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Unformatted text preview: Approach to Chaos 0. Introduction There are two major reasons for studying non-linear systems. The first and most basic is that the equations of motion of almost all real systems are non-linear. The second reason is that even a relatively simple system which obeys a non-linear equation of motion can exhibit unusual and surprisingly complex behavior for certain ranges of the system parameters. In addition, in a wide variety of dramatically different non-linear systems, identical features show up. Much of the existing knowledge of non-linear behavior has been obtained from numerical solutions of the equations. The traditional methods, which lead to analytic (explicit equations) expressions for the motion, fail for most problems in non-linear systems. Numerical integration of the equations of motion is usually necessary. From the time of Newton until the 20th century, physicists and philosophers viewed the universe as a sort of enormous clock which, once wound up, behaves in a predictable manner. This idea was dramatically shaken by the discovery of quantum mechanics and the Heisenberg uncertainty principle, but physicists still thought that the motion of classical systems (macroscopic systems) that obey Newton's equations of motion would exhibit predictable or deterministic behavior. It turns out, however, that even macroscopic systems obeying Newton's equations can exhibit so-called chaotic motion or motion that seems very difficult to predict (or is even unpredictable). The main difference between a chaotic system and a non-chaotic system is the degree of predictability of the motion given the initial conditions to some level of accuracy (note that we are introducing the idea that we might not be able to specify the initial conditions exactly). Let us look at a particular system, namely, the linear, damped oscillator with sinusoidal driving force. This system satisfies the equation ma = F = D cos t bv cx where Page 1 a = acceleration , m = mass , D = amplitude of driving force F = force , b = damping strength , c = oscillation constant Examples of such systems are a spring and pendulum. Let us do a simulation. In the simulation we plot the motion of two oscillators with starting points(initial conditions) on the same diagram. The plot is in phase space , where we plot velocity (y-axis) versus position (x-axis). The system at any time is represented by a point in phase space(a point specifies the position and velocity at an instant of time). Different initial conditions correspond to different points in phase space. Class demonstration of spring and pendulum . From the demonstrations we see that the motion corresponds to both the position and the velocity oscillating in time. It corresponds to the values D=0.0 (no driving force), b=0.0(little or no damping (friction)), c 0 and =2/3 in a simulation. The simulation gives (nlo_1.m) The frequency of this motion corresponds to the natural frequency....
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This note was uploaded on 09/16/2011 for the course ME 563 taught by Professor Staff during the Spring '11 term at Auburn University.
- Spring '11