Non-convex-valued differential inclusions in Banach spaces

Digitalitzat per Artypla

Estimating long term behavior of flows without trajectory integration: the infinitesimal generator approach

The long-term distributions of trajectories of a flow are described by invariant densities, i.e. fixed points of an associated transfer operator. In addition, global slowly mixing structures, such as almost-invariant sets, which partition phase space into regions that are almost dynamically disconnected, can also be identified by certain eigenfunctions of this operator. Indeed, these structures are often hard to obtain by brute-force trajectory-based analyses. In a wide variety of applications, transfer operators have proven to be very efficient tools for an analysis of the global behavior of a dynamical system. The computationally most expensive step in the construction of an approximate transfer operator is the numerical integration of many short term trajectories. In this paper, we propose to directly work with the infinitesimal generator instead of the operator, completely avoiding trajectory integration. We propose two different discretization schemes; a cell based discretization and a spectral collocation approach. Convergence can be shown in certain circumstances. We demonstrate numerically that our approach is much more efficient than the operator approach, sometimes by several orders of magnitude

Comparison of averages of flows and maps

It is shown that in transient chaos there is no direct relation between averages in a continuos time dynamical system (flow) and averages using the analogous discrete system defined by the corresponding Poincare map. In contrast to permanent chaos, results obtained from the Poincare map can even be qualitatively incorrect. The reason is that the return time between intersections on the Poincare surface becomes relevant. However, after introducing a true-time Poincare map, quantities known from the usual Poincare map, such as conditionally invariant measure and natural measure, can be generalized to this case. Escape rates and averages, e.g. Liapunov exponents and drifts can be determined correctly using these novel measures. Significant differences become evident when we compare with results obtained from the usual Poincare map.Comment: 4 pages in Revtex with 2 included postscript figures, submitted to Phys. Rev.

Diffusion in normal and critical transient chaos

In this paper we investigate deterministic diffusion in systems which are spatially extended in certain directions but are restricted in size and open in other directions, consequently particles can escape. We introduce besides the diffusion coefficient D on the chaotic repeller a coefficient ${\hat D}$ which measures the broadening of the distribution of trajectories during the transient chaotic motion. Both coefficients are explicitly computed for one-dimensional models, and they are found to be different in most cases. We show furthermore that a jump develops in both of the coefficients for most of the initial distributions when we approach the critical borderline where the escape rate equals the Liapunov exponent of a periodic orbit.Comment: 4 pages Revtex file in twocolumn format with 2 included postscript figure

Bursts in the Chaotic Trajectory Lifetimes Preceding the Controlled Periodic Motion

The average lifetime ($\tau(H)$) it takes for a randomly started trajectory to land in a small region ($H$) on a chaotic attractor is studied. $\tau(H)$ is an important issue for controlling chaos. We point out that if the region $H$ is visited by a short periodic orbit, the lifetime $\tau(H)$ strongly deviates from the inverse of the naturally invariant measure contained within that region ($\mu_N(H)^{-1}$). We introduce the formula that relates $\tau(H)/\mu_N(H)^{-1}$ to the expanding eigenvalue of the short periodic orbit visiting $H$.Comment: Accepted for publication in Phys. Rev. E, 3 PS figure

marker tracking for local strain measurement in mechanical testing of biomedical materials

Local strain measurement is one of the key aspects in tensile tests of biomedical materials and biological tissues, especially if aimed at developing appropriate constitutive formulations to describe mechanical behavior. The measurement of strain as the ratio between the current and the initial length of the sample can be coupled with markers recognition via non-contact video extensometer for characterizing the local mechanical behavior. A crucial point in video extensometer measurement is the selection of the most appropriate markers and technique of their application on the sample surface. This work promotes understanding the effect of markers on material mechanical response. Different solutions were taken into account, as paint markers, namely a commercial lacquer and an acrylic paint, or physical markers attached with the use of adhesives, i.e. cyanoacrylate or medical spray band. Tensile tests revealed that markers can modify the mechanical response of the tested materials, inducing a local stiffening of the samples. The use of cyanoacrylate, as marker adhesive, affects not only the local but also the overall mechanical response, at least for the sample size considered in this work. These effects are more pronounced with higher material compliance. Based on these results, caution is recommended with the use of cyanoacrylate for attaching markers on biomedical polymers

