) Orbits of Hamiltonian (6) with e = 0.004. The upper curve is for a chaotic orbit with initial conditions -Zi(O) = 0.2849, the second one is for a weak chaotic orbit with /1(0) = 0.2309, the third one is for a non resonant orbit with /i(0) = 0.2204 and the lowest one is for a resonant orbit with /1(0) = 0.342. The other initial conditions are I2(0) = 0.16, M0) = 1, >i(0) =

Froeschle C. and Lega, E. Twist angles a fast method for distinguishing islands, tori and weak chaotic orbits. Comparison with other methods of analysis. AA, 334 355-362, (1998). 

The linear response function [3], R(r, r ) = (hp(r)/hv(r ))N, is used to study the effect of varying v(r) at constant N. If the system is acted upon by a weak electric field, polarizability (a) may be used as a measure of the corresponding response. A minimum polarizability principle [17] may be stated as, the natural direction of evolution of any system is towards a state of minimum polarizability. Another important principle is that of maximum entropy [18] which states that, the most probable distribution is associated with the maximum value of the Shannon entropy of the information theory. Attempts have been made to provide formal proofs of these principles [19-21], The application of these concepts and related principles vis-a-vis their validity has been studied in the contexts of molecular vibrations and internal rotations [22], chemical reactions [23], hydrogen bonded complexes [24], electronic excitations [25], ion-atom collision [26], atom-field interaction [27], chaotic ionization [28], conservation of orbital symmetry [29], atomic shell structure [30], solvent effects [31], confined systems [32], electric field effects [33], and toxicity [34], In the present chapter, will restrict ourselves to mostly the work done by us. For an elegant review which showcases the contributions from active researchers in the field, see [4], Atomic units are used throughout this chapter unless otherwise specified. 

Abstract In order to describe the motion of two weakly interacting satellites of a central body we suggest to use orbital elements based on the the linear theory of Kepler motion in Levi-Civita s regularizing coordinates. The basic model is the planar three-body problem with two small masses, a model in which both regular (e.g. quasi-periodic) as well as chaotic motion can occur. 

Thus, the time-dependence of the flow generates chaotic trajectories that will enhance the mixing of fluid within these regions. However, the KAM tori formed by the remaining quasiperiodic orbits separate the domain into a set of disconnected regions with no advec-tive transport between them. Therefore, when the time-dependence is weak the fluid is only mixed within narrow layers around the resonant streamlines of the original time-independent flow. The areas... 

Figure 20 Schematic drawing of two cylindrical manifolds within isomer A in the weak-coupling limit (refer to Figure 8 for an explanation of the symbols). The two-dimensional cylinders will intersect each other along one-dimensional lines. These lines are two homoclinic orbits. The small, thin tube spanning both isomers corresponds to a reactive KAM torus. Note that although we have stopped drawing the cylinders beyond a certain point for clarity, in reality the cylinders continue to wind about and explore the entire accessible region of chaotic phase space. Reprinted with permission from Ref. 108.

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