Computer Simulation Of Dynamic Systems Research Articles

A System Dynamics Model for Basin Level Forecasting on the Basis of a Digital Elevation Model

Geographic Information System (GIS) is an information technology that is readily applied as a decision aid for a variety of water resource applications. System dynamics as a method of computer simulation of dynamic systems is based on the basic concept of stock with incoming and outgoing flows. In most real-world problems the change in the level of the basin is considered to be the result of summation of incoming and outgoing flows over a certain period of time, taking into account the initial level of the basin. This makes system dynamics an appropriate tool for modelling the behaviour of water basins. A digital terrain model consists of a digital elevation model and a digital model of the situation. For the needs of system dynamics modelling in this case-study only the digital elevation model (DEM) is used. GIS product - DEM -usage in system dynamics models advances simulation-based analysis and decision making for local and regional operational forecasting. The task of the article is to formulate both necessary and sufficient requirements for DEM usage in effective forecasting of water basin behaviour with system dynamics model. Within the proposed approach simple system dynamics models of a basin are developed and verified. These models include incoming and outgoing flow models in form of time-series and basin level DEMbased model. The incoming flow is defined with a hydrograph. The water regime of the basin is determined by its geometrical properties. The outgoing flow is formed at the exit from the basin. The models are formalized as a set of first-order difference equations, which are solved sequentially after constant time steps. The solutions provide simulation progress over time. The concept of constant time step is associated with the periodicity of observations of the natural water objects. The basin level function of the volume makes it possible to develop the system dynamics model of the basin suitable for simulation of the level changes in the basin. The developed system dynamics models are used for experimentation with various scenarios of the behaviour of the system under investigation and for forecasting the consequences of changes in parameters or structure of the system. The requirements for DEM usage in system dynamics models are formulated. The application of system dynamics with incorporated DEMbased basin models makes it possible to analyse the behaviour of the investigated basins and to obtain operational forecasts of the situation for various scenarios. Further research is outlined on the accuracy of simulation results depending on the complexity increase in the simulated object for closer similarity to real reservoirs.

Dimension Reduction methods for Infinite Dimensional Systems

AbstractFrom experiments and also from computer simulation of dynamical systems it is well known that for many dynamical phenomena in physics or engineering, which are modelled by infinite dimensional dynamical systems, the asymptotic behavior can be accurately described by replacing the original infinite dimensional system by a low dimensional system represented by so‐called essential variables. Such a dimension reduction of a dynamical system turns out to be central, both for a qualitative and quantitative understanding of its behaviour.In [1] Approximate Inertial Manifolds are presented, which perform extremely well for nonlinear evolution equations, but don't work as expected for the dynamics of a fluid conveying tube. By comparing the results for different internal damping values it can be seen that the larger gaps and the location of the cluster point in the spectrum for the weaker damping improve the approximation quality considerably.

Set-Valued Markov Chains and Negative Semitrajectories of Discretized Dynamical Systems

Computer simulation of dynamical systems involves a phase space which is the finite set of machine arithmetic. Rounding state values of the continuous system to this grid yields a spatially discrete dynamical system, often with different dynamical behaviour. Discretization of an invertible smooth system gives a system with set-valued negative semitrajectories. As the grid is refined, asymptotic behaviour of the semitrajectories follows probabilistic laws which correspond to a set-valued Markov chain, whose transition probabilities can be explicitly calculated. The results are illustrated for two-dimensional dynamical systems obtained by discretization of fractional linear transformations of the unit disc in the complex plane.

Mixed moments of random mappings and chaotic dynamical systems

Some statistical characteristics of completely random mappings and of random mappings with an absorbing or an attracting centre are calculated. Results are applied to validation of some phenomenological models of computer simulations of dynamical systems.

