Takahiro Yajima

Finsler geometry of seismic ray path in anisotropic media

The seismic ray theory in anisotropic inhomogeneous media is studied based on non-Euclidean geometry called Finsler geometry. For a two-dimensional ray path, the seismic wavefront in anisotropic media can be geometrically expressed by Finslerian parameters. By using elasticity constants of a real rock, the Finslerian parameters are estimated from a wavefront propagating in the rock. As a result, the anisotropic parameters indicate that the shape of wavefront is expressed not by a circle but by a convex curve called a superellipse. This deviation from the circle as an isotropic wavefront can be characterized by a roughness of wavefront. The roughness parameter of the real rock shows that the shape of the wavefront is expressed by a fractal curve. From an orthogonality of the wavefront and the ray, the seismic wavefront in anisotropic media relates to a fractal structure of the ray path.

Differential geometric structures of stream functions: incompressible two-dimensional flow and curvatures

The Okubo–Weiss field, frequently used for partitioning incompressible two-dimensional (2D) fluids into coherent and incoherent regions, corresponds to the Gaussian curvature of the stream function. Therefore, we consider the differential geometric structures of stream functions and calculate the Gaussian curvatures of some basic flows. We find the following. (I) The vorticity corresponds to the mean curvature of the stream function. Thus, the stream-function surface for an irrotational flow and that for a parallel shear flow correspond to the minimal surface and a developable surface, respectively. (II) The relationship between the coherency and the magnitude of the vorticity is interpreted by the curvatures. (III) Using the Gaussian curvature, stability of single and double point vortex streets is analyzed. The results of this analysis are compared with the well-known linear stability analysis. (IV) Conformal mapping in fluid mechanics is the physical expression of the geometric fact that the sign of the Gaussian curvature does not change in conformal mapping. These findings suggest that the curvatures of stream functions are useful for understanding the geometric structure of an incompressible 2D flow.

Jacobi stability for dynamical systems of two-dimensional second-order differential equations and application to overhead crane system

Geometric structures of dynamical systems are investigated based on a differential geometric method (Jacobi stability of KCC-theory). This study focuses on differences of Jacobi stability of two-dimensional second-order differential equation from that of one-dimensional second-order differential equation. One of different properties from a one-dimensional case is the Jacobi unstable condition given by eigenvalues of deviation curvature with different signs. Then, this geometric theory is applied to an overhead crane system as a two-dimensional dynamical system. It is shown a relationship between the Hopf bifurcation of linearized overhead crane and the Jacobi stability. Especially, the Jacobi stable trajectory is found for stable and unstable spirals of the two-dimensional linearized system. In case of the linearized overhead crane system, the Jacobi stable spiral approaches to the equilibrium point faster than the Jacobi unstable spiral. This means that the Jacobi stability is related to the resilience of deviated trajectory in the transient state. Moreover, for the nonlinear overhead crane system, the Jacobi stability for limit cycle changes stable and unstable over time.

Geometry of surfaces with Caputo fractional derivatives and applications to incompressible two-dimensional flows

Geometric structures of surfaces are formulated based on Caputo fractional derivatives. The Gauss frame of a surface with fractional order is introduced. Then, the non-locality of the fractional derivative characterizes the asymmetric second fundamental form. The mean and Gaussian curvatures of the surface are defined in the case of fractional order. Based on the fractional curvatures, incompressible two-dimensional flows are discussed. The stream functions are obtained from a fractional continuity equation. The asymmetric second fundamental form of stream-function surface is related to the path dependence of flux. Moreover, the fractional curvatures are calculated for the stream-function surfaces of uniform and corner flows. The uniform flow with fractional order is characterized by the non-vanishing mean curvature. The non-locality of corner flow is expressed by the mean and Gaussian curvatures with fractional order. In particular, the fractional order within the stream-function of corner flow determines the change of sign of Gaussian curvature. Therefore, the non-local property of incompressible flows can be investigated by the fractional curvatures.

