Algorithm for rigorous integration of Delay Differential Equations and the computer-assisted proof of periodic orbits in the Mackey-Glass equation

We present an algorithm for the rigorous integration of Delay Differential Equations (DDEs) of the form $x'(t)=f(x(t-\tau),x(t))$. As an application, we give a computer assisted proof of the existence of two attracting periodic orbits (before and after the first period-doubling bifurcation) in the Mackey-Glass equation

High-order Lohner-type algorithm for rigorous computation of Poincar\'e maps in systems of Delay Differential Equations with several delays

We present a Lohner-type algorithm for rigorous integration of systems of Delay Differential Equations (DDEs) with multiple delays and its application in computation of Poincar\'e maps to study the dynamics of some bounded, eternal solutions. The algorithm is based on a piecewise Taylor representation of the solutions in the phase-space and it exploits the smoothing of solutions occurring in DDEs to produces enclosures of solutions of a high order. We apply the topological techniques to prove various kinds of dynamical behavior, for example, existence of (apparently) unstable periodic orbits in Mackey-Glass Equation (in the regime of parameters where chaos is numerically observed) and persistence of symbolic dynamics in a delay-perturbed chaotic ODE (the R\"ossler system)

A computer assisted proof of multiple periodic orbits in some first order non-linear delay differential equation

We present an application of a recently developed algorithm for rigorous integration forward in time of delay differential equations (DDEs) to a computer assisted proof of the existence of several periodic orbits in a DDE obtained by a singular perturbation limit method from the classical logistic map. The proofs are done near the parameter value for which multistability was numerically observed

Network of lipid interconnections at the interfaces of galactolipid and phospholipid bilayers

Interactions among lipid head groups at the bilayer/water interface do, to a large extent, determine membrane properties. In this study graph theory is employed to objectively describe and compare the pattern of the interactions at the interfaces of computer models of 128- and 512-lipid monogalactolipid (MGDG) and phosphatidylcholine (DOPC) bilayers. Both MGDG and DOPC have polar head groups but of different chemical structures so at the bilayer interfaces they participate in different types of interaction. Nevertheless, at both interfaces these interactions and the lipid molecules they link make networks. In graph theory, a network of interconnected objects (nodes) is described by well-defined quantities which define its topology and can be used to assess inner properties of the network, its strength and density, etc. In this study, several topological properties of the networks in the DOPC and MGDG bilayers are determined. A comparison of these properties indicates that the topologies of both networks differ significantly but are stable during the simulation time. The networks in the MGDG bilayers are more extended, branched, stable, and stronger than those in the DOPC bilayers. This is consistent with the smaller surface area per lipid and higher rigidity of the MGDG than the DOPC bilayers as well as the tendency of MGDG to form an inverse hexagonal phase in water. The scale of the systems is an important factor when assessing the properties of the network; the system scaling is more evident in the DOPC bilayers where several quantities increase directly proportional to the increasing size of the system than in the MGDG bilayers where this is rarely the case

Lipid/water interface of galactolipid bilayers in different lyotropic liquid-crystalline

In this study, carried out using computational methods, the organisation of the lipid/water interface of bilayers composed of galactolipids with both α-linolenoyl acyl chains is analysed and compared in three different lyotropic liquid-crystalline phases. These systems include the monogalactosyldiglyceride (MGDG) and digalactosyldiglyceride (DGDG) bilayers in the lamellar phase, the MGDG double bilayer during stalk phase formation and the inverse hexagonal MGDG phase. For each system, lipid-water and direct and water-mediated lipid-lipid interactions between the lipids of one bilayer leaflet and those of two apposing leaflets at the onset of new phase (stalk) formation, are identified. A network of interactions between DGDG molecules and its topological properties are derived and compared to those for the MGDG bilayer

Existence of homoclinic orbit in some hydrodynamical system describing relaxing media. A computer assisted proof.

W pracy udowadniono istnienie orbity homoklinicznej dopewnego punktu sta.ego dla sparametryzowanej rodzinyrowna. ro.niczkowych zwyczajnych:{đîńHđßRđŕ R'=FńH đßR,đňđŕđîńH đßRđŕđň'=HńH đßR,đňđŕ (1)gdzie R:.đÍ. , đň:.đÍ. s. szukanymi funkcjami,đîńH :.đÍ. , FńH:.2đÍ. , HńH :.2đÍ. s. wielomianamiw zmiennych R and đň oraz punkt sta.y orbity le.y nalinii osobliwych punktow đîńH đßRđŕ=0 . Indeks ńH oznaczazale.no.. od parametru.Analizowane rownanie jest otrzymane z rownaniaro.niczkowego cz.stkowego poprzez odpowiedniepodstawienia. Wyj.ciowe rownanie cz.stkowe jest tematempracy .Compacton-like solutionsof the hydrodynamic system describing relaxing media hdra Vsevoloda Vladimirova.Dowod istnienia orbity homoklinicznej jest wspieranykomputerowo

A homoclinic orbit in a planar singular ODE : a computer assisted proof

We consider a family of 2-dimensional ODEs of the form $\Delta_\xi(z)z' = f_\xi(z)$ depending on a real parameter $\xi$ which was investigated by Vladimirov [Rep. Math. Phys., 61 (2008), pp. 381--400]. In this system, there exist stationary points $p_\xi$ which belong to the set of zeros of $\Delta_\xi$. We prove, using rigorous numerics, the existence of a homoclinic orbit to $p_\xi$ for some parameter value $\xi = \xi_h$. Due to the singularity of the system it takes a finite time to travel along this orbit, and this property gives rise to a compacton-like traveling wave in some hydrodynamic system describing relaxing media. Our approach could be used to prove similar results in other singular systems as well

DMG-αA Computational Geometry Library for Multimolecular Systems

The DMG-α library grants researchers in the field of computational biology, chemistry, and biophysics access to an open-sourced, easy to use, and intuitive software for performing fine-grained geometric analysis of molecular systems. The library is capable of computing power diagrams (weighted Voronoi diagrams) in three dimensions with 3D periodic boundary conditions, computing approximate projective 2D Voronoi diagrams on arbitrarily defined surfaces, performing shape properties recognition using α-shape theory and can do exact Solvent Accessible Surface Area (SASA) computation. The software is written mainly as a template-based C++ library for greater performance, but a rich Python interface (pydmga) is provided as a convenient way to manipulate the DMG-α routines. To illustrate possible applications of the DMG-α library, we present results of sample analyses which allowed to determine nontrivial geometric properties of two Escherichia coli-specific lipids as emerging from molecular dynamics simulations of relevant model bilayers