Bifurcations of homoclinic tangency in two-dimensional area-preserving and reversible maps

Report for the scientific sojourn at the Research Institute for Applied Mathematics and Cybernetics, Nizhny Novgorod, Russia, from July to September 2006. Within the project, bifurcations of orbit behavior in area-preserving and reversible maps with a homoclinic tangency were studied. Finitely smooth normal forms for such maps near saddle fixed points were constructed and it was shown that they coincide in the main order with the analytical Birkhoff-Moser normal form. Bifurcations of single-round periodic orbits for two-dimensional symplectic maps close to a map with a quadratic homoclinic tangency were studied. The existence of one- and two-parameter cascades of elliptic periodic orbits was proved.

Non-hyperbolic boundary equilibrium bifurcations in planar Filippov systems: a case study approach

Boundary equilibrium bifurcations in piecewise smooth discontinuous systems are characterized by the collision of an equilibrium point with the discontinuity surface. Generically, these bifurcations are of codimension one, but there are scenarios where the phenomenon can be of higher codimension. Here, the possible collision of a non-hyperbolic equilibrium with the boundary in a two-parameter framework and the nonlinear phenomena associated with such collision are considered. By dealing with planar discontinuous (Filippov) systems, some of such phenomena are pointed out through specific representative cases. A methodology for obtaining the corresponding bi-parametric bifurcation sets is developed.

Bifurcations and chaos in single-roll natural convection with low Prandtl number

Convective flows of a small Prandtl number fluid contained in a two-dimensional cavity subject to a lateral thermal gradient are numerically studied by using different techniques. The aspect ratio (length to height) is kept at around 2. This value is found optimal to make the flow most unstable while keeping the basic single-roll structure. Two cases of thermal boundary conditions on the horizontal plates are considered: perfectly conducting and adiabatic. For increasing Rayleigh numbers we find a transition from steady flow to periodic oscillations through a supercritical Hopf bifurcation that maintains the centrosymmetry of the basic circulation. For a Rayleigh number of about ten times that of the Hopf bifurcation the system initiates a complex scenario of bifurcations. In the conductive case these include a quasiperiodic route to chaos. In the adiabatic one the dynamics is dominated by the interaction of two Neimark-Sacker bifurcations of the basic periodic solutions, leading to the stable coexistence of three incommensurate frequencies, and finally to chaos. In all cases, the complex time-dependent behavior does not break the basic, single-roll structure.

Interlinkages and forecasting made possible by the use of the DECOIN analytical tool kit

This technical report is a document prepared as a deliverable [D4.3 Report of the Interlinkages and forecasting prototype tool] of a EU project – DECOIN Project No. 044428 - FP6-2005-SSP-5A. The text is divided into 4 sections: (1) this short introductory section explains the purpose of the report; (2) the second section provides a general discussion of a systemic problem found in existing quantitative analysis of sustainability. It addresses the epistemological implications of complexity, which entails the need of dealing with the existence of Multiple-Scales and non-equivalent narratives (multiple dimensions/attributes) to be used to define sustainability issues. There is an unavoidable tension between a “steady-state view” (= the perception of what is going on now – reflecting a PAST --& PRESENT view of the reality) versus an “evolutionary view” (= the unknown transformation that we have to expect in the process of becoming of the observed reality and in the observer – reflecting a PRESENT --& FUTURE view of the reality). The section ends by listing the implications of these points on the choice of integrated packages of sustainability indicators; (3) the third section illustrates the potentiality of the DECOIN toolkit for the study of sustainability trade-offs and linkages across indicators using quantitative examples taken from cases study of another EU project (SMILE). In particular, this section starts by addressing the existence of internal constraints to sustainability (economic versus social aspects). The narrative chosen for this discussion focuses on the dark side of ageing and immigration on the economic viability of social systems. Then the section continues by exploring external constraints to sustainability (economic development vs the environment). The narrative chosen for this discussion focuses on the dark side of current strategy of economic development based on externalization and the “bubbles-disease”; (4) the last section presents a critical appraisal of the quality of energy data found in energy statistics. It starts with a discussion of the general goal of statistical accounting. Then it introduces the concept of multipurpose grammars. The second part uses the experience made in the activities of the DECOIN project to answer the question: how useful are EUROSTAT energy statistics? The answer starts with an analysis of basic epistemological problems associated with accounting of energy. This discussion leads to the acknowledgment of an important epistemological problem: the unavoidable bifurcations in the mechanism of accounting needed to generate energy statistics. By using numerical example the text deals with the following issues: (i) the pitfalls of the actual system of accounting in energy statistics; (ii) a critical appraisal of the actual system of accounting in BP statistics; (iii) a critical appraisal of the actual system of accounting in Eurostat statistics. The section ends by proposing an innovative method to represent energy statistics which can result more useful for those willing develop sustainability indicators.

