Spectral signature of the pitchfork bifurcation: Liouville equation approach

Journal Article · Sun Jan 01 04:00:00 EST 1995 · Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics; (United States)

DOI:https://doi.org/10.1103/PhysRevE.51.74· OSTI ID:6702195

Gaspard, P; Nicolis, G; Provata, A ^[1]; Tasaki, S ^[2]

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1. Faculte des Sciences and Center for Nonlinear Phenomena and Complex Systems, Universite Libre de Bruxelles, Campus Plaine, Code Postal 231, Boulevard du Triomphe, B-1050 Bruxelles (Belgium)
2. Institute for Fundamental Chemistry, 34-4 Takano-Nishihiraki-cho, Sakyo-ku, Kyoto 606 (Japan)

The time evolution of probability densities of one-dimensional nonlinear vector fields is studied using a Liouville equation approach. It is shown that the Liouville operator admits a discrete spectrum of eigenvalues of decaying type if the vector field is far from bifurcation. The associated right and left eigenvectors are explicitly constructed for simple models and shown to be distributions rather than regular functions. On the other hand, the spectrum of the Liouville operator may become continuous at the bifurcation point, a phenomenon illustrated explicitly in the paper in the case of the pitchfork bifurcation. The relationship between the spectral decompositions of the Liouville and of the Fokker-Planck equations is discussed. In particular, the spectral decompositions constructed here for the Liouville equation are obtained as the noiseless limit of the well known spectral decompositions of the Fokker-Planck equation of the associated stochastic process.

• OSTI ID: 6702195
• Journal Information: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics; (United States), Journal Name: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics; (United States) Vol. 51:1; ISSN PLEEE8; ISSN 1063-651X
• Country of Publication: United States
• Language: English

Related Subjects

661300* -- Other Aspects of Physical Science-- (1992-)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
DIFFERENTIAL EQUATIONS
DISTRIBUTION FUNCTIONS
DYNAMICS
EIGENVALUES
EIGENVECTORS
EQUATIONS
FOKKER-PLANCK EQUATION
FUNCTIONS
MATHEMATICAL OPERATORS
MECHANICS
PARTIAL DIFFERENTIAL EQUATIONS
STATISTICAL MECHANICS
STOCHASTIC PROCESSES
VECTOR FIELDS