Unstable evolution of pointwise trajectory solutions to chaotic maps. (English) Zbl 1055.37538
Summary: Simple chaotic maps are used to illustrate the inherent instability of trajectory solutions to the Frobenius-Perron equation. This is demonstrated by the difference in the behavior of \(\delta\)-function solutions and of extended densities. Extended densities evolve asymptotically and irreversibly into invariant measures on stationary attractors. Pointwise trajectories chaotically roam over these attractors forever. Periodic Gaussian distributions on the unit interval are used to provide insight. Viewing evolving densities as ensembles of unstable pointwise trajectories gives densities a stochastic interpretation.
MSC:
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 28D10 | One-parameter continuous families of measure-preserving transformations |
| 39B12 | Iteration theory, iterative and composite equations |
| 82C05 | Classical dynamic and nonequilibrium statistical mechanics (general) |
| 37A10 | Dynamical systems involving one-parameter continuous families of measure-preserving transformations |
References:
| [1] | DOI: 10.1103/PhysRevLett.64.1196 · Zbl 0964.37501 · doi:10.1103/PhysRevLett.64.1196 |
| [2] | DOI: 10.1103/PhysRevLett.65.3211 · doi:10.1103/PhysRevLett.65.3211 |
| [3] | DOI: 10.1103/PhysRevLett.68.1259 · doi:10.1103/PhysRevLett.68.1259 |
| [4] | DOI: 10.1103/PhysRevA.46.7401 · doi:10.1103/PhysRevA.46.7401 |
| [5] | DOI: 10.1103/PhysRevE.50.1781 · doi:10.1103/PhysRevE.50.1781 |
| [6] | DOI: 10.1063/1.165950 · doi:10.1063/1.165950 |
| [7] | DOI: 10.1063/1.165950 · doi:10.1063/1.165950 |
| [8] | DOI: 10.1063/1.165950 · doi:10.1063/1.165950 |
| [9] | DOI: 10.1088/0951-7715/3/2/005 · Zbl 0702.58064 · doi:10.1088/0951-7715/3/2/005 |
| [10] | DOI: 10.1088/0951-7715/3/2/005 · Zbl 0702.58064 · doi:10.1088/0951-7715/3/2/005 |
| [11] | DOI: 10.1088/0951-7715/3/2/005 · Zbl 0702.58064 · doi:10.1088/0951-7715/3/2/005 |
| [12] | DOI: 10.1088/0951-7715/3/2/005 · Zbl 0702.58064 · doi:10.1088/0951-7715/3/2/005 |
| [13] | DOI: 10.1103/PhysRevLett.64.249 · Zbl 1050.82539 · doi:10.1103/PhysRevLett.64.249 |
| [14] | DOI: 10.1103/PhysRevA.41.2969 · doi:10.1103/PhysRevA.41.2969 |
| [15] | DOI: 10.1103/PhysRevA.43.1709 · doi:10.1103/PhysRevA.43.1709 |
| [16] | Fox R. F., Chaos 3 pp 313– (1993) · Zbl 1055.37588 · doi:10.1063/1.165940 |
| [17] | Fox R. F., Phys. Rev. E 49 pp 3683– (1994) · doi:10.1103/PhysRevE.49.3683 |
| [18] | Fox R. F., Phys. Rev. E 50 pp 2553– (1994) · doi:10.1103/PhysRevE.50.2553 |
| [19] | DOI: 10.1103/PhysRev.40.749 · Zbl 0004.38201 · doi:10.1103/PhysRev.40.749 |
| [20] | Husimi K., Proc. Phys. Math. Soc. Jpn. 22 pp 264– (1940) |
| [21] | DOI: 10.1016/0375-9601(81)90881-1 · doi:10.1016/0375-9601(81)90881-1 |
| [22] | DOI: 10.1103/PhysRevA.34.7 · doi:10.1103/PhysRevA.34.7 |
| [23] | Nakamura K., Phys. Rev. B 39 pp 12– (1989) |
| [24] | Fox R. F., Phys. Rev. A 41 pp 2952– (1990) · doi:10.1103/PhysRevA.41.2952 |
| [25] | Gaspard P., J. Phys. A: Math. Gen. 25 pp L483– (1992) · Zbl 0762.58013 · doi:10.1088/0305-4470/25/8/017 |
| [26] | Hutchinson J., Ind. Univ. J. Math. 30 pp 713– (1981) · Zbl 0598.28011 · doi:10.1512/iumj.1981.30.30055 |
| [27] | DOI: 10.1007/BF01010528 · Zbl 0624.58037 · doi:10.1007/BF01010528 |
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