Statistical properties of some spatially discretized dynamical systems

Computer simulations of dynamical systems are {spatial discretizations} in which the finite machine arithmetic space replaces the continuum state space of the original system. Any trajectory of a spatial discretization of a dynamical system is thus eventually periodic, so the dynamical behaviour of such computations are essentially determined by the cycles of the discretized map. Such dynamical behaviour depends seemingly randomly on the fineness of the discretization mesh. In this paper statistical properties of the maximal cycles of spatial discretizations are investigated for some systems such as the tent map, rotations on a circle and toral endomorphisms.

Asymptotic Independence and Uniform Distribution of Quantization Errors for Spatially Discretized Dynamical Systems

Computer simulation of dynamical systems involves a state space which is the finite set of computer arithmetic. Restricting state values to this grid produces roundoff effects which can be studied by replacing the original system with a spatially discretized dynamical system. Study of the deviation of the discretized trajectories from those of the original system reduces to that of appropriately defined quantization errors. As the grid is refined, the asymptotic behavior of these quantization errors follows probabilistic laws. These results are applied to discretized polynomial mappings of the unit interval.

A model for roundoff and collapse in computation of chaotic dynamical systems

Computer simulations of dynamical systems contain discretizations, where finite machine arithmetic replaces continuum state space. For chaotic dynamical systems, the main features of this discretization are stochastically related to the parameters both, of the underlying continuous system and of the computer arithmetic. A model of this process is required to describe and analyze its statistical properties and this is carried out for the family of mappings f l ( x) = 1 − |1 − 2 x| l , x ∈ [0, 1], l > 2. Computer modeling results are presented.

Statistical properties of discretizations of a class of chaotic dynamical systems

Computer simulations of dynamical systems contain discretizations , where finite machine arithmetic replaces continuum state spaces. In some circumstances, complicated theoretical behavior has a tendency to collapse to trivial and degenerate behavior as a result of discretizations. Various statistical estimators associated with such collapsing effects often seem to depend on the corresponding discretization in a random way. Behavior of some statistical properties of collapse is discussed. Results of computer modeling of mappings x 1 − ¦1 − 2x¦ l , x ϵ [0, 1], l > 2 are presented.

Collapsing effects in numerical simulation of a class of chaotic dynamical systems and random mappings with a single attracting centre

Computer simulations of dynamical systems contain discretizations, where finite machine arithmetic replaces continuum state spaces. In some circumstances, complicated theoretical behaviour has a tendency to collapse to trivial and degenerate behaviour as a result of discretizations. Characteristics of such collapsing effects often seem to depend on the corresponding discretization only in a random way. We describe a procedure to construct a stochastic process with similar statistical characteristics. Results of computer modelling are discussed.

Discussion on the Meeting on Chaos

Discussion on the Meeting on Chaos

Rigorous verification of trajectories for the computer simulation of dynamical systems

The authors present a new technique for constructing a computer-assisted proof of the reliability of a long computer-generated trajectory of a dynamical system. Auxiliary calculations made along the noise-corrupted computer trajectory determine whether there exists a true trajectory which follows the computed trajectory closely for long times. A major application is to verify trajectories of chaotic differential equations and discrete systems. They apply the main results to computer simulations of the Henon map and the forced damped pendulum.

Isolating blocks and epsilon chains for maps

The relationship between the outputs and computer simulations of dynamical systems and their true orbit structures is explored. Epsilon chains are used as a mathematical representation of computer generated pseudo-orbits. The question “When is a pseudo-orbit shadowed by a true orbit” is discussed. Isolated invariant sets are defined and are shown to be contained in special neighborhoods called isolating blocks. The Conley index is extended and applied to maps.

Model-Building and the Use of Computer Simulation of Dynamic Systems

The Systems Thinking and Curriculum Innovation (STACI) Project is a multi-year research effort intended to examine the cognitive impact of learning from a systems thinking approach to instruction and from using simulation-modeling software. Systems thinking is an analytic problem solving tool that can be integrated into courses to supplement and enhance instruction in a variety of content areas. The purpose of the study is to test the potentials and effects of using the technology-based approach in secondary school curricula to teach content-specific knowledge as well as general problem solving skills. The research focuses on the effects of introducing a software environment that enables students to learn from and make concrete multiple representations of scientific, mathematical, and historical phenomena.

