M. A. Gonzalez Leon, J. L. Hernandez Pastora
We investigate the exact relation existing between the stability equation for the solutions of a mechanical system and the geodesic deviation equation of the associated geodesic problem in the Jacobi metric constructed via the Maupertuis-Jacobi Principle. We conclude that the dynamical and geometrical approaches to the stability/instability problem are not equivalent.
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We establish a topological criterion for connection between reducibility to constant rotations and dual localization, for the general family of analytic quasiperiodic Jacobi operators. As a corollary, we obtain the sharp arithmetic phase transition for the extended Harper's model in the positive Lyapunov exponent region.
We present a new oscillation criterion to determine whether the number of eigenvalues below the essential spectrum of a given Jacobi operator is finite or not. As an application we show that Kenser's criterion for Jacobi operators follows as a special case.
The asymptotic stability of a global solution satisfying Hamilton-Jacobi equations with jumps will be analyzed in dependence on the strong dissipativity of the jump control function and using orbits of the differentiable flows to describe the corresponding characteristic system.
When the coefficients of a Jacobi operator are finitely supported perturbations of the 1 and 0 sequences, respectively, the left reflection coefficient is a rational function whose poles inside, respectively outside, the unit disk correspond to eigenvalues and resonances. By including the zeros of the reflection coefficient, we have a set of data that determines the Jacobi coefficients up to a translation as long as there is at most one half-bound state. We prove that the coefficients of two Jacobi operators are pointwise close assuming that the zeros and poles of their left reflection coeffic…
This work studies the interplay between Green functions, the index of determinacy of spectral measures and interior finite rank perturbations of Jacobi operators. The index of determinacy quantifies the stability of uniqueness of solutions of the moment problem. We give results on the constancy of this index in terms of perturbations of the corresponding Jacobi operators. The permanence of the $N$-extremality of a measure is also studied. A measure $μ$ is $N$-extremal when the polynomials are dense in $L_2(\mathbb{R},μ)$. As a by-product, we give a characterization of the index in terms of cyc…
Examples of non-standard construction of Hamiltonian structures for dynamical systems and the respective Hamilton-Jacobi (H-J) equations, without using Lagrangians, are presented. Alternative H-J equations for Euler top are explicitly exhibited and solved. We demonstrate that some stability criterion, relating the slope of a Casimir function parametrized by the Lagrange multiplier to critical point type, depends on the used Hamiltonian structure and it is inadequate for this reason.
When the classical Hamburger moment problem has solutions, it has either exactly one solution or infinitely many solutions. Correspondingly, the moment problem is said to be either determinate or indeterminate. In terms of Jacobi operators, this dichotomy translates into the operator being either selfadjoint or symmetric nonselfadjoint. In this work, we present a new criterion for the determinate-indeterminate classification which hinges on bases of representation (in Akhiezer-Glazman terminology) for Jacobi operators so that the corresponding matrices have a certain structure.
Geometrization of dynamics using (non)-affine parametization of arc length with time is investigated. The two archetypes of such parametrizations, the Eisenhart and the Jacobi metrics, are applied to a system of linear harmonic oscillators. Application of the Jacobi metric results in positive values of geometrical lyapunov exponent. The non-physical instabilities are shown to be due to a non-affine parametrization. In addition the degree of instability is a monotonically increasing function of the fluctuations in the kinetic energy. We argue that the Jacobi metric gives equivalent results as E…
We analyze the stability of the geodesic curves in the geometry of the Gibbons-Maeda-Garfinkle-Horowitz-Strominger black hole, describing the space time of a charged black hole in the low energy limit of the string theory. The stability analysis is performed by using both the linear (Lyapunov) stability method, as well as the notion of Jacobi stability, based on the Kosambi-Cartan-Chern theory. Brief reviews of the two stability methods are also presented. After obtaining the geodesic equations in spherical symmetry, we reformulate them as a two-dimensional dynamic system. The Jacobi stability…
The Kosambi-Cartan-Chern (KCC) theory represents a powerful mathematical method for the analysis of dynamical systems. In this approach one describes the evolution of a dynamical system in geometric terms, by considering it as a geodesic in a Finsler space. By associating a non-linear connection and a Berwald type connection to the dynamical system, five geometrical invariants are obtained, with the second invariant giving the Jacobi stability of the system. The Jacobi (in)stability is a natural generalization of the (in)stability of the geodesic flow on a differentiable manifold endowed with …
Contraction theory is a recently developed dynamic analysis and nonlinear control system design tool based on an exact differential analysis of convergence. This paper extends contraction theory to local and global stability analysis of important classes of nonlinear distributed dynamics, such as convection-diffusion-reaction processes, Lagrangian and Hamilton-Jacobi dynamics, and optimal controllers and observers. The Hamilton-Jacobi-Bellman controller and a similar optimal nonlinear observer design are studied. Explicit stability conditions are given, which extend the well-known conditions o…