Differential geometric approach to the stress aspect of a fault: Airy stress function surface and curvatures

We considered the two-dimensional stress aspect of a fault from the viewpoint of differential geometry. For this analysis, we concentrated on the curvatures of the Airy stress function surface. We found the following: (i) Because the principal stresses are the principal curvatures of the stress function surface, the first and the second invariant quantities in the elasticity correspond to invariant quantities in differential geometry; specifically, the mean and Gaussian curvatures, respectively; (ii) Coulomb’s failure criterion shows that the coefficient of friction is the physical expression of the geometric energy of the stress function surface; (iii) The differential geometric expression of the Goursat formula shows that the fault (dislocation) type (strike-slip or dip-slip) corresponds to the stress function surface type (elliptic or hyperbolic). Finally, we discuss the need to use non-biharmonic stress tensor theory to describe the stress aspect of multi-faults or an earthquake source zone.

Differential geometry of viscoelastic models with fractional-order derivatives

Viscoelastic materials with memory effect are studied based on the fractional rheonomic geometry. The geometric objects are regarded as basic quantities of fractional viscoelastic models, i.e. the metric tensor and torsion tensor are interpreted as the strain and the fractional strain rate, respectively. The generalized viscoelastic equations are expressed by the geometric objects. Especially, the basic constitutive equations such as Voigt and Maxwell models can be derived geometrically from the generalized equation. This leads to the fact that various viscoelastic models can be unified into one geometric expression.

Fractional-order derivative and time-dependent viscoelastic behaviour of rocks and minerals

A general constitutive equation for viscoelastic behaviour of rocks and minerals with fractional-order derivative is investigated. This constitutive law is derived based on differential geometry and thermodynamics of rheology, and the fractional order of derivative represents the degree of time delay. Analyzing some laboratory experimental data of high temperature deformation of rocks and minerals such as halite, marble and orthopyroxene, we propose how to determine the orders of fractional derivative for viscoelastic behaviours of rocks and minerals. The order is related to the exponents for the temporal scaling in the relaxation modulus and the stress power-law of strain rate, i.e., the non-Newtonian flow law, and considered as an indicator representing the macroscopic behaviour and microscopic dynamics of rocks.

Geometry of stress function surfaces for an asymmetric continuum

A two-dimensional stress field of dislocation or fault is geometrically studied for an asymmetric continuum. For geometric surfaces of the stress and couple-stress functions, the mean and Gaussian curvatures are derived. The mean curvature of couple-stress function surface is connected with the asymmetric of stress tensor. Moreover, the Gaussian curvature of stress function surface is characterized by both the stress and couple-stress. On the other hand, the mean curvature of stress function surface is not affected by the asymmetry of stress. Based on these geometric expressions, the Coulomb’s failure criterion and the friction coefficient are expressed by the curvatures of couple-stress function surface. Moreover, geometric structures of stress and couple stress function surfaces are shown for edge and wedge dislocations as faults. The curvatures of these surfaces show that the effect of couple-stress is constrained around the dislocations only.

Jacobi stability analysis and chaotic behavior of nonlinear double pendulum

In this paper, we study the Jacobi stability on the nonlinear double pendulum by the Kosambi–Cartan–Chern (KCC) theory. We assume that the mass and length of rods of two kinds of pendulums are equal, respectively. Moreover, we consider the case that initial angles of the double pendulum are equal. Under these conditions, we obtain the boundary between Jacobi stable and unstable trajectories for initial angles. It is shown that the condition of Jacobi stable or unstable depends only on deflection angles of the nonlinear double pendulum. Then, we discuss relationships between Jacobi stability, physical parameters and other concepts of stability such as Lyapunov stability and chaos. We suggest that the ratio of length of rods and the mass ratio of pendulums of the double pendulum do not affect the Jacobi stability. It is suggested that the equilibrium points in the Jacobi stable region and in the Jacobi unstable region are Lyapunov stable and Lyapunov unstable, respectively, and that the motions in the Jacob...

KCC Analysis of the Normal Form of Typical Bifurcations in One-Dimensional Dynamical Systems: Geometrical Invariants of Saddle-Node, Transcritical, and Pitchfork Bifurcations

The Jacobi stability of the normal form of typical bifurcations in one-dimensional dynamical systems is analyzed by introducing the concept of the production process (time-like potential) to KCC th...