This article proposes a topological method that extracts hierarchical structures of various amorphous solids. The method is based on the persistence diagram (PD), a mathematical tool for capturing shapes of multiscale data. The input to the PDs is given by an atomic configuration and the output is expressed as 2D histograms. Then, specific distributions such as curves and islands in the PDs identify meaningful shape characteristics of the atomic configuration. Although the method can be applied to a wide variety of disordered systems, it is applied here to silica glass, the Lennard-Jones system, and Cu-Zr metallic glass as standard examples of continuous random network and random packing structures. In silica glass, the method classified the atomic rings as short-range and medium-range orders and unveiled hierarchical ring structures among them. These detailed geometric characterizations clarified a real space origin of the first sharp diffraction peak and also indicated that PDs contain information on elastic response. Even in the Lennard-Jones system and Cu-Zr metallic glass, the hierarchical structures in the atomic configurations were derived in a similar way using PDs, although the glass structures and properties substantially differ from silica glass. These results suggest that the PDs provide a unified method that extracts greater depth of geometric information in amorphous solids than conventional methods.T he atomic configurations of amorphous solids are difficult to characterize. Because they have no periodicity as found in crystalline solids, only local structures have been analyzed in detail. Although short-range order (SRO) defined by the nearest neighbor is thoroughly studied, it is not sufficient to fully understand the atomic structures of amorphous solids. Therefore, medium-range order (MRO) has been discussed to properly characterize amorphous solids (1-3). Many experimental and simulation studies (4-7) have suggested signatures of MRO such as a first sharp diffraction peak (FSDP) in the structure factor of the continuous random network structure, and a split second peak in the radial distribution function of the random packing structure. However, in contrast to SRO, the geometric interpretation of MRO and the hierarchical structures among different ranges are not yet clear.Among the available methods, the distributions of bond angle and dihedral angle are often used to identify the geometry beyond the scale of SRO. They cannot, however, provide a complete description of MRO because they only deal with the atomic configuration up to the third nearest neighbors. Alternatively, ring statistics are also applied as a conventional combinatorial topological method (2, 8, 9). However, this method is applicable only for the continuous random network or crystalline structures, and furthermore it cannot describe length scale. Therefore, methodologies that precisely characterize hierarchical structures beyond SRO and are applicable to a wide variety of amorphous solids are highly desired.In recent years, ...

Icosahedral order has been suggested as the prevalent atomic motif of supercooled liquids and metallic glasses for more than half a century, because the icosahedron is highly close-packed but is difficult to grow, owing to structure frustration and the lack of translational periodicity. By means of angstrom-beam electron diffraction of single icosahedra, we report experimental observation of local icosahedral order in metallic glasses. All the detected icosahedra were found to be distorted with partially face-centered cubic symmetry, presenting compelling evidence on geometric frustration of local icosahedral order in metallic glasses.

We provide explicit criteria for blow-up solutions of autonomous ordinary differential equations. Ideas are based on the quasi-homogeneous desingularization (blowing-up) of singularities and compactifications of phase spaces, which suitably desingularize singularities at infinity. We derive several type of compactifications and show that dynamics at infinity is qualitatively independent of the choice of such compactifications. We also show that hyperbolic invariant sets, such as equilibria and periodic orbits, at infinity induce blow-up solutions with specific blow-up rates. In particular, blow-up solutions can be described as trajectories on stable manifolds of equilibria at infinity for associated vector fields. Finally, we demonstrate blow-up solutions of several differential equations.

This paper focuses on blow-up solutions of ordinary differential equations (ODEs). We present a method for validating blow-up solutions and their blow-up times, which is based on compactifications and the Lyapunov function validation method. The necessary criteria for this construction can be verified using interval arithmetic techniques. Some numerical examples are presented to demonstrate the applicability of our method.

