Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic FlowThe author uses mathematical techniques along with observations and experiments to give an in-depth look at models for mechanical vibrations, population dynamics, and traffic flow. Equal emphasis is placed on the mathematical formulation of the problem and the interpretation of the results. In the sections on mechanical vibrations and population dynamics, the author emphasizes the nonlinear aspects of ordinary differential equations and develops the concepts of equilibrium solutions and their stability. He introduces phase plane methods for the nonlinear pendulum and for predator-prey and competing species models. Haberman develops the method of characteristics to analyze the nonlinear partial differential equations that describe traffic flow. Fan-shaped characteristics describe the traffic situation that occurs when a traffic light turns green and shock waves describe the effects of a red light or traffic accident. Although it was written over 20 years ago, this book is still relevant. It is intended as an introduction to applied mathematics, but can be used for undergraduate courses in mathematical modeling or nonlinear dynamical systems or to supplement courses in ordinary or partial differential equations. |
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Page 56
... unstable! Any mathematical solution must illustrate this striking difference between these two equilibrium positions ... unstable. For example, even if only one initial condition exists for which the solution tends “away” from the ...
... unstable! Any mathematical solution must illustrate this striking difference between these two equilibrium positions ... unstable. For example, even if only one initial condition exists for which the solution tends “away” from the ...
Page 57
... unstable. Thus the stability of the equilibrium solution is determined by the time dependence of the displacement from equilibrium. A simple analysis of equation 18.3, known as a linearized stability analysis, shows that: 1. If. 57 Sec ...
... unstable. Thus the stability of the equilibrium solution is determined by the time dependence of the displacement from equilibrium. A simple analysis of equation 18.3, known as a linearized stability analysis, shows that: 1. If. 57 Sec ...
Page 58
... unstable. If displaced from equilibrium, the force will push it further away. 3. If f '(x,,) I 0, then this linearized stability analysis is inconclusive as additional terms of the Taylor series need to be calculated. As an example, let ...
... unstable. If displaced from equilibrium, the force will push it further away. 3. If f '(x,,) I 0, then this linearized stability analysis is inconclusive as additional terms of the Taylor series need to be calculated. As an example, let ...
Page 61
... unstable. What is the exponential behavior of the angle in the neighborhood of this unstable equilibrium position? Suppose 18.10. 18.11. (a) (b) (9) d 2x fix . mw=ot(e —1)w1thot>0andfl>0. What are the dimensions of at and ,6 ? Determine ...
... unstable. What is the exponential behavior of the angle in the neighborhood of this unstable equilibrium position? Suppose 18.10. 18.11. (a) (b) (9) d 2x fix . mw=ot(e —1)w1thot>0andfl>0. What are the dimensions of at and ,6 ? Determine ...
Page 64
... at an equilibrium position has an extremum point dF/dx = 0 (a relative minimum if f '(x,,) > 0, a relative F(X) Unstable equilibrium Unstable equilibrium Stable equilibrium Stable equilibrium /x1 64 Mechanical Vibrations.
... at an equilibrium position has an extremum point dF/dx = 0 (a relative minimum if f '(x,,) > 0, a relative F(X) Unstable equilibrium Unstable equilibrium Stable equilibrium Stable equilibrium /x1 64 Mechanical Vibrations.
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Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic ... Richard Haberman No preview available - 1998 |
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amplitude analysis applied approximately Assume birth calculated called cars characteristics Consider constant continuous corresponding curve decreases delay depends derived described determine differential equation discussed distance energy equal equilibrium population equilibrium position equivalent example exercise experiments expression Figure first fish flow force formulate friction function given growth rate hence highway illustrated increases initial initial conditions integral isoclines known length light limit linear manner mass mathematical model maximum measured method motion moving nonlinear number of cars observer obtained occurs oscillation partial differential equation pendulum period phase plane possible probability problem region result roots sharks shock Show shown in Fig simple sketched sketched in Fig solution solve species spring spring-mass system stable straight line Suppose tion traffic density traflic trajectories unstable variables velocity yields zero