Behavior of the Escape Rate Function in Hyperbolic Dynamical Systems

For a fixed initial reference measure, we study the dependence of the escape rate on the hole for a smooth or piecewise smooth hyperbolic map. First, we prove the existence and Holder continuity of the escape rate for systems with small holes admitting Young towers. Then we consider general holes for Anosov diffeomorphisms, without size or Markovian restrictions. We prove bounds on the upper and lower escape rates using the notion of pressure on the survivor set and show that a variational principle holds under generic conditions. However, we also show that the escape rate function forms a devil's staircase with jumps along sequences of regular holes and present examples to elucidate some of the difficulties involved in formulating a general theory.Comment: 21 pages. v2 differs from v1 only by additions to the acknowledgment

Spectral degeneracy and escape dynamics for intermittent maps with a hole

We study intermittent maps from the point of view of metastability. Small neighbourhoods of an intermittent fixed point and their complements form pairs of almost-invariant sets. Treating the small neighbourhood as a hole, we first show that the absolutely continuous conditional invariant measures (ACCIMs) converge to the ACIM as the length of the small neighbourhood shrinks to zero. We then quantify how the escape dynamics from these almost-invariant sets are connected with the second eigenfunctions of Perron-Frobenius (transfer) operators when a small perturbation is applied near the intermittent fixed point. In particular, we describe precisely the scaling of the second eigenvalue with the perturbation size, provide upper and lower bounds, and demonstrate $L^1$ convergence of the positive part of the second eigenfunction to the ACIM as the perturbation goes to zero. This perturbation and associated eigenvalue scalings and convergence results are all compatible with Ulam's method and provide a formal explanation for the numerical behaviour of Ulam's method in this nonuniformly hyperbolic setting. The main results of the paper are illustrated with numerical computations.Comment: 34 page

Evaluation of a microbiological screening and acceptance procedure for cryopreserved skin allografts based on 14 day cultures

Viable donor skin is still considered the gold standard for the temporary covering of burns. Since 1985, the Brussels military skin bank supplies cryopreserved viable cadaveric skin for therapeutic use. Unfortunately, viable skin can not be sterilised, which increases the risk of disease transmission. On the other hand, every effort should be made to ensure that the largest possible part of the donated skin is processed into high-performance grafts. Cryopreserved skin allografts that fail bacterial or fungal screening are reworked into ‘sterile’ non-viable glycerolised skin allografts. The transposition of the European Human Cell and Tissue Directives into Belgian Law has prompted us to install a pragmatic microbiological screening and acceptance procedure, which is based on 14 day enrichment broth cultures of finished product samples and treats the complex issues of ‘acceptable bioburden’ and ‘absence of objectionable organisms’. In this paper we evaluate this procedure applied on 148 skin donations. An incubation time of 14 days allowed for the detection of an additional 16.9% (25/148) of contaminated skin compared to our classic 3 day incubation protocol and consequently increased the share of non-viable glycerolised skin with 8.4%. Importantly, 24% of these slow-growing microorganisms were considered to be potentially pathogenic. In addition, we raise the issue of ‘representative sampling’ of heterogeneously contaminated skin. In summary, we feel that our present microbiological testing and acceptance procedure assures adequate patient safety and skin availability. The question remains, however, whether the supposed increased safety of our skin grafts outweighs the reduced overall clinical performance and the increase in work load and costs

Open Mushrooms: Stickiness revisited

We investigate mushroom billiards, a class of dynamical systems with sharply divided phase space. For typical values of the control parameter of the system $\rho$, an infinite number of marginally unstable periodic orbits (MUPOs) exist making the system sticky in the sense that unstable orbits approach regular regions in phase space and thus exhibit regular behaviour for long periods of time. The problem of finding these MUPOs is expressed as the well known problem of finding optimal rational approximations of a real number, subject to some system-specific constraints. By introducing a generalized mushroom and using properties of continued fractions, we describe a zero measure set of control parameter values $\rho\in(0,1)$ for which all MUPOs are destroyed and therefore the system is less sticky. The open mushroom (billiard with a hole) is then considered in order to quantify the stickiness exhibited and exact leading order expressions for the algebraic decay of the survival probability function $P(t)$ are calculated for mushrooms with triangular and rectangular stems.Comment: 21 pages, 11 figures. Includes discussion of a three-dimensional mushroo