Two standard methods for numerically estimating Lyapunov exponents are reviewed and it is noted that a numerical integration scheme that preserves orthonormality is required. A procedure is introduced for modifying arbitrary rth order numerical schemes to preserve orthonormality. Convergence is shown for the particular case when explicit Euler's method is taken as the arbitrary method. This mot...

In this paper it is shown that the implicit Euler time-discretization of some classes of switching systems with sliding modes, yields a very good stabilization of the trajectory and of its derivative on the sliding surface. Therefore the spurious oscillations which are pointed out elsewhere when an explicit method is used, are avoided. Moreover the method (an event-capturing, or time-stepping a...

When simulating trajectories by integrating time-continuous car-following models, standard integration schemes such as the forth-order Runge-Kutta method (RK4) are rarely used while the simple Euler’s method is popular among researchers. We compare four explicit methods: Euler’s method, ballistic update, Heun’s method (trapezoidal rule), and the standard forth-order RK4. As performance metrics,...

A glycolytic model, which is an autocatalytic biochemical system, is solved numerically using two numerical methods based on finite-difference schemes. Method 1, the well known Euler method, is an explicit method, whereas method 2 is implicit. Although the implicit method, method 2, is first-order accurate in time it converges to the fixed point(s) for large time step, ‘. 2005 Elsevier Inc. All...

Two midpoint-trapezoid pairs of dynamically equivalent (conjugate) algorithms are derived as compositions of first-order forward Euler and backward Euler integrators as applied to an incremental form of the initial-value problem of three-dimensional rigid body rotation. The algorithms are related to the recently developed methodology of the so-called Munthe-Kaas Runge–Kutta methods. Selected ex...

The emerging field of modern biosciences and biotechnologies ask for appropriate and refined methods from applied mathematics. Modeling and prediction of gene-expression patterns have an important place here. In our paper, we deepen the analytical understanding and the algorithmical treatment by including affine terms on the right-hand side of the nonlinear differential equations and using Rung...

Abstract Size effects in metal plasticity are widely accepted, and different theoretical approaches to handle the phenomenon developing literature. The present work considers Fleck Willis (2009b) framework, creates a new gradient enhanced rate-independent crystal FE-implementation. study both energetic dissipative hardening strengthening, adopts general form of effective slip rate. Monotonic cy...