Solving stiff differential equations in the simulation of physical systems

Computer simulation of dynamic systems often requires the solution of a set of stiff ordinary differential equations. The solution involves the eigenvalues of the Jacobian matrix. The greater the spread in eigenvalues, the more time-consuming the solutions become if one uses standard numerical techniques. In many cases, the differential equations are so stiff that accurate numerical solutions are unecon omic. In this paper we propose new techniques for solving stiff differential equations. These algo rithms are based on implicit Runge-Kutta procedures with complete error estimates. Experiments show that the new techniques are superior to Gear's method for solving mathematical models for real sys tems in chemical engineering, biology, and laser kinetics.

Solving stiff differential equations for simulation

Computer simulation of dynamic systems very often leads to the solution of a set of stiff ordinary differential equations. The solution of this set of equations involves the eigenvalues of its Jacobian matrix. The greater the spread in eigenvalues, the more time consuming the solutions become when existing numerical methods are employed. Extremely stiff differential equations can become a very serious problem for some systems, rendering accurate numerical solutions completely uneconomic. In this paper, we propose new techniques for solving extremely stiff systems of differential equations. These algorithms are based on a class of implicit Runge-Kutta procedure with complete error estimate. The new techniques are applied to solving mathematical models of the relaxation problem behind blast waves.

Numerical methods for extremely stiff systems of ordinary differential equations

Computer simulation of dynamic systems very often leads to the solution of a set of stiff ordinary differential equations. The solution of this set of equations involves the eigenvalues of its Jacobian matrix. The greater the spread in eigenvalues, the more time consuming the solutions become when existing numerical methods are employed. Extremely stiff differential equations can become a very serious problem for some systems, rendering accurate numerical solutions completely uneconomic. In this paper, we propose new techniques for solving extremely stiff systems of differential equations. These algorithms are based on a class of implicit Runge-Kutta procedure with complete error estimate. The new techniques are applied to solving mathematical models of the relaxation problem behind blast waves.

Error analysis in hybrid computer simulation of dynamic systems

Abstract A state-space oriented approach to analyze dynamic errors in a hybrid computer simulation loop is presented. The approach helps to predict the value of the sampling interval that achieves stable simulation with a prescribed accuracy. A pre-perturbation technique to compensate for the serious digital execution time delay error is then developed. It has the advantages that it does not require additional computer hardware and/or software and its utilization does not result in generation of any extraneous modes. The approach points out the contribution of each individual error source in the hybrid computer loop to the overall simulation accuracy.

Sequential Z-transform analysis of a multirate sampled data system

An ordered sequential analysis of multirate sampled data systems is developed after presenting Ragazzini-Zadeh type identities for fast and slow samplers. These is no need for the uniform decomposition of the fast sampler or for the often used Kranc decomposition. A Kranc decomposition critique is presented. An example of a multirate sampler and hold is given. The Discrete Fourier Transform is shown to be properly based on the Z-transform. Digital computer simulation of dynamical systems requires three Fast Fourier Transform passes.

Computer Simulation of Dynamic Systems

Computer Simulation of Dynamic Systems

Error Analysis in the Computer Simulation of Dynamic Systems: Variational Aspects of the Problem

Error analysis in the computer simulation of dynamic systems is fundamentally a variational problem. The computing errors are small variations of the computed solutions with respect to the exact solution of the differential equations being integrated. It is not surprising, therefore, that many of the mathematical tools used to perform error analysis in the computer simulation of dynamic systems are similar to those used in the variational or perturbational analysis of those systems themselves. Fundamental papers in this direction have been published previously [8], [9], [13]. The present paper, however derives the error-propagation equations in a more basic form, which makes it easier to apply such variational mathematical tools as Liapunoff's second method to analyze error stability, and Pontryagin's maximum principle to study ``worst-case'' errors in computation.