Valter Franceschini

Sequences of Infinite Bifurcations and Turbulence in a Five-Mode Truncation of the Navier-Stokes Equations

Two infinite sequences of orbits leading to turbulence in a five-mode truncation of the Navier-Stokes equations for a 2-dimensional incompressible fluid on a torus are studied in detail. Their compatibility with Feigenbaums theory of universality in certain infinite sequences of bifurcations is verified and some considerations on their asymptotic behavior are inferred. An analysis of the Poincaré map is performed, showing how the turbulent behavior is approached gradually when, with increasing Reynolds number, no stable fixed point or periodic orbit is present and all the unstable ones become more and more unstable, in close analogy with the Lorenz model.

Bifurcations of tori and phase locking in a dissipative system of differential equations

Abstract In a system of ordinary differential equations, obtained through a seven-mode truncation of the plane incompressible Navier-Stokes equations, a two-dimensional torus undergoes first two period-doubling bifurcations and then a transition to a strange attractor. This strange attractor, of Liapunov dimension larger than three in a wide parameter interval, is characterized by a power spectrum which retains the two fundamental frequencies of the original torus superimposed on a broad, jagged background. As the Liapunov dimension goes down towards two, an interesting phenomenon of phase locking occurs, which gives rise to an alternation of chaotic and periodic behavior.

A Feigenbaum sequence of bifurcations in the Lorenz model

For some high values of the Rayleigh numberr, the Lorenz model exhibits laminar behavior due to the presence of a stable periodic orbit. A detailed numerical study shows that, forr decreasing, the turbulent behavior is reached via an infinite sequence of bifurcations, whereas forr increasing, this is due to a collapse of the stable orbit to a hyperbolic one. The infinite sequence of bifurcations is found to be compatible with Feigenbaums conjecture.

Stable and unstable manifolds of the Hénon mapping

By using a parametric representation of the stable and unstable manifolds, we prove that for some given values of the parameter (in particular in the case first investigated by Hénon) the Hénon mapping has a transversal homoclinic orbit.

A seven-mode truncation of the plane incompressible Navier-Stokes equations

A model obtained by a seven-mode truncation of the Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is studied. This model, extending a previously studied five-mode one, exhibits a very rich and varied phenomenology including some remarkable properties of hysteresis (i.e., coexistence of attractors). A stochastic behavior is found for high values of the Reynolds number, when no stable fixed points, closed orbits, or tori are present.

Common periodic behavior in larger and larger truncations of the Navier-Stokes equations

The periodic behavior ofN-mode truncations of the Navier-Stokes equations on a two-dimensional torus is studied forN=44, 60, 80, and 98. Significant common features are found, particularly for not too high Reynolds numbers. In all models periodicity ends, giving rise, though at quite different parameter values, to quasiperiodicity.

Fixed point limit behavior ofN-mode truncated Navier-Stokes equations asN increases

The fixed point behavior ofN-mode truncations of the Navier-Stokes equations on a two-dimensional torus is investigated asN increases. FromN=44 on the behavior does not undergo any qualitative change. Furthermore, the bifurcations occur at critical parameter values which clearly tend to stabilize asN approaches 100.

Two models of truncated Navier--Stokes equations on a two-dimensional torus

Two truncations of the Navier–Stokes equations for an incompressible fluid on a two‐dimensional torus are numerically investigated. The two models, an eight‐mode truncation and a nine‐mode extension of it, exhibit almost completely different behaviors. Only two aspects appear to be common to both. First, in contrast to all the previously studied models, turbulence takes place at low Reynolds numbers and for critical values close to each other. Second, there is a parameter interval for which the same stable periodic orbit is present in both models. The main feature of the two truncations is a very rich phenomenology, with many different bifurcations of closed orbits. Turbulence is reached through a sequence of period‐doubling bifurcations in the eight‐mode model, and via a tangent bifurcation in the nine‐mode one. Sequences of infinite bifurcations, the presence in a very narrow range of a stable closed orbit with a quite long period, a homoclinic bifurcation, and the coexistence of strange attractors with...

Truncations to 12, 14 and 18 modes of the Navier-Stokes equations on a two-dimensional torus

SommarioTre troncamenti delle equazioni bidimensionali di Navier-Stokes con condizioni periodiche al contorno sono studiati numericamente facendo uso di tecniche basate sulla teoria della biforcazione. I tre troncamenti, rispettivamente a 12, 14 e 18 modi, sono ottenuti considerando tutti i modi contenuti in bocce di raggio crescente. Per quanto un dettagliato confronto delle fenomenologie sia privo di significato, i tre modelli mostrano alcune caratteristiche globali comuni. Il comportamento di ciascun modello è infatti descritto attraverso tre differenti storie che partono da tre punti fissi distinti e si sviluppano parallelamente. Due storie sono caratterizzate dalla presenza di punti fissi ed orbite periodiche, mentre la terza coinvolge anche tori bidimensionali. I fenomeni di maggior interesse, soprattutto nel contesto dei sistemi dinamici, sembrano essere la rottura dei tori e la scomparsa, per crisi, di attrattori strani.SummaryThree truncations of the Navier-Stokes equations on a two-dimensional torus are numerically investigated by making use of techniques based on bifurcation theory. The three truncations, to 12, 14 and 18 modes respectively, are obtained by taking into account all the modes contained in balls of increasing radius. While a comparison of the details of the phenomenologies is meaningless, the three models show common global qualitative features. In fact the behaviour of each model is described by three different stories which start from three distinct fixed points and develop parallely. Two stories are characterized by the presence of fixed points and periodic orbits, the third one involves also two-dimensional tori. The three truncations exhibit a surprisingly rich collection of bifurcations. Breaking of tori and disappearance of strange attractors by crisis seem to be the phenomena of greatest interest, particularly in the framework of dynamical systems.

Truncated Navier-Stokes Equations on a Two-Dimensional Torus

The behavior of complex truncated Navier-Stokes equations is investigated. A significant role in the phenomenology of these equations is played by a set of infinitely many invariant N-dimensional hyperplanes (subspaces of the 2N-dimensional phase space) which are symmetrically placed by virtue of a symmetry group. There is numerical evidence that, at least for Reynolds numbers below a certain critical threshold, every random initial point is captured first by one of these hyperplanes, then by an attractor on it. Particular bifurcations can take place in the 2N-phase space. A fixed point can bifurcate into a closed curve consisting entirely of fixed points; in addition, a closed curve of fixed points can bifurcate into a torus covered entirely by either fixed points or periodic orbits. As long as the invariant hyperplanes are attracting, a complete correspondence is found between the phenomenologies of some complex models and their previously studied real versions.