Shankar C. Venkataramani

Diffusion, attraction and collapse

We study a parabolic-elliptic system of partial differential equations that arises in modelling the overdamped gravitational interaction of a cloud of particles or chemotaxis in bacteria. The system has a rich dynamics and the possible behaviours of the solutions include convergence to time-independent solutions and the formation of finite-time singularities. Our goal is to describe the different kinds of solutions that lead to these outcomes. We restrict our attention to radial solutions and find that the behaviour of the system depends strongly on the space dimension d. For 2<d<10 there are two stable blowup modalities (self-similar and Burgers-like) and one stable steady state. On unbounded domains, there exists a one-parameter family of unstable steady solutions and a countable number of unstable blowup behaviours. We document connections between one unstable blowup behaviour and both a stable steady state and a stable blowup, as well as connections between one unstable blowup and two different stable blowups. There is a topological and stability correspondence between the various asymptotic behaviours and this suggests the possibility of constructing a global phase portrait for the system that treats the global in time solutions and the blowing up solutions on an equal footing.

On-off intermittency: power spectrum and fractal properties of time series

Abstract Some dynamical systems possess invariant submanifolds such that the dynamics restricted to the invariant submanifold is chaotic. This situation arises in systems with a spatial symmetry or in the synchronization of chaotic oscillators. The invariant submanifold could become unstable to perturbations in the transverse directions when a parameter of the system is changed through a critical blow-out value. This could result in an extreme form of temporally intermittent bursting called on-off intermittency . We propose a model that incorporates the universal features of systems that display on-off intermittency. We study this model both with and without additive noise and we derive scaling results for the power spectral density of the on-off intermittent process and for the box counting dimension for the set of time intervals when the process takes on values above a given threshold. We then present numerical simulations realizing these results.

Topology of event horizons and topological censorship

We prove that, under certain conditions, the topology of the event horizon of a four-dimensional asymptotically flat black-hole spacetime must be a 2-sphere. No stationarity assumption is made. However, in order for the theorem to apply, the horizon topology must be unchanging for long enough to admit a certain kind of cross section. We expect this condition is generically satisfied if the topology is unchanging for much longer than the light-crossing time of the black hole. More precisely, let M be a four-dimensional asymptotically flat spacetime satisfying the averaged null energy condition, and suppose that the domain of outer communication to the future of a cut K of is globally hyperbolic. Suppose further that a Cauchy surface for is a topological 3-manifold with compact boundary in M, and is a compact submanifold of with spherical boundary in (and possibly other boundary components in ). Then we prove that the homology group must be finite. This implies that either consists of a disjoint union of 2-spheres, or is non-orientable and contains a projective plane. Furthermore, , and will be a cross section of the horizon as long as no generator of becomes a generator of . In this case, if is orientable, the horizon cross section must consist of a disjoint union of 2-spheres.

Lower bounds for the energy in a crumpled elastic sheet—a minimal ridge

We study the linearized Foppl–von Karman theory of a long, thin rectangular elastic membrane that is bent through an angle 2 α. We prove rigorous bounds for the minimum energy of this configuration in terms of the plate thickness, σ, and the bending angle. We show that the minimum energy scales as σ5/3 α7/3. This scaling is in sharp contrast with previously obtained results for the linearized theory of thin sheets with isotropic compression boundary conditions, where the energy scales as σ.We study the linearized Fopl - von Karman theory of a long, thin rectangular elastic membrane that is bent through an angle

Characterization of on-off intermittent time series

2 \alpha

Singularities, structures, and scaling in deformed m-dimensional elastic manifolds.

. We prove rigorous bounds for the minimum energy of this configuration in terms of the plate thickness

Shape selection in non-Euclidean plates

\sigma

A Variational Theory for Point Defects in Patterns

and the bending angle. We show that the minimum energy scales as

Limitations on the smooth confinement of an unstretchable manifold

\sigma^{5/3} \alpha^{7/3}

Discrete charges on a two dimensional conductor

. This scaling is in sharp contrast with previously obtained results for the linearized theory of thin sheets with isotropic compression boundary conditions, where the energy scales as