widening of a chaotic attractor and .. extinguishing of a chaotic attractor. ..sudden death.. live and ..prosper or die ..

The world in ..crisis .. economic crisis .. 

..crisis and ..sudden death ..or widening of chaotic attractors .. systems ..in context .. all of their own ..

..however ..

..the difference in ..context ..irrelevant .. what holds for one .. holds for the other .. 

..what applies for one ..applies for the other ..too .. self-similarity ..employed ..

..the  passage from one context to another easily ..accomplished ..  ..

..

..

..

CRISES, SUDDEN CHANGES IN CHAOTIC ATTRACTORS, AND TRANSIENT CHAOS.

------[u1] 

 [u1]  under ..new light ..afforded .. between .. widening of an attractor and .. extinguishing of an attractor ..

In section 2 we treat the case of the one-dimensional quadratic map. This simple paradigmatic example is used to introduce and distinguish two types of crises[u2] ,..

[u2]  .. crises ..

identifying the crisis .. the ensuing crisis .. to determine the results ..on the chaotic attractors ..

 .. the boundary crisis and the interior crisis[u3].  

  [u3]two types of crises, the boundary crisis and the interior crisis..

.. the distinction into ..kinds ..types of crises .. significant .. as ..each leads into completely different ..outcomes .. one of the types ..leads into the destruction …the sudden death of the attractor ..and the other into the ..proliferation of the attractors .. bigger and more inclusive attractors .. a ..bifurcation to a ..dead-end ..and a bifurcation ..capable of ..branching out ..

more entities ..included ..allowed ..in ..

This ..intuitively .. I find as more significant in the ..social context .. attractors having to do ..with societies ..

Note: intuitively ..I mis-conceived ..mis-construed ..which crisis would lead to what outcome .. a point I should ..delve upon ..

The former leads to sudden destruction of the chaotic attractor and its basin of attraction[u4] ,..

 [u4]The former leads to sudden destruction of the chaotic attractor and its basin of attraction ..

namely the ..boundary crisis ..

..while the latter can cause sudden changes in the size of the chaotic attractor.[u5] 

[u5]the latter can cause sudden changes in the size of the chaotic attractor…

namely the ..interior crisis ..

(The former and the latter.. relationship ..explained ..fathomed .. ambiguity ..dissolved .. vanished ..)

Section 2 also discusses the associated characteristic statistical behavior [u6]...

 [u6]... associated characteristic statistical behavior..  

understanding ..fully .. the meaning of this sentence ..being crucial ..   

... occurring for parameter values [u7] ...

 [u7]  ..parameter values  [u7].. 

 paying attention to ..associated parameters ..identifying the parameters .. since we are talking ..dealing with chaotic attractors .. namely ..chaos .. it is very hard to ..identify the parameters involved .. chaos ..in the mind .. proper .. suits the human individual .. unable ..problematic  to discern the parameters ..concerned ..an inability to distinguish the parameters .. that are crucial in any given system .. 

a drawback but ..at the same time ..a gift .. 

....just past that at which the crisis takes place. In the case of the boundary crisis, for parameter values just past the crisis point, the attractor no longer exists. Nevertheless, typical trajectories initialized in the region formerly occupied by the destroyed attractor appear to move about in this region chaotically[u8],...

 [u8] typical trajectories initialized in the region formerly occupied by the destroyed attractor appear to move about in this region chaotically..

understandable .. processes that ..persevere ..persist ..habits in the behaviour of entities .. being put into the social context .. at the back of my mind .. one of the attributes under which this current theme occupies my mind .. being part of my search .. what drives to look for answers .. what triggers my ..efforts ..

 ... as before the crisis occurred, but only for a finite time after which the orbit rapidly leaves the region [u9]...

 [u9]as before the crisis occurred, but only for a finite time after which the orbit rapidly leaves the region..

aimless ..wandering about.. the attractor ..has ..gone .. what driven them in the first place is no more .. the remnant ..processes that their purpose ..cease to be .. it brings into my mind that excerpt from the book about poverty from Broniclav(the name elludes me) the historical periods of Europe ..overlapping ..the vestiges of behaviour still persisting into the next historical period till them vanishing ..completely ..

 

... (a chaotic transient[u10] ). 

 [u10]a chaotic transient ..  

how chaos is perceived .. assumed as its main ingredient which still perseveres .. 

aimless ..purposeless ..state of being and ..doing ..

In the case of an interior crisis, the attractor suddenly increases in size, and we study the dependence of the fraction of time a typical orbit on an attractor of increased size spends in the formerly empty region as a function of closeness of the parameter to its crisis value[u11]. 

 [u11]the dependence of the fraction of time a typical orbit on an attractor of increased size spends in the formerly empty region as a function of closeness of the parameter to its crisis value..

what is the significance ..found in this ..sentence ..

the formerly ..empty region .. a region that the ..previous attractor ..did not enter .. and now ..new ..ground ..and ..the new attractor ..enters .. a relationship ..looked upon ..the closeness of the parameters to the crisis value .. determining the crisis value ..significant .. in identifying the ..extend .. the inclusiveness ..of the new wider ..attractor .. what processes have been added ..or augmented ..at what degree the new attractor ..corresponds ..to the entities that are ..affected by its processes .. satisfied within its ..orbits ..needs, wants satisfied ..fulfilled ..

 pose the question .. to find the answer ..find out the question ..significant ..crucial ..

 In this case a new type of scaling phenomenon [u12]...

 [u12]scaling phenomenon..

 scaling phenomenon.. does that ..remind of .. butterfly flap .. reminiscent of .. implicated in chaos ..

.. the sudden widening of three chaotic bands to form one broader band ..

.. is found and discussed. [The example of an interior crisis used in section 2, the sudden widening of three chaotic bands to form one broader band, was first extensively discussed by Chang and Wright [4] (see also Grebogi et al. [3]).]

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