Numerical Simulations Of Dynamical Systems Research Articles

A review of structure-preserving numerical methods for engineering applications

Accurate numerical simulation of dynamical systems is essential in applications ranging from particle physics to geophysical fluid flow to space hazard analysis. However, most traditional numerical methods do not account for the underlying geometric structure of the physical system, leading to simulation results that may suggest nonphysical behavior. The field of geometric numerical integration (GNI) is concerned with numerical methods that respect the fundamental physics of a problem by preserving the geometric properties of the governing differential equations. Research over the past two decades has produced GNI methods that are so accurate that they are now used for benchmarking purposes for long-time simulation of conservative dynamical systems. However, their utility for large-scale engineering problems is still an open question. This paper presents a review of structure-preserving numerical methods with focus on their engineering applications. The purpose of this paper is to provide an overview of different classes of GNI methods for mechanical systems while providing a survey of practical examples from numerical simulation of realistic engineering problems.

PHASE SPACE ERROR CONTROL WITH VARIABLE TIME-STEPPING ALGORITHMS APPLIED TO THE FORWARD EULER METHOD FOR AUTONOMOUS DYNAMICAL SYSTEMS

We consider a phase space stability error control for numerical simulation of dynamical systems. Standard adaptive algorithm used to solve the linear systems perform well during the finite time of integration with fixed initial condition and performs poorly in three areas. To overcome the difficulties faced the Phase Space Error control criterion was introduced. A new error control was introduced by R. Vigneswaran and Tony Humbries which is generalization of the error control first proposed by some other researchers. For linear systems with a stable hyperbolic fixed point, this error control gives a numerical solution which is forced to converge to the fixed point. In earlier, it was analyzed only for forward Euler method applied to the linear system whose coefficient matrix has real negative eigenvalues. In this paper we analyze forward Euler method applied to the linear system whose coefficient matrix has complex eigenvalues with negative large real parts. Some theoretical results are obtained and numerical results are given.

Quasilinearized Scale-3 Haar wavelets-based algorithm for numerical simulation of fractional dynamical systems

PurposeThe main purpose of this work is to develop a novel algorithm based on Scale-3 Haar wavelets (S-3 HW) and quasilinearization for numerical simulation of dynamical system of ordinary differential equations.Design/methodology/approachThe first step in the development of the algorithm is quasilinearization process to linearize the problem, and then Scale-3 Haar wavelets are used for space discretization. Finally, the obtained system is solved by Gauss elimination method.FindingsSome numerical examples of fractional dynamical system are considered to check the accuracy of the algorithm. Numerical results show that quasilinearization with Scale-3 Haar wavelet converges fast even for small number of collocation points as compared of classical Scale-2 Haar wavelet (S-2 HW) method. The convergence analysis of the proposed algorithm has been shown that as we increase the resolution level of Scale-3 Haar wavelet error goes to zero rapidly.Originality/valueTo the best of authors’ knowledge, this is the first time that new Haar wavelets Scale-3 have been used in fractional system. A new scheme is developed for dynamical system based on new Scale-3 Haar wavelets. These wavelets take less time than Scale-2 Haar wavelets. This approach extends the idea of Jiwari (2015, 2012) via translation and dilation of Haar function at Scale-3.

Inconsistencies in Numerical Simulations of Dynamical Systems Using Interval Arithmetic

Over the past few decades, interval arithmetic has been attracting widespread interest from the scientific community. With the expansion of computing power, scientific computing is encountering a noteworthy shift from floating-point arithmetic toward increased use of interval arithmetic. Notwithstanding the significant reliability of interval arithmetic, this paper presents a theoretical inconsistency in a simulation of dynamical systems using a well-known implementation of arithmetic interval. We have observed that two natural interval extensions present an empty intersection during a finite time range, which is contrary to the fundamental theorem of interval analysis. We have proposed a procedure to at least partially overcome this problem, based on the union of the two generated pseudo-orbits. This paper also shows a successful case of interval arithmetic application in the reduction of interval width size on the simulation of discrete map. The implications of our findings on the reliability of scientific computing using interval arithmetic have been properly addressed using two numerical examples.

Quasi-static transient and mixed mode oscillations induced by fractional derivatives effect on the slow flow near folded singularity

This paper aims at offering an insight into the dynamical behaviors of incommensurate fractional-order singularly perturbed van der Pol oscillators subjected to constant forcing, especially when the forcing is close to Andronov–Hopf bifurcation points. These bifurcation points are predicted thanks to the theorem on stability of incommensurate fractional-order systems, as functions of the forcing and fractional derivative orders. When the forcing is chosen near Andronov–Hopf bifurcation, the dynamics of fractional-order systems show a static-looking transient regime whose length increases exponentially with the closeness to the bifurcation point. This peculiar phenomenon is not common in numerical simulation of dynamical systems. We show that this quasi-static transient behavior is due to the combine action of the slow passage effect at folded saddle-node singularity and fractional derivation memory effect on the slow flow around this singularity; this forces the system to remain for a long time in the vicinity of its equilibrium point, though unstable. The system frees oneself from this quasi-static transient state by spiraling before entering relaxation oscillation. Such a situation results in mixed mode oscillations in the oscillatory regime. One obtains mixed mode oscillations from a very simple system: A two-variable system subjected to constant forcing.

