T. M. Rocha Filho

Superinflation, quintessence, and nonsingular cosmologies

The dynamics of a universe dominated by a self-interacting nonminimally coupled scalar field are considered. The structure of the phase space and complete phase portraits are given. New dynamical behaviors include superinflation (

Boundedness of solutions and Lyapunov functions in quasi-polynomial systems

\dot{H}>0

Algebraic structures and invariant manifolds of differential systems

), avoidance of big bang singularities through classical birth of the universe, and spontaneous entry into and exit from inflation. This model is promising for describing quintessence as a nonminimally coupled scalar field.

The dynamical system approach to scalar field cosmology

Abstract In this Letter we establish sufficient conditions for the existence of a Lyapunov function for a large class of non-linear systems, the Quasi-Polynomial systems [Figueredo et al., J. Math. Phys. 39 (1998) 2929; Figueredo et al., Phys. A 262 (1999) 158; Brenig, Phys. Lett. A 133 (1988) 378] . We also present sufficient conditions such that the solutions are bounded and bounded away from zero componentwise.

Ergodicity and central-limit theorem in systems with long-range interactions

Algebraic tools are applied to find integrability properties of ODEs. Bilinear nonassociative algebras are associated to a large class of polynomial and nonpolynomial systems of differential equations, since all equations in this class are related to a canonical quadratic differential system: the Lotka–Volterra system. These algebras are classified up to dimension 3 and examples for dimension 4 and 5 are given. Their subalgebras are associated to nonlinear invariant manifolds in the phase space. These manifolds are calculated explicitly. More general algebraic invariant surfaces are also obtained by combining a theorem of Walcher and the Lotka–Volterra canonical form. Applications are given for Lorenz model, Lotka, May–Leonard, and Rikitake systems.

A numerical method for the stability analysis of quasi-polynomial vector fields

A spatially flat FLRW universe (motivated by inflation) is studied; by a dimensional reduction of the dynamical equations of scalar field cosmology, it is demonstrated that a spatially flat universe cannot exhibit chaotic behaviour. The result holds when the source of gravity is a non-minimally coupled scalar field, for any self-interaction potential and for arbitrary values of the coupling constant with the Ricci curvature. The phase space of the dynamical system is studied, and regions inaccessible to the evolution are found. The topology of the forbidden regions, their dependence on the parameters, the fixed points and their stability character, and the asymptotic behaviour of the solutions are studied. New attractors are found, in addition to those known from the minimal coupling case, certain exact solutions are presented and the implications for inflation are discussed. The equation of state is not prescribed a priori , but rather is deduced self-consistently from the field equations.

Necessary conditions for the existence of quasi-polynomial invariants: the quasi-polynomial and Lotka–Volterra systems

In this letter we discuss the validity of the ergodicity hypothesis in theories of violent relaxation in long-range interacting systems. We base our reasoning on the Hamiltonian mean-field model and show that the lifetime of quasi-stationary states resulting from the violent relaxation does not allow the system to reach a complete mixed state. We also discuss the applicability of a generalization of the central-limit theorem. In this context, we show that no attractor exists in distribution space for the sum of velocities of a particle other than the Gaussian distribution. The long-range nature of the interaction leads in fact to a new instance of sluggish convergence to a Gaussian distribution.

On the stability of a class of general non-linear systems

This paper shows the sufficient conditions for the existence of a Lyapunov function in the class of quasi-polynomial dynamical systems. We focus on the cases where the systems parameters are numerically specified. A numerical algorithm to analyze this problem is presented, which involves the resolution of a linear matrix inequality (LMI). This LMI is collapsed to a linear programming problem. From the numerical viewpoint, this computational method is very useful to search for sufficient conditions for the stability of non-linear systems of ODEs. The results of this paper greatly enlarge the scope of applications of a method previously presented by the authors.

Stability properties of a general class of nonlinear dynamical systems

We show that any quasi-polynomial invariant of a quasi-polynomial dynamical system can be transformed into a quasi-polynomial invariant of a homogeneous quadratic Lotka–Volterra dynamical system. We show how this quasi-polynomial invariant can be decomposed in a simple manner. This decomposition permits to conclude that the existence of polynomial semi-invariants in Lotka–Volterra systems is a necessary condition for the existence of quasi-polynomial invariants. We derive a method which allows to construct the necessary conditions for existence of semi-invariants on Lotka–Volterra dynamical systems. Applications are given.

Superinflation, quintessence, and the avoidance of the initial singularity

Abstract In a previous work (Phys. Lett. A 268 (2000) 335) the authors established sufficient conditions for the boundedness of solutions and stability of the interior fixed points for a class of general non-linear systems—the quasi-polynomial (QP) systems—and an algebraic method to analyse the problem. In this Letter we present an extension of our results based on a numeric approach which greatly enlarges the scope of applications of the method.