Tippe Top Inversion as a Dissipation-Induced Instability

By treating tippe top inversion as a dissipation-induced instability, we explain tippe top inversion through a system we call the modified Maxwell--Bloch equations. We revisit previous work done on this problem and follow Or's mathematical model [SIAM J. Appl. Math., 54 (1994), pp. 597--609]. A linear analysis of the equations of motion reveals that the only equilibrium points correspond to the inverted and noninverted states of the tippe top and that the modified Maxwell--Bloch equations describe the linear/spectral stability of these equilibria. We supply explicit criteria for the spectral stability of these states. A nonlinear global analysis based on energetics yields explicit criteria for the existence of a heteroclinic connection between the noninverted and inverted states of the tippe top. This criteria for the existence of a heteroclinic connection turns out to agree with the criteria for spectral stability of the inverted and noninverted states. Throughout the work we support the analysis with numerical evidence and include simulations to illustrate the nonlinear dynamics of the tippe top

Approximate probabilistic verification of hybrid systems

Hybrid systems whose mode dynamics are governed by non-linear ordinary differential equations (ODEs) are often a natural model for biological processes. However such models are difficult to analyze. To address this, we develop a probabilistic analysis method by approximating the mode transitions as stochastic events. We assume that the probability of making a mode transition is proportional to the measure of the set of pairs of time points and value states at which the mode transition is enabled. To ensure a sound mathematical basis, we impose a natural continuity property on the non-linear ODEs. We also assume that the states of the system are observed at discrete time points but that the mode transitions may take place at any time between two successive discrete time points. This leads to a discrete time Markov chain as a probabilistic approximation of the hybrid system. We then show that for BLTL (bounded linear time temporal logic) specifications the hybrid system meets a specification iff its Markov chain approximation meets the same specification with probability $1$. Based on this, we formulate a sequential hypothesis testing procedure for verifying -approximately- that the Markov chain meets a BLTL specification with high probability. Our case studies on cardiac cell dynamics and the circadian rhythm indicate that our scheme can be applied in a number of realistic settings

A ghost-stabilised material point method for large deformation geotechnical analysis

The Material Point Method (MPM) is advertised as the method for large deformation analysis of geotechnical problems. However, the method suffers from several instabilities which are widely documented in the literature, such as: material points crossing between elements, different number of points when projecting quantities between the grid and points, etc. A key issue that has received relatively little attention in the literature is the conditioning of the linear system of equations due to the arbitrary nature of the interaction between the physical body (represented by material points) and the background grid (used to solve the governing equations). This arbitrary interaction can cause significant issues when solving the linear system, making some systems unsolvable or causing them to predict spurious results. This paper presents a cut-FEM (Finite Element Method) inspired ghost-stabilised MPM that removes this issue

A ghost-stabilised material point method for large deformation geotechnical analysis

The Material Point Method (MPM) is advertised as the method for large deformation analysis of geotechnical problems. However, the method suffers from several instabilities which are widely documented in the literature, such as: material points crossing between elements, different number of points when projecting quantities between the grid and points, etc. A key issue that has received relatively little attention in the literature is the conditioning of the linear system of equations due to the arbitrary nature of the interaction between the physical body (represented by material points) and the background grid (used to solve the governing equations). This arbitrary interaction can cause significant issues when solving the linear system, making some systems unsolvable or causing them to predict spurious results. This paper presents a cut-FEM (Finite Element Method) inspired ghost-stabilised MPM that removes this issue

Dynamical Systems Approach to Magnetised Cosmological Perturbations

Assuming a large-scale homogeneous magnetic field, we follow the covariant and gauge-invariant approach used by Tsagas and Barrow to describe the evolution of density and magnetic field inhomogeneities and curvature perturbations in a matter-radiation universe. We use a two parameter approximation scheme to linearize their exact non-linear general-relativistic equations for magneto-hydrodynamic evolution. Using a two-fluid approach we set up the governing equations as a fourth order autonomous dynamical system. Analysis of the equilibrium points for the radiation dominated era lead to solutions similar to the super-horizon modes found analytically by Tsagas and Maartens. We find that a study of the dynamical system in the dust-dominated era leads naturally to a magnetic critical length scale closely related to the Jeans Length. Depending on the size of wavelengths relative to this scale, these solutions show three distinct behaviours: large-scale stable growing modes, intermediate decaying modes, and small-scale damped oscillatory solutions.Comment: 15 pages RevTeX, 5 figures. Accepted for publication in Physical Review