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]

On the Dynamics of the Unified Chaotic System Between Lorenz and Chen Systems

PublishedA one-parameter family of differential systems that bridges the gap between the Lorenz and the Chen systems was proposed by Lu, Chen, Cheng and Celikovsy. The goal of this paper is to analyze what we can say using analytic tools about the dynamics of this one-parameter family of differential systems. We shall describe its global dynamics at infinity, and for two special values of the parameter a we can also describe the global dynamics in the whole ℝ3 using the invariant algebraic surfaces of the family. Additionally we characterize the Hopf bifurcations of this family.The first author is partially supported by a MINECO/FEDER grant MTM2008-03437 and MTM2013-40998-P, an AGAUR grant number 2014SGR-568, an ICREA Academia, the grants FP7-PEOPLE-2012-IRSES 318999 and 316338, and UNAB 13-4E-1604

Energetics, skeletal dynamics and long-term predictions in Kolmogorov-Lorenz systems

We study a particular return map for a class of low dimensional chaotic models called Kolmogorov Lorenz systems, which received an elegant general Hamiltonian description and includes also the famous Lorenz63 case, from the viewpoint of energy and Casimir balance. In particular it is considered in detail a subclass of these models, precisely those obtained from the Lorenz63 by a small perturbation on the standard parameters, which includes for example the forced Lorenz case in Ref.[6]. The paper is divided into two parts. In the first part the extremes of the mentioned state functions are considered, which define an invariant manifold, used to construct an appropriate Poincare surface for our return map. From the experimental observation of the simple orbital motion around the two unstable fixed points, together with the circumstance that these orbits are classified by their energy or Casimir maximum, we construct a conceptually simple skeletal dynamics valid within our sub class, reproducing quite well the Lorenz map for Casimir. This energetic approach sheds some light on the physical mechanism underlying regime transitions. The second part of the paper is devoted to the investigation of a new type of maximum energy based long term predictions, by which the knowledge of a particular maximum energy shell amounts to the knowledge of the future (qualitative) behaviour of the system. It is shown that, in this respect, a local analysis of predictability is not appropriate for a complete characterization of this behaviour. A perspective on the possible extensions of this type of predictability analysis to more realistic cases in (geo)fluid dynamics is discussed at the end of the paper.Comment: 21 pages, 14 figure

Quadratic three-dimensional differential systems having invariant planes with total multiplicity nine

In this paper we consider all the quadratic polynomial differential systems in R having exactly nine invariant planes taking into account their multiplicities. This is the maximum number of invariant planes that these kind of systems can have, without taking into account the infinite plane. We prove that there exist thirty possible configurations for these invariant planes, and we study the realization and the existence of first integrals for each one of these configurations. We show that at least twenty three of these configurations are realizable and provide explicit examples for each one of them