Previously we wrote about randomly generated attractors. That time we have used Wolfram CDF technology to power the interactive applet. This technology has a serious drawback that you have to have installed specific additional software to be able to use it. As of now we have replaced the old app with HTML5-based interactive applet. This applet can be run on almost any modern web browser without a need to have any additional software preinstalled.
Main topic: Dynamical chaos
Physicists love mathematics. Especially and most usually differential calculus. Mathematical description of any system using differential calculus is usually referenced as dynamical description. Thus in this case equations themselves are called dynamical equations. Purely theoretically it appears that if we know precise form dynamical equations we know everything about the system, we can make precise predictions of it’s evolution.
The problem is that this impression is only theoretical. In practice one must take measurements of reality. As we all know it measurements are just approximation – every measurement has some error related to it, no matter how small or apparently insignificant it appears. Measurement bias in the linear systems may play no role at all – system evolves as predicted with small or no corrections at all. But most of the interesting systems are not linear! In fact most of them are strongly non-linear. Non-linearities in dynamical equations may cause accumulating differences between solutions with apparently insignificant variation in initial conditions. Thus over the time in non-linear system prediction of evolution and observed evolution may start to disagree by far more than just by primary measurement bias. This behavior is known as dynamical chaos.
Most widely known example of dynamical chaos is the Butterfly effect – butterfly flapping it’s wings may cause hurricane on next week, but thousand kilometers away. In this section of Physics of Risk website we will attempt to show that even smallest bias of initial conditions, of reality itself, might be the reason for unexpected results and chaos.
In classical physics differential equations is the main tool to mathematically describe dynamical systems. Having obtained the mathematical descriptions of the system we should be able to predict the evolution of the system. It is noted that the evolution of the classical systems is pretty trivial – no matter what the initial condition is the system will “find” the stable state. Usually dissipative forces (such as frictions) are to be blamed for this. Though not all systems are so simple… Continue reading “Randomly generated strange attractors” »