Monday, 29 April 2013

the concept of phase space extended

Understanding phase space ... using phase space to understand processes

p.134 James Gleick's "Chaos making a new science"

"In phase space the complete state of knowledge about a dynamical system at a single instant in time collapses to a point. The point is the dynamical system-at that instant. At the next instant, though, the system will have changed, ever so slightly, and so the point moves. The history of the system time can be charted by the moving point, tracing its orbit through phase space with the passage of time."

.. phase space provides the complete state of knowledge about a dynamical system ... the complete set of states of a dynamical system .. any state that a dynamical system can attain is included in the phase space for that particular system ... a phase space that can be defined by the variables and parameters of the dynamical system .. something that allows the mathematical treatment .. manipulation .. of the phase space of a system .. and can be used to explain dynamical systems

any dynamical system .. whatever its origin and field upon which it develops .. unfolds, it possesses its very own phase space and it will unravel its course within that particular phase space .. it will not fall outside the boundaries .. or limits of its phase space .. that phase space as determined by the variables and parameters of the dynamical system

what falls outside the phase space does not influence the progress of the dynamical system .. whatever falls outside the limits of the phase space is ignored .. it is not taken into consideration

where are you driving at .. what do you want to say, be specific .. phase space can be thought as being isolated ..from one another .. whatever happens within the phase space of a dynamical system .. has not any relevance any connection with the going-ons in another phase space of another dynamical system

phase spaces might intersect .. happen to intersect .. by virtue of unfolding at the same space or even time or a combination of space and time put together as spacetime .. should the two dimensions coincide for an intersection to occur? .. maybe should extend further to include of any variables and parameters that dynamical systems are defined by .. hence whenever there is coincidence .. then the phase spaces mix

when mixing occur the phase spaces .. are shared? .. the extend that each phase space possess becomes the phase space of their dynamical systems .. each dynamical system of the mixed phase spaces surpass its limits .. something that attributes nonlinearity in dynamical systems?

nonlinear functions hard to solve .. solvable only for special cases .. the only approach to solve nonlinear equations .. that is because nonlinear happens when the phase spaces of dynamical systems mix .. the particular phase spaces that mix should be taken into account in order to account for the nonlinearity in the development

the isolation of phase space used to explain why humans as dynamical systems have limited perceptions as the extend of their perceptions are determined by the phase spaces their dynamical systems unfold .. our perceptions are limited by the phase spaces that the variables and parameters set have set .. variables and parameters as are influenced by knowledge ..limited knowledge limited scope variables and parameters .. determine the extend of the phase spaces . and these phase spaces do not mix .. strangers but close together