Regionalization Method for Nonlinear Differential Equation Systems In a Cartesian Plan

We propose a regionalization technique for analyzing nonlinear differential equation systems where coefficients are standard and nonzero. The present work starts with the study of a natural object which was the magic squares providing us with a new way to partition the plan in regions.


INTRODUCTION
In the present work we study the effects of delinearity of box type in orbital geometry and in orbital dynamics. We start this work with a first natural object around which we organize our plan , namely the magic square (MS) giving the partition of the plan 2 ℜ into 25 external sets depicted in the following heuristic diagram . 15 22 9  16  3  2  14  21  8  20  19  1  13  25  7  6  18  5  12  24  23  10  17  4  11 Also, it plays an essential role in the nonclassic partition (regionalization) as following ( Here are some of the questions that can be asked The apparition of the magic square The behaviour of the orbits in the regions of the MS The transition of the orbits of a region have the other The nature of the singular place The strut with the linear case. The objects: We are interested by non-standard systems of nonlinear differential equations in 2 ℜ provided with cartesian coordinates ( X 1 , X 2 ).   (1) where the reals a 11 , a 12 , a 21 , a 22 , b 1, b 2 (resp α 1 > 0 ,α 2 >0) are standard and nonzero (resp infinitely great ) and for i= 1,2 we proceed to a sharp delinearization " of box type delinearization " of differential system with constant coefficients With conserving the linear equations. Then the problem is to evaluat the effects.
We can also see a problem of transient to the limit ( ) +∞ → +∞ → 2 1 , α α In some families which are linear when . 1 2 1 = = α α The differential system (1) is divided into three families :F 1 , F 2 , F 3 F 1 : (a 21, a 22, b 2 ) = m(a 11, a 12, b 1 ) F 2 : (a 21, a 22, b 2 ) = m (a 11, a 12, b 1 ) + (0,0,d) ; d ≠ 0 The technique: we use the technique of regionalization to get some predictions relative to the macroscopic behaviour of orbits and the dynamics along the orbits. Some predictions i. The magic square (MS) in the first analysis the plan 2 ℜ is partitioned in 25 external regions where we denote by (15) (resp (3)) the region defined by the conditions 1 X pp -1 and 2 X ff 1 (resp 1 X ff 1 and 2 X ff 1) (22) (resp (16)) the region defined by the conditions 1 X ≈ -1 and 2 X ff 1 (resp 1 X ≈ 1 and 2 X ff 1) ' is a first integral. The non-singular orbits are rectiligne and with the same slope m, the singular position is given by the equation a differential system of the family F1 induce a family with one real parameter C (C specify the level straight) of differential the form of singular place of F 1 implique some number of bifurcations in the equations D c when C follow ℜ .

The family F 2
The singular place is empty If d was null we obtain a system of 1 F with a singular emptiness in the case to add 0 ≠ d hunts the singular place, but it stay a witness that create a river phenomena.

The family F 3 :
we distinguish three cases j. the vectors ( ) ( ) ( )  (1)) is infinitely near to the rectiligne field vectors Proof of proposition 1: By the lemma of short shadows, the orbits of (1) have in the region (13) the same halo as the system orbits In the regions (19) and (7) (resp (9) and (17) are infinitely small .

Proof of proposition 2:
The proposition 2 is an immediate consequence of the lemma of the short shadows as long as we have lemma 2.  , the curves C 1 = (X' 1 = 0) and C 2 =(X' 2 =0) don't intersect.