We perform the study of the stability of the Lorenz system by using the Jacobi stability analysis, or the Kosambi-Cartan-Chern (KCC) theory. The Lorenz model plays an important role for understanding hydrodynamic instabilities and the nature of the turbulence, also representing a non-trivial testing object for studying non-linear effects. The KCC theory represents a powerful mathematical method for the analysis of dynamical systems. In this approach we describe the evolution of the Lorenz system in geometric terms, by considering it as a geodesic in a Finsler space. By associating a non-linear…
We analyze the stability of the structure equations of the vacuum in the brane world models, by using both the linear (Lyapunov) stability analysis, and the Jacobi stability analysis, the Kosambi-Cartan-Chern (KCC) theory. In the brane world models the four dimensional effective Einstein equations acquire extra terms, called dark radiation and dark pressure, respectively, which arise from the embedding of the 3-brane in the bulk. Generally, the spherically symmetric vacuum solutions of the brane gravitational field equations, have properties quite distinct as compared to the standard black hol…
We perform the study of the stability of the cosmological scalar field models, by using the Jacobi stability analysis, or the Kosambi-Cartan-Chern (KCC) theory. In the KCC approach we describe the time evolution of the scalar field cosmologies in geometric terms, by performing a "second geometrization", by considering them as paths of a semispray. By introducing a non-linear connection and a Berwald type connection associated to the Friedmann and Klein-Gordon equations, five geometrical invariants can be constructed, with the second invariant giving the Jacobi stability of the cosmological mod…
The circular restricted three body problem, which considers the dynamics of an infinitesimal particle in the presence of the gravitational interaction with two massive bodies moving on circular orbits about their common center of mass, is a very useful model for investigating the behavior of real astronomical objects in the Solar System. In such a system, there are five Lagrangian equilibrium points, and one important characteristic of the motion is the existence of linearly stable equilibria at the two equilibrium points that form equilateral triangles with the primaries, in the plane of the …
Two entropic dynamical models are considered. The geometric structure of the statistical manifolds underlying these models is studied. It is found that in both cases, the resulting metric manifolds are negatively curved. Moreover, the geodesics on each manifold are described by hyperbolic trajectories. A detailed analysis based on the Jacobi equation for geodesic spread is used to show that the hyperbolicity of the manifolds leads to chaotic exponential instability. A comparison between the two models leads to a relation among statistical curvature, stability of geodesics and relative entropy-…
We numerically solve the equations of motion (EOM) for two models of circular cosmic string loops with windings in a simply connected internal space. Since the windings cannot be topologically stabilized, stability must be achieved (if at all) dynamically. As toy models for realistic compactifications, we consider windings on a small section of $\mathbb{R}^2$, which is valid as an approximation to any simply connected internal manifold if the winding radius is sufficiently small, and windings on an $S^2$ of constant radius $\mathcal{R}$. We then use Kosambi-Cartan-Chern (KCC) theory to analyze…
A variational principle is proposed for obtaining the Jacobi equations in systems admitting a Lagrangian description. The variational principle gives simultaneously the Lagrange equations of motion and the Jacobi variational equations for the system. The approach can be of help in finding constants of motion in the Jacobi equations as well as in analysing the stability of the systems and can be related to the vertical extension of the Lagrangian formalism. To exemplify two of such aspects, we uncover a constant of motion in the Jacobi equations of autonomous systems and we recover the well-kno…
A more restrictively general stability criterion of two-dimensional inviscid parallel flow is obtained analytically. First, a sufficient criterion for stability is found as either $-μ_1<\frac{U''}{U-U_s}<0$ or $0<\frac{U''}{U-U_s}$ in the flow, where $U_s$ is the velocity at inflection point, $μ_1$ is the eigenvalue of Poincaré's problem. Second, this criterion is generalized to barotropic geophysical flows in $β$ plane. Based on the criteria, the flows are are divided into different categories of stable flows, which may simplify the further investigations. And the connections between present …
In this article, we propose a Lyapunov stability approach to analyze the convergence of the density operator of a quantum system. In analog to the classical probability measure for Markovian processes, we show that the set of invariant density operators is both closed and convex. We then show how to analyze the stability of this set via a candidate Lyapunov operator. We complete our analysis of the set of invariant density operators by introducing an analog of the Barbashin-Krasovskii-La Salle theorem on the dynamics of quantum systems.
We study classically the problem of two relativistic particles with an invariant Duffing-like potential which reduces to the usual Duffing form in the nonrelativistic limit. We use a special relativistic generalization (RGEM) of the geometric method (GEM) developed for the analysis of nonrelativistic Hamiltonian systems to study the local stability of a relativistic Duffing oscillator. Poincar'e plots of the simulated motion are consistent with the RGEM. We found a threshold for the external driving force required for chaotic behavior in the Minkowski spacetime.
We provide a criterion in order to decide the stability of non-degenerate equilibrium states of completely integrable systems. More precisely, given a Hamilton-Poisson realization of a completely integrable system generated by a smooth $n-$ dimensional vector field, $X$, and a non-degenerate regular (in the Poisson sense) equilibrium state, $\overline{x}_e$, we define a scalar quantity, $\mathcal{I}_{X}(\overline{x}_e)$, whose sign determines the stability of the equilibrium. Moreover, if $\mathcal{I}_{X}(\overline{x}_e)>0$, then around $\overline{x}_e$ there exist one-parameter families of pe…
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