arXiv subjects

S Lakshmivarahan

Publications and source records attributed to S Lakshmivarahan.

3 recordsLinked to original sources

arXiv2026

Quadratic invariants and Hamiltonian structure in coupled gyrostat low-order model hierarchies

Coupled gyrostat low-order models (GLOMs) are energy-conserving cores of Galerkin-truncated fluid and geophysical systems, including Rayleigh-Benard convection and vorticity dynamics. A single gyrostat always possesses two quadratic invariants; when gyrostats are coupled, the number and geometry of invariants vary sensitively with model configuration, influencing the effective dimension of the dynamics, nonlinear stability, and statistical equilibria. We provide a systematic theory of this dependence. For sparse nested hierarchies of K gyrostats (M=2K+1 modes, no linear feedback), the number of independent quadratic invariants is exactly (M+1)/2; for general GLOMs with all parameters nonzero, energy is the only guaranteed invariant. The standard algebraic approach to finding invariants does not scale with model size. We show instead that many GLOMs admit a non-canonical Hamiltonian structure, with quadratic invariants recoverable as Casimir functions of an explicitly constructible Poisson matrix. The Hamiltonian structure imposes precise, computationally verifiable constraints on the nonlinear coefficients. For Hamiltonian hierarchies, Casimir gradients project consistently across models of increasing complexity, so that invariants are compatible under restriction to subspaces. The clear geometric interpretation of these models enables consistent application of Hamiltonian dynamics across low-order model hierarchies.

math.DS↗

arXiv2023

Minimal chaotic models from the Volterra gyrostat

Low-order models obtained through Galerkin projection of several physically important systems (e.g., Rayleigh-Bénard convection, mid-latitude quasi-geostrophic dynamics, and vorticity dynamics) appear in the form of coupled gyrostats. Forced dissipative chaos is an important phenomenon in these models, and this paper introduces and identifies 'minimal chaotic models' (MCMs), in the sense of having the fewest external forcing and linear dissipation terms, for the class of models arising from an underlying gyrostat core. The identification of MCMs reveals common conditions for chaos across a wide variety of physical systems. It is shown here that a critical distinction is whether the gyrostat core (without forcing or dissipation) conserves energy, depending on whether the sum of the quadratic coefficients is zero. The paper demonstrates that, for the energy-conserving condition of the gyrostat core, the requirement of a characteristic pair of fixed points that repel the chaotic flow dictates placement of forcing and dissipation in the minimal chaotic models. In contrast if the core does not conserve energy, the forcing can be arranged in additional ways for chaos to appear in the subclasses where linear feedbacks render fewer invariants in the gyrostat core. In all cases, the linear mode must experience dissipation for chaos to arise. The Volterra gyrostat presents a clear example where the arrangement of fixed points circumscribes more complex dynamics.

physics.flu-dyn↗

arXiv2023

Invariants and chaos in the Volterra gyrostat without energy conservation

The model of the Volterra gyrostat (VG) has not only played an important role in rigid body dynamics but also served as the foundation of low-order models of many naturally occurring systems. It is well known that VG possesses two invariants, or constants of motion, corresponding to kinetic energy and squared angular momentum, giving oscillatory solutions to its equations of motion. Nine distinct subclasses of the VG have been identified, two of which the Euler gyroscope and Lorenz gyrostat are each known to have two constants. This paper characterizes quadratic invariants of the VG and each of its subclasses, showing how these enjoy two invariants even when rendered in terms of a non-invertible transformation of parameters, leading to a transformed Volterra gyrostat (TVG). If the quadratic coefficients of the TVG sum to zero, as they do for the VG, the system conserves energy. In all of these cases, the flows preserve volume. However, physical models where the quadratic coefficients do not sum to zero are ubiquitous, and characterization of invariants and the resulting dynamics for this more general class of models with volume conservation but without energy conservation is lacking. This paper provides the first such characterization for each of the subclasses of the VG in the absence of energy conservation, showing how the number of invariants depends on the number of linear feedback terms. It is shown that the gyrostat with three linear feedback terms has no invariants. The number of invariants circumscribes the possible dynamics for these three-dimensional flows, and those without any invariants are shown to admit rich dynamics including chaos. This gives rise to a broad class of three-dimensional volume conserving chaotic flows, arising naturally from model reduction techniques.

physics.ao-ph↗