N-dimensional dynamical systems exploiting instabilities in full

We report experimental and numerical results showing how certain N-dimensional dynamical systems are able to exhibit complex time evolutions based on the nonlinear combination of N-1 oscillation modes. The experiments have been done with a family of thermo-optical systems of effective dynamical dimension varying from 1 to 6. The corresponding mathematical model is an N-dimensional vector field based on a scalar-valued nonlinear function of a single variable that is a linear combination of all the dynamic variables. We show how the complex evolutions appear associated with the occurrence of successive Hopf bifurcations in a saddle-node pair of fixed points up to exhaust their instability capabilities in N dimensions. For this reason the observed phenomenon is denoted as the full instability behavior of the dynamical system. The process through which the attractor responsible for the observed time evolution is formed may be rather complex and difficult to characterize. Nevertheless, the well-organized structure of the time signals suggests some generic mechanism of nonlinear mode mixing that we associate with the cluster of invariant sets emerging from the pair of fixed points and with the influence of the neighboring saddle sets on the flow nearby the attractor. The generation of invariant tori is likely during the full instability development and the global process may be considered as a generalized Landau scenario for the emergence of irregular and complex behavior through the nonlinear superposition of oscillatory motions

Mophological analysis of the human vasculature from medical imaging. Application to cerebral and coranary arteries

In this project, we have investigated new ways of modelling and analysis of human vasculature from Medical images. The research was divided in two main areas: cerebral vasculature analysis and coronary arteries modeling. Regarding cerebral vasculature analysis, we have studed cerebral aneurysms, internal carotid and the Circle of Willis (CoW). Aneurysms are abnormal vessel enlargements that can rupture causing important cerebral damages or death. The understanding of this pathology, together with its virtual treatment, and image diagnosis and prognosis, includes identification and detailed measurement of the aneurysms. In this context, we have proposed two automatic aneurysm isolation method, to separate the abnormal part of the vessel from the healthy part, to homogenize and speed-up the processing pipeline usually employed to study this pathology, [Cardenes2011TMI, arrabide2011MedPhys]. The results obtained from both methods have been also compared and validatied in [Cardenes2012MBEC]. A second important task here the analysis of the internal carotid [Bogunovic2011Media] and the automatic labelling of the CoW, Bogunovic2011MICCAI, Bogunovic2012TMI]. The second area of research covers the study of coronary arteries, specially coronary bifurcations because there is where the formation of atherosclerotic plaque is more common, and where the intervention is more challenging. Therefore, we proposed a novel modelling method from Computed Tomography Angiography (CTA) images, combined with Conventional Coronary Angiography (CCA), to obtain realistic vascular models of coronary bifurcations, presented in [Cardenes2011MICCAI], and fully validated including phantom experiments in [Cardene2013MedPhys]. The realistic models obtained from this method are being used to simulate stenting procedures, and to investigate the hemodynamic variables in coronary bifurcations in the works submitted in [Morlachi2012, Chiastra2012]. Additionally, another preliminary work has been done to reconstruct the coronary tree from rotational angiography, and published in [Cardenes2012ISBI].

