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# set4 - 4 Bifurcations and stability in 1-D Consider the...

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1 4. Bifurcations and stability in 1-D Consider the system: Steady-state solns: W e study singular points of the curves in . Suppose is a solution. Implicit function theorem: Let and let F be C 1 in some open interval containing of the plane. Then, if there exists an ! >0 and a ! >0 such that " # \$ # \$ \$ R R, C R R k du F( ,u), u , F ( , ) (1) dt Let u F( , ) 0 (2) " % & # % " F( , ) # % ( , ) plane # % ’ 0 0 ( , ) # % 0 0 F( , ) 0 # % " 0 0 ( , ) # % ( , ) # % 0 0 0 0 F ( , ) F ( , ) 0, % ( # % ) # % * (%

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2 4. Bifurcations and stability in 1-D… i. has a unique solution when such that . ii. The function for iii. Note: W hen , we can solve for . If F is analytic in so is or . # % " F( , ) 0 % " % # ( ) # ’ + , # , # - + 0 0 % ’! , % , % -! 0 0 % # \$ 1 ( ) C # ’ + , # , # - + 0 0 # # % % % # ) " ’ # % # # % # # d ( ) F ( , ( ))/F ( , ( )) (3) d % # # % " * 0 0 F ( , ) 0 but F 0, # " # % ( ) # % ( , ), % # ( ) # % ( )
3 Bifurcations and stability in 1-D… Classification of points 1. Regular point: of - it is a point at which the Implic it Function Theorem works, i.e. , either . That is, there ex ists a unique solution curve or through # % " F( , ) 0 # % # 0 % 0 % ( # ) # % 0 0 ( , ) # % * * F 0 or F 0 % # ( ) # % ( ) # % 0 0 ( , )

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4 Bifurcations and stability in 1-D… 2. Regular turning point: of - it is a point at which changes sign and . The graph of the curve looks like: # % " F( , ) 0 # # % * F ( , ) 0 % # % ( )
5 Bifurcations and stability in 1-D… 3. Singular point: I t is a point at which (4) Consider a point on the curve . Let us consider perturbations in # 0 and % 0 , .# and .% , that is, ( # 0 + .# , % 0 + .% ) also is on the curve F and satisf ies . Assuming .# and .% to be small , we can expand the function in a power series # % 0 0 ( , ) # % # % " # % " 0 0 0 0 F ( , ) F ( , ) 0 # % 0 0 ( , ) # % " F( , ) 0 # - .# % -.% " 0 0 F( , ) 0

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6 Bifurcations and stability in 1-D… Noting that we get (5) # % # % " # % " # % " 0 0 0 0 0 0 F( , ) 0, F ( , ) F ( , ) 0, # % ## #% %% # - .# % - .% " ) # % - # % .# - # % .% - # % .# - # % .# .% - # % .% - .# - .% " 0 0 0 0 0 0 0 0 2 0 0 0 0 2 3 0 0 F( , ) 0 F( , ) F ( , ) F ( , ) F ( , )( ) 2F ( , )( )( ) F ( , )( ) O( ) 0 ## #% %% # -.# % -.% # % .# - # % .# .% - # % .% - .# - .% " 0 0 2 0 0 0 0 2 3 0 0 F( , ) F ( , )( ) 2F ( , )( )( ) F ( , )( ) O( ) 0
7 Bifurcations and stability in 1-D… If we consider % as a function of # , we get in the limit as d #& 0, (6) This is an equation in that gives slopes of possible solution curves through the point .

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set4 - 4 Bifurcations and stability in 1-D Consider the...

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