Scalar Field Cosmologies With Inverted Potentials

Regular bouncing solutions in the framework of a scalar-tensor gravity model were found in a recent work. We reconsider the problem in the Einstein frame (EF) in the present work. Singularities arising at the limit of physical viability of the model in the Jordan frame (JF) are either of the Big Bang or of the Big Crunch type in the EF. As a result we obtain integrable scalar field cosmological models in general relativity (GR) with inverted double-well potentials unbounded from below which possess solutions regular in the future, tending to a de Sitter space, and starting with a Big Bang. The existence of the two fixed points for the field dynamics at late times found earlier in the JF becomes transparent in the EF.Comment: 18 pages, 4 figure

Bouncing Universes in Scalar-Tensor Gravity Models admitting Negative Potentials

We consider the possibility to produce a bouncing universe in the framework of scalar-tensor gravity models in which the scalar field potential may be negative, and even unbounded from below. We find a set of viable solutions with nonzero measure in the space of initial conditions passing a bounce, even in the presence of a radiation component, and approaching a constant gravitational coupling afterwards. Hence we have a model with a minimal modification of gravity in order to produce a bounce in the early universe with gravity tending dynamically to general relativity (GR) after the bounce.Comment: 12 pages, Improved presentation with 4 figures, Results and conclusions unchange

Sufficient conditions for the existence of a center in polynomial systems of arbitrary degree

In this paper, we consider polynomial systems of the form $\dot x=y+P (x,y)$, $\dot y=-x+Q(x,y)$, where $P$ and $Q$ are polynomials of degree $n$ wihout linear part. For the case $n=3$, we have found new sufficient conditions for a center at the origin, by proposing a first integral linear in certain coefficient of the system. The resulting first integral is in the general case of Darboux type. By induction, we have been able to generalize these results for polynomial systems of arbitrary degree

New sufficient conditions for a center and global phase portraits for polynomial systems

In this paper we consider cubic polynomial systems of the form: $\dot x=y+P(x,y)$, $\dot y=-x+Q(x,y)$, where $P$ and $Q$ are polynomials of degree 3 without linear part. If $M(x,y)$ is an integrating factor of the system, we propose its reciprocal $V(x,y)=\frac{1}{M(x,y)}$ as a linear function of certain coefficients of the system. We find in this way several new sets of sufficient conditions for a center. The resulting integrating factors are of Darboux type and the first integrals are in the Liouville form. By induction, we have generalized these results for polynomials systems of arbitrary degree. Moreover, for the cubic case, we have constructed all the phase portraits for each new family with a center

Integrals of motion and the shape of the attractor for the Lorenz model

In this paper, we consider three-dimensional dynamical systems, as for example the Lorenz model. For these systems, we introduce a method for obtaining families of two-dimensional surfaces such that trajectories cross each surface of the family in the same direction. For obtaining these surfaces, we are guided by the integrals of motion that exist for particular values of the parameters of the system. Nonetheless families of surfaces are obtained for arbitrary values of these parameters. Only a bounded region of the phase space is not filled by these surfaces. The global attractor of the system must be contained in this region. In this way, we obtain information on the shape and location of the global attractor. These results are more restrictive than similar bounds that have been recently found by the method of Lyapunov functions.Comment: 17 pages,12 figures. PACS numbers : 05.45.+b / 02.30.Hq Accepted for publication in Physics Letters A. e-mails : [email protected] & [email protected]

The null divergence factor

Let $(P, Q)$ be a $C^{1}$ vector field defined in a open subset $U \subset R^{2}$. We call a null divergence factor a $C^{1}$ solution $V(x, y)$ of the equation $P \frac{\partial V}{\partial x} + Q \frac{\partial V}{\partial y} = \left(\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}\right) \, V$. In previous works it has been shown that this function plays a fundamental role in the problem of the center and in the determination of the limit cycles. In this paper we show how to construct systems with a given null divergence factor. The method presented in this paper is a generalization of the classical Darboux method to generate integrable systems

Localizing limit cycles : from numeric to analytical results

Presentation given by participants of the joint international multidisciplinary workshop MURPHYS-HSFS-2016 (MUltiRate Processes and HYSteresis; Hysteresis and Slow-Fast Systems), which was dedicated to the mathematical theory and applications of multiple scale systems and systems with hysteresis, and held at the Centre de Recerca Matemàtica (CRM) in Barcelona from June 13th to 17th, 2016This note presents the results of [4]. It deals with the problem of location and existence of limit cycles for real planar polynomial differential systems. We provide a method to construct Poincaré-Bendixson regions by using transversal curves, that enables us to prove the existence of a limit cycle that has been numerically detected. We apply our results to several known systems, like the Brusselator one or some Liénard systems, to prove the existence of the limit cycles and to locate them very precisely in the phase space. Our method, combined with some other classical tools can be applied to obtain sharp bounds for the bifurcation values of a saddle-node bifurcation of limit cycles, as we do for the Rychkov syste

Experimental Realization of Optimal Noise Estimation for a General Pauli Channel

We present the experimental realization of the optimal estimation protocol for a Pauli noisy channel. The method is based on the generation of 2-qubit Bell states and the introduction of quantum noise in a controlled way on one of the state subsystems. The efficiency of the optimal estimation, achieved by a Bell measurement, is shown to outperform quantum process tomography

On the number of limit cycles of the Lienard equation

In this paper, we study a Lienard system of the form dot{x}=y-F(x), dot{y}=-x, where F(x) is an odd polynomial. We introduce a method that gives a sequence of algebraic approximations to the equation of each limit cycle of the system. This sequence seems to converge to the exact equation of each limit cycle. We obtain also a sequence of polynomials R_n(x) whose roots of odd multiplicity are related to the number and location of the limit cycles of the system.Comment: 10 pages, 5 figures. Submitted to Physical Review

Periodic analytic approximate solutions for the Mathieu equation

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