A subcritical instability of wave-driven alongshore currents

The development of shear instabilities of a wave-driven alongshore current is investigated. In particular, we use weakly nonlinear theory to investigate the possibility that such instabilities, which have been observed at various sites on the U.S. coast and in the laboratory, can grow in linearly stable flows as a subcritical bifurcation by resonant triad interaction, as first suggested by Shrira eta/. [1997]. We examine a realistic longshore current profile and include the effects of eddy viscosity and bottom friction. We show that according to the weakly nonlinear theory, resonance is possible and that these linearly stable flows may exhibit explosive instabilities. We show that this phenomenon may occur also when there is only approximate resonance, which is more likely in nature. Furthermore, the size of the perturbation that is required to trigger the instability is shown in some circumstances to be consistent with the size of naturally occurring perturbations. Finally, we consider the differences between the present case examined and the more idealized case of Shrira et a/. [ 1997]. It is shown that there is a possibility of coupling between triads, due to the richer modal structure in more realistic flows, which may act to stabilize the flow and act against the development of subcritical bifurcations. Extensive numerical tests are called for.

A survey on the inverse integrating factor

The relation between limit cycles of planar differential systems and the inverse integrating factor was first shown in an article of Giacomini, Llibre and Viano appeared in 1996. From that moment on, many research articles are devoted to the study of the properties of the inverse integrating factor and its relationwith limit cycles and their bifurcations. This paper is a summary of all the results about this topic. We include a list of references together with the corresponding related results aiming at being as much exhaustive as possible. The paper is, nonetheless, self-contained in such a way that all the main results on the inverse integrating factor are stated and a complete overview of the subject is given. Each section contains a different issue to which the inverse integrating factor plays a role: the integrability problem, relation with Lie symmetries, the center problem, vanishing set of an inverse integrating factor, bifurcation of limit cycles from either a period annulus or from a monodromic ω-limit set and some generalizations.

A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: explorations and mechanisms for the breakdown of hyperbolicity

In two previous papers [J. Differential Equations, 228 (2006), pp. 530 579; Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), pp. 1261 1300] we have developed fast algorithms for the computations of invariant tori in quasi‐periodic systems and developed theorems that assess their accuracy. In this paper, we study the results of implementing these algorithms and study their performance in actual implementations. More importantly, we note that, due to the speed of the algorithms and the theoretical developments about their reliability, we can compute with confidence invariant objects close to the breakdown of their hyperbolicity properties. This allows us to identify a mechanism of loss of hyperbolicity and measure some of its quantitative regularities. We find that some systems lose hyperbolicity because the stable and unstable bundles approach each other but the Lyapunov multipliers remain away from 1. We find empirically that, close to the breakdown, the distances between the invariant bundles and the Lyapunov multipliers which are natural measures of hyperbolicity depend on the parameters, with power laws with universal exponents. We also observe that, even if the rigorous justifications in [J. Differential Equations, 228 (2006), pp. 530-579] are developed only for hyperbolic tori, the algorithms work also for elliptic tori in Hamiltonian systems. We can continue these tori and also compute some bifurcations at resonance which may lead to the existence of hyperbolic tori with nonorientable bundles. We compute manifolds tangent to nonorientable bundles.

Full instability behavior of N-dimensional dynamical systems with a one-directional nonlinear vector field

We show how certain N-dimensional dynamical systems are able to exploit the full instability capabilities of their fixed points to do Hopf bifurcations and how such a behavior produces complex time evolutions based on the nonlinear combination of the oscillation modes that emerged from these bifurcations. For really different oscillation frequencies, the evolutions describe robust wave form structures, usually periodic, in which selfsimilarity with respect to both the time scale and system dimension is clearly appreciated. For closer frequencies, the evolution signals usually appear irregular but are still based on the repetition of complex wave form structures. The study is developed by considering vector fields with a scalar-valued nonlinear function of a single variable that is a linear combination of the N dynamical variables. In this case, the linear stability analysis can be used to design N-dimensional systems in which the fixed points of a saddle-node pair experience up to N21 Hopf bifurcations with preselected oscillation frequencies. The secondary processes occurring in the phase region where the variety of limit cycles appear may be rather complex and difficult to characterize, but they produce the nonlinear mixing of oscillation modes with relatively generic features