Makino and Berz introduced the Taylor model approach for validated integration of initial value problems (IVPs) for ordinary differential equations (ODEs). Especially, they invented preconditioning of Taylor models for stabilizing the integration and proposed the following different types: parallelepiped preconditioning (with and without blunting), QR preconditioning, and curvilinear preconditioning. We review these types of preconditioning and show how they are implemented in INTLAB's verified ODE solver verifyode by stating explicit MATLAB code. Finally, we test our implementation with several examples.

Since the numerical verification method for solving boundary value problems for elliptic partial differential equations (PDEs) was first developed in 1988, many methods have been devised. In this paper, existing verification methods are reformulated using a convergence theorem for simplified Newton-like methods in the direct product space V_h × V_⊥ of a computable finite-dimensional space V_h and its orthogonal complement space V_⊥. Additionally, the Verified Computation for PDEs (VCP) library is provided, which is a software library written in the C++ programming language. The VCP library is introduced as a software library for numerical verification methods of solutions to PDEs. Finally, numerical examples are presented using the reformulated verification methods and VCP library.

A discrete-time two-dimensional dynamical system appears in a Golden Ratio Encoder (GRE), a type of analog-to-digital converter. One of the essential elements in analyzing a given dynamical system is identifying the invariant set of that system. The invariant set of dynamics of GREs is not known, except in special cases. We herein determine the invariant set of the dynamics of GREs with an amplification factor α and a threshold θ for a wide range of parameters (α, θ). The invariant set is separated into six sub-regions and the transition probabilities between the sub-regions are defined. We show that the uniform distribution on the invariant set is an invariant density for this dynamical system.

In a previous study [Inaba et al., Physica D (2020), web on-line], we discovered nested mixed-mode oscillations (MMOs) generated by a Bonhoeffer-van der Pol (BVP) oscillator under weak periodic perturbations near a subcritical Hopf bifurcaton point. The dynamics of BVP oscillators are equivalent to FitzHugh-Nagumo dynamics and have been studied extensively for more than five decades. In this study, we focus on the singly nested MMOs that occur between the 1⁴- and 1⁵-generating regions in a piecewise-smooth BVP oscillator with an idealized diode where 1^s indicates alternating time-series waveforms that consist of a large amplitude oscillation followed by s small peaks, and we confirm 400 nested mixed-mode oscillation-incrementing bifurcations (MMOIBs). Our numerical results suggest that the universal constant converges to one, which was predicted because MMOIBs increment and terminate toward an MMO increment-terminating tangent bifurcation point and the gradient of the tangent points is one.

Oscillators - dynamical systems with stable periodic orbits - arise in many systems of physical, technological, and biological interest. The standard phase reduction, a model reduction technique based on isochrons, can be unsuitable for oscillators which have a small-magnitude negative nontrivial Floquet exponent. This necessitates the use of the augmented phase reduction, a recently devised two-dimensional reduction technique based on isochrons and isostables. In this article, we calculate analytical expressions for the augmented phase reduction for two dynamically different planar systems: periodic orbits born out of homoclinic bifurcation, and relaxation oscillators. To validate our calculations, we simulate models in these dynamic regimes, and compare their numerically computed augmented phase reduction with the derived analytical expressions. These analytical and numerical calculations help us to understand conditions for which the use of augmented phase reduction over the standard phase reduction can be advantageous.