2002 |
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| 2. | Ramaswamy, R Symmetry-breaking in local lyapunov exponents Journal Article European Physical Journal B, 29 (2), pp. 339–343, 2002, ISSN: 14346028. Abstract | Links | BibTeX | Tags: 05.45.-a Nonlinear dynamics and nonlinear dynamical systems, 05.45.Pq Numerical simulations of chaotic models, 71.30.+h Metal-insulator transitions and other electronic transitions @article{Ramaswamy2002, title = {Symmetry-breaking in local lyapunov exponents}, author = {R Ramaswamy}, url = {https://ramramaswamy.org/papers/088.pdf}, doi = {10.1140/epjb/e2002-00313-8}, issn = {14346028}, year = {2002}, date = {2002-01-01}, journal = {European Physical Journal B}, volume = {29}, number = {2}, pages = {339–343}, abstract = {Integrable dynamical systems, namely those having as many independent conserved quantities as freedoms, have all Lyapunov exponents equal to zero. Locally, the instantaneous or finite time Lyapunov exponents are nonzero, but owing to a symmetry, their global averages vanish. When the system becomes nonintegrable, this symmetry is broken. A parallel to this phenomenon occurs in mappings which derive from quasiperiodic Schrödinger problems in 1-dimension. For values of the energy such that the eigenstate is extended, the Lyapunov exponent is zero, while if the eigenstate is localized, the Lyapunov exponent becomes negative. This occurs by a breaking of the quasiperiodic symmetry of local Lyapunov exponents, and corresponds to a breaking of a symmetry of the wavefunction in extended and critical states.}, keywords = {05.45.-a Nonlinear dynamics and nonlinear dynamical systems, 05.45.Pq Numerical simulations of chaotic models, 71.30.+h Metal-insulator transitions and other electronic transitions}, pubstate = {published}, tppubtype = {article} } Integrable dynamical systems, namely those having as many independent conserved quantities as freedoms, have all Lyapunov exponents equal to zero. Locally, the instantaneous or finite time Lyapunov exponents are nonzero, but owing to a symmetry, their global averages vanish. When the system becomes nonintegrable, this symmetry is broken. A parallel to this phenomenon occurs in mappings which derive from quasiperiodic Schrödinger problems in 1-dimension. For values of the energy such that the eigenstate is extended, the Lyapunov exponent is zero, while if the eigenstate is localized, the Lyapunov exponent becomes negative. This occurs by a breaking of the quasiperiodic symmetry of local Lyapunov exponents, and corresponds to a breaking of a symmetry of the wavefunction in extended and critical states. |
1993 |
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| 1. | K Someda, Ramaswamy R; Nakamura, H Decoupling surface analysis of classical irregular scattering and classification of its icicle structure Journal Article The Journal of Chemical Physics, 98 (2), pp. 1156–1169 , 1993, ISSN: 0021-9606. Abstract | Links | BibTeX | Tags: 05.45.-a Nonlinear dynamics and nonlinear dynamical systems, Coupled Oscillators, Stochastic dynamics @article{Someda1993b, title = {Decoupling surface analysis of classical irregular scattering and classification of its icicle structure}, author = {K Someda, R Ramaswamy and H Nakamura}, url = {https://pubs.aip.org/aip/jcp/article-pdf/98/2/1156/11047349/1156\_1\_online.pdf}, doi = {10.1063/1.464339}, issn = {0021-9606}, year = {1993}, date = {1993-01-15}, journal = {The Journal of Chemical Physics}, volume = {98}, number = {2}, pages = {1156–1169 }, abstract = {Irregular scattering in molecular inelastic collision is analyzed classical mechanically by a novel method called ‘‘decoupling surface analysis.’’ Effective Hamiltonian of this analysis provides a phase space view of collision processes analogous to the Poincaré section of coupled‐oscillator systems. In this phase space view irregular scattering occurs in a stochastic layer formed around separatrix connected to resonance structure of the effective Hamiltonian. This circumstance is parallel to that in the coupled‐oscillator systems, in which stochastic motion is known to be connected to nonlinear resonance. The resonance structure in collision indicates trapping of classical trajectories in a certain dynamical well. The decoupling surface analysis suggests that the dynamical well is formed by a dip of stability exponents of trajectories as a function of time. By using a prototypical model exhibiting irregular scattering, a formal theoretical treatment is developed to analyze the structure of the fractal, termed icicle structure, observed in the plot of final vibrational action against the initial vibrational phase angle.}, keywords = {05.45.-a Nonlinear dynamics and nonlinear dynamical systems, Coupled Oscillators, Stochastic dynamics}, pubstate = {published}, tppubtype = {article} } Irregular scattering in molecular inelastic collision is analyzed classical mechanically by a novel method called ‘‘decoupling surface analysis.’’ Effective Hamiltonian of this analysis provides a phase space view of collision processes analogous to the Poincaré section of coupled‐oscillator systems. In this phase space view irregular scattering occurs in a stochastic layer formed around separatrix connected to resonance structure of the effective Hamiltonian. This circumstance is parallel to that in the coupled‐oscillator systems, in which stochastic motion is known to be connected to nonlinear resonance. The resonance structure in collision indicates trapping of classical trajectories in a certain dynamical well. The decoupling surface analysis suggests that the dynamical well is formed by a dip of stability exponents of trajectories as a function of time. By using a prototypical model exhibiting irregular scattering, a formal theoretical treatment is developed to analyze the structure of the fractal, termed icicle structure, observed in the plot of final vibrational action against the initial vibrational phase angle. |