Generalized Laguerre pseudospectral method based Laguerre interpolation

In this paper, we develop the generalized Laguerre pseudospectral method for the second-order ordinary differential equation based on the modified generalized Laguerre approximation, which is very efficient for the long-time numerical simulations of dynamical systems. The numerical solution possesses the spectral accuracy. The global convergence of the proposed algorithm is proved. Numerical results demonstrate the effectiveness of the proposed method and coincide well with theoretical analysis.

Book Review: Vincent Acary, Bernard Brogliato, Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and electronics, Lecture Notes in Applied and Computations Mechanics, Vol. 35

“This book concerns the numerical simulation of dynamical systems whose trajectories may not be differentiable everywhere. They are named nonsmooth dynamical systems.” This characterization – which precisely describes the content of the book – is given by the authors at the very beginning. Whereas the convex and nonsmooth analysis has a long tradition (in particular in France) and many basic problems at least for nonsmooth ODEs are solved, the development of efficient algorithms is still an active area of research; the current state-of-the-art is collected in this monograph. This book consists of four parts (and an appendix on background material in mathematical analysis). The first part (covering nearly half of the book) is devoted to modeling aspects (presenting nonsmooth models in mechanics, electronics, and control). In terms of convex and nonsmooth analysis they are formulated as differential inclusions, variational inequalities, complementarity systems, or as a projected dynamical system. The next two parts consider numerical aspects. In Part II numerical time integration schemes are introduced, and Part III considers one-step numerical methods for the arising incremental nonsmooth problems (containing a summary of 70 pages on the basics of mathematical programming). Finally, the implementation of algorithms for nonsmooth ODEs is discussed by introducing the software platform Siconos. The features, interfaces, and methods of this software are explained, and its performance is demonstrated by a couple of examples. This monograph is a valuable and concise contribution to the numerics of nonsmooth ODEs with an up-to-date reference list of 27 pages. It is clearly designed for engineering applications in computational mechanics, and the main concern is the discussion of the construction of algorithms and their properties; it is not intended to cover topics of numerical analysis as well. Nevertheless, many parts are written in a formal mathematical style with precise definitions, followed by results stated as Lemma, Proposition, or Theorem (and references to the literature for the proofs). This makes the overall work accessible for both, researchers in mechanics and in mathematics.

Integration processes of ordinary differential equations based on Laguerre-Radau interpolations

In this paper, we propose two integration processes for ordinary differential equations based on modified Laguerre-Radau interpolations, which are very efficient for long-time numerical simulations of dynamical systems. The global convergence of proposed algorithms are proved. Numerical results demonstrate the spectral accuracy of these new approaches and coincide well with theoretical analysis.

Phase Space Stability Error Control with Variable Time‐stepping Runge‐Kutta Methods for Dynamical Systems

AbstractWe consider a phase space stability error control for numerical simulation of dynamical systems. We illustrate how variable time‐stepping algorithms perform poorly for long time computations which pass close to a fixed point. A new error control was introduced in [9], which is a generalization of the error control first proposed in [8]. In this error control, the local truncation error at each step is bounded by a fraction of the solution arc length over the corresponding time interval. We show how this error control can be thought of either a phase space or a stability error control. For linear systems with a stable hyperbolic fixed point, this error control gives a numerical solution which is forced to converge to the fixed point. In particular, we analyze the forward Euler method applied to the linear system whose coefficient matrix has real and negative eigenvalues. We also consider the dynamics in the neighborhood of saddle points. We introduce a step‐size selection scheme which allows this error control to be incorporated within the standard adaptive algorithm as an extra constraint at negligible extra computational cost. Theoretical and numerical results are presented to illustrate the behavior of this error control. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

One-dimensional dynamical systems and Benford’s law

Near a stable fixed point at 0 or $\infty$, many real-valued dynamical systems follow Benford’s law: under iteration of a map $T$ the proportion of values in $\{x, T(x), T^2(x),\dots , T^n(x)\}$ with mantissa (base $b$) less than $t$ tends to $\log _bt$ for all $t$ in $[1,b)$ as $n\to \infty$, for all integer bases $b>1$. In particular, the orbits under most power, exponential, and rational functions (or any successive combination thereof), follow Benford’s law for almost all sufficiently large initial values. For linearly-dominated systems, convergence to Benford’s distribution occurs for every $x$, but for essentially nonlinear systems, exceptional sets may exist. Extensions to nonautonomous dynamical systems are given, and the results are applied to show that many differential equations such as $\dot x=F(x)$, where $F$ is $C^2$ with $F(0)=0>F’(0)$, also follow Benford’s law. Besides generalizing many well-known results for sequences such as $(n!)$ or the Fibonacci numbers, these findings supplement recent observations in physical experiments and numerical simulations of dynamical systems.