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The solutions to Eq. (6) possess a homogeneity property: rescaling all variables by the same factor (Formula presented): (Formula presented) is equivalent to rescaling the time by (Formula presented): (Formula presented)
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The solutions to Eq. (6) possess a homogeneity property: rescaling all variables by the same factor (Formula presented): (Formula presented) is equivalent to rescaling the time by (Formula presented): (Formula presented).
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44
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85037250777
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defines a direction in the phase space (the set of geometrically similar configurations) along which no local divergence takes place. Thus, besides the translations along the trajectories, there is a second direction in which the local Lyapunov exponent is zero. Because of the conservation of phase-space volume, the sum of Lyapunov exponents must be zero and hence the last exponent must be (Formula presented)
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defines a direction in the phase space (the set of geometrically similar configurations) along which no local divergence takes place. Thus, besides the translations along the trajectories, there is a second direction in which the local Lyapunov exponent is zero. Because of the conservation of phase-space volume, the sum of Lyapunov exponents must be zero and hence the last exponent must be (Formula presented).
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46
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85037204458
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T. Tél, in Directions in Chaos, edited by Hao Bai-Lin (World Scientific, Singapore, 1990), Vol. 3, pp. 149–221; in STATPHYS’19, edited by Hao Bai-lin (World Scientific, Singapore, 1996), pp. 346–362
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T. Tél, in Directions in Chaos, edited by Hao Bai-Lin (World Scientific, Singapore, 1990), Vol. 3, pp. 149–221;in STATPHYS’19, edited by Hao Bai-lin (World Scientific, Singapore, 1996), pp. 346–362.
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85037195636
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Due to the reversibility property of Eq. (6), it is true that the time reversed dynamics of a starting configuration (Formula presented), (Formula presented), (Formula presented) is the same as the direct dynamics of the mirror configuration (Formula presented), (Formula presented), (Formula presented). Therefore, if a point (Formula presented) in the relative coordinate representation [see Eq. (6)] is on the stable manifold of the chaotic saddle at a fixed (Formula presented) and (Formula presented), then the point (Formula presented) belonging to (Formula presented), (Formula presented) is on the unstable manifold. Thus, the unstable manifold on the (Formula presented) plane is obtained, in general, by rotating the plot of the stable manifold by 180(Formula presented) and interchanging the positions of particles 1 and 2. Due to the specific choice (Formula presented) used in Figs. 44 and 55(a), the unstable manifold is just the mirror image of the stable one with respect to the axis (Formula presented)
-
Due to the reversibility property of Eq. (6), it is true that the time reversed dynamics of a starting configuration (Formula presented), (Formula presented), (Formula presented) is the same as the direct dynamics of the mirror configuration (Formula presented), (Formula presented), (Formula presented). Therefore, if a point (Formula presented) in the relative coordinate representation [see Eq. (6)] is on the stable manifold of the chaotic saddle at a fixed (Formula presented) and (Formula presented), then the point (Formula presented) belonging to (Formula presented), (Formula presented) is on the unstable manifold. Thus, the unstable manifold on the (Formula presented) plane is obtained, in general, by rotating the plot of the stable manifold by 180(Formula presented) and interchanging the positions of particles 1 and 2. Due to the specific choice (Formula presented) used in Figs. 44 and 55(a), the unstable manifold is just the mirror image of the stable one with respect to the axis (Formula presented).
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57
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85037203700
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(private communication)
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R.J. Phillips (private communication).
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