From Last Time Gravitational forces are apparent at a wide range of scales. Obeys F gravity (Mass of object 1) (Mass of object 2) square of distance between them F = 6.7 10-11 m 1 m 2 d 2 Gravitational Constant
The deterministic solar system
Simple Planetary motion http://galileoandeinstein.phys.virginia.edu/ more_stuff/flashlets/kepler6.htm
Newtonian Determinism Newton s laws seem to determine all future motion. All future behavior exactly known.
The three-body problem Newton could not solve any problem past a single planet orbiting the sun. Prize offered for solution of 3-body problem. Poincare in 1896 showed problem not analytically solvable.
Complicated motion of 3 bodies
Sensitivity to initial conditions in the 3-body problem Flash simulation http://faraday.physics.utoronto.ca/pvb/harrison/flash/chaos/ ThreeBody/ThreeBody.html Two planets start out with almost identical positions and velocities. Resulting motions due to gravitational attraction to the two suns are very different Sensitive to initial conditions
How can we summarize this motion? The three-body problem exhibits all of the hallmarks of chaos. In particular, the outcome of any given interaction depends sensitively on the initial conditions. The following image shows how the final state of a scattering encounter between a binary star system and another star depends on the initial phase (horizontal axis) of the binary and the impact parameter (vertical axis) of the incomer. Color represents the angle at which the star that eventually escapes leave the interaction region. Note the alternating regions of regular (smooth) and irregular (chaotic, resonant) behavior. Each pixel in this image coresponds to a complete three-body encounter. The particular series of calculations shown here has relative velocity at infinity equal to 10% of the binary orbit speed. Encounters such as these are believed to be important in determining the dynamical evolution of globular star clusters in the Milky Way galaxy.
Dynamical Systems The system evolves in time according to a set of rules. The present conditions determine the future. The rules are usually nonlinear. There may be many interacting variables.
Examples of Dynamical Systems The Solar System The atmosphere (the weather) The economy (stock market) The human body (heart, brain, lungs,...) Ecology (plant and animal populations) Cancer growth Spread of epidemics Chemical reactions The electrical power grid The Internet
A double pendulum One way to drive a pendulum is to hang it from another that is swinging. http://www.treasuretroves.com/physics/ DoublePendulum.ht ml
The weather The strange behavior of nonlinear systems was not fully appreciated until computers permitted extensive numerical simulations of motions not susceptible to analytic methods. 1961 - Edward Lorenz discovered that a rather simple model of atmospheric processes exhibited erratic behavior.
Lorenz model Lorenz studied a simple model of the evolution of temperature and pressure and found a small change in initial value led to ultimately wildly different results.
Lorentz attractor in 3D
Simple sensitive systems Released balloon Air hose (fire hose instability)
The magnetic pendulum Pendulum comes to rest above one of the stationary magnets (attractors) Result depends sensitively on point of release. Two similar release points Different trajectory and rest point
Quantify the dependence on initial conditions Blue and white regions show initial positions which correspond to the magnet coming to equilibrium around either the blue or white fixed magnet.
Fractal structure of the boundary If we could blow up the region around the boundaries between blue and white areas, we would find that they are not infinitely sharp. Instead, we would see a complex structure which is termed a fractal. Fractals have fractional dimensions and the unique property of self-similarity to all levels of magnification. If you magnify any part of a fractal, you see a miniature copy of the overall fractal structure repeated on the small scale.
Three magnet pendulum Now have three attractors for the magnet on the pendulum. Release pendulum at diff. Points & see where it comes to rest Three almost identical starting positions, but three different final positions.
Basins of attraction for 3-magnet pendulum Color coding indicates final rest position of magnet on pendulum Green is above green stationary magnet, etc. Region of solid color is called a basin of attraction This shows a fractal, selfsimilar structure. y release position x release position
Fractal structure in a similar problem By fractal, or selfsimilar, we mean similar on all length scales. I.e. the picture looks the same after zooming in much closer. http://www.sekine-lab.ei.tuat.ac.jp/~kanamaru/chaos/e/newton/
Driven systems The magnet pendula were examples of systems attracted to a fixed, stable position after some time. The stable positions are attractors, and the final position depends sensitively on the initial release point of the pendulum. In this case the initial motion was damped out by frictional forces. But if the pendulum were continually driven, it would continue to oscillate forever. We could say it is attracted to a fixed, stable motion rather than a final position.
The driven pendulum Pendulum driven (pushed) with a particular strength, and at a particular frequency. Drive mechanism ω θ θ = angle of pendulum ω = angular velocity (e.g. rotation rate) If we know both θ and ω at a particular instant of time, we know the motion of the pendulum.
Describe motion with phase-space plot Can plot θ vs ω for all times instead of θ vs time. Makes a compact, self-contained description of the motion. If we take a strobe photograph of the pendulum, once per drive cycle, we will get pairs of θ and ω that we can plot in phase space This is a Poincare plot.
Small drive amplitude: periodic motion Drive amplitude Poincare plot Angular velocity (ω) Phase-space plot Angle (θ)
Increase the drive amplitude Drive amplitude = 0.665, period doubling
Period four oscillation Drive amplitude = 0.667, period doubles again, to four times the drive period Angular velocity Angle
Chaotic motion Drive amplitude = 0.68, motion is now chaotic
More chaotic motion Drive amplitude = 0.69, chaotic motion Angular velocity Angle
Route to chaos in the driven pendulum Increasing drive amplitude induces period doubling, quadrupling, then chaotic behavior. For each value of drive, the pendulum angular velocity is plotted at intervals of the drive period (vert. axis of Poincare plot).
Driven pendulum, basins of attraction Even something as simple as a periodically forced damped pendulum can have complex behavior. The computer-generated images below show initial positions that asymptote to one of different behaviors (one color for each behavior). For example, orbits starting at points in the blue region would yield a different type of motion from orbits starting in the red region. The brighter the shade of color, the longer it takes to settle into the corresponding motion. The different regions are separated by fractal basin boundaries.
Everyday chaos: A dripping faucet Water dripping from a faucet can show chaotic behavior. Small average flow rate results in water droplets falling perfectly periodically from the faucet. Faster drip rate leads to period doubling, then chaotic falling droplets. When drop detaches, it leaves vibrations on remaining water For slow enough rates, these vibrations die out, so each drop independent. At high drop rate, vibrations influence next drop detachment
Natural Fractals
Fractal coastline
Basins of attraction Release position green - pendulum comes to rest above green magnet Diagram indicates relation between initial conditions and final motion. y release position x release position
Fractals as art Many different systems show chaotic or fractal behavior. They range from physical systems, to purely mathematical ones. In a phase space plot, the color coding usually indicates the basin of attraction of some attractor (dynamical state). The saturation of the color might indicate the rate at which the system is attracted to that dynamical state. Following are several fractal images from the web. See sprott.physics.wisc.edu/fractals.htm for many other fascinating phase space plots.