We provide a rigorous numerical computation method to validate periodic, homoclinic and heteroclinic orbits as the continuation of singular limit orbits for the fast-slow system x = f (x, y, ), y = g(x, y, ) with one-dimensional slow variable y. Our validation procedure is based on topological tools called isolating blocks, cone condition and covering relations. Such tools provide us with existence theorems of global orbits which shadow singular orbits in terms of a new concept, the covering-exchange. Additional techniques called slow shadowing and m-cones are also developed. These techniques give us not only generalized topological verification theorems, but also easy implementations for validating trajectories near slow manifolds in a wide range, via rigorous numerics. Our procedure is available to validate global orbits not only for sufficiently small > 0 but all in a given half-open interval (0, 0]. Several sample verification examples are shown as a demonstration of applicability. Kaname MatsueRigorous Numerics for Fast-Slow Systems and the dynamics of the reduced problem 0 = f (x, y, 0), y = g(x, y, 0), (1.4) which are the limiting problems for = 0 on the fast and the slow time scale, respectively. Notice that (1.4) makes sense only on f (x, y, 0) = 0, while (1.3) makes sense in whole R n+1 as the yparameter family of x-systems. The meaning of the " → 0-limit" is thus different between (1.1) and (1.2). This is why (1.1) or (1.2) is a kind of singular perturbation problems. In particular, (1.1) or (1.2) is known as fast-slow systems (or slow-fast systems), where x dominates the behavior in the fast time scale and y dominates the behavior in the slow time scale. When we study the dynamical system of the form (1.1), we often consider limit systems (1.3) and (1.4) independently at first. Then ones try to match them in an appropriate way to obtain trajectories for the full system (1.1). One of major methods for completely solving singularly perturbed systems like (1.1) is the geometric singular perturbation theory formulated by Fenichel [10], Jones-Kopell [20], Szmolyan [32] and many researchers. A series of theories are established so that formally constructed singular limit orbits of (1.3) and (1.4) can perturb to true orbits of (1.1) for sufficiently small > 0. In geometric singular perturbation theory, there are mainly two key points to consider. One is the description of slow dynamics for sufficiently small near the nullcline {(x, y) | f (x, y, 0) = 0}. The other is the matching of fast and slow dynamics. As for the former, Fenichel [10] provided the Invariant Manifold Theorem for describing the dynamics on and around locally invariant manifolds, called slow manifolds, for sufficiently small > 0. Such manifolds can be realized as the perturbation of normally hyperbolic invariant manifolds at = 0, which are often given by submanifolds of nullcline in (1.3). As for the latter, Jones and Kopell [20] originally formulated the geometric answer for the matching problem deriving Exchange Lemma. This lemma infor...

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Abstract. Computer assisted procedures of Lyapunov functions defined in given neighborhoods of fixed points for flows and maps are discussed. We provide a systematic methodology for constructing explicit ranges where quadratic Lyapunov functions exist in two stages; negative definiteness of associating matrices and direct approach. We note that the former is equivalent to the procedure of cones describing enclosures of the stable and the unstable manifolds of invariant sets, which gives us flexible discussions of asymptotic behavior not only around equilibria for flows but also fixed points for maps. Additionally, our procedure admits a re-parameterization of trajectories in terms of values of Lyapunov functions. Several verification examples are shown for discussions of applicability.

Our concerns here are blow-up solutions for ODEs with exponential nonlinearity from the viewpoint of dynamical systems and their numerical validations. As an example, the finite difference discretization of ut = uxx + e u m with the homogeneous Dirichlet boundary condition is considered. Our idea is based on compactification of phase spaces and time-scale desingularization as in previous works. In the present case, treatment of exponential nonlinearity is the main issue. Fortunately, under a kind of exponential homogeneity of vector field, we can treat the problem in the same way as polynomial vector fields. In particular, we can characterize and validate blow-up solutions with their blow-up times for differential equations with such exponential nonlinearity in the similar way to previous works. A series of technical treatments of exponential nonlinearity in blow-up problems is also shown with concrete validation examples.