Definitions Numerical continuation

1 definitions

1.1 solution component
1.2 numerical continuation
1.3 regular point
1.4 singular point

definitions
solution component

a solution component



Γ
(


u


0


,

λ

0


)


{\displaystyle \gamma (\mathbf {u} _{0},\lambda _{0})}

of nonlinear system



f


{\displaystyle f}

set of points



(

u

,
λ
)


{\displaystyle (\mathbf {u} ,\lambda )}

satisfy



f
(

u

,
λ
)
=
0


{\displaystyle f(\mathbf {u} ,\lambda )=0}

, connected initial solution



(


u


0


,

λ

0


)


{\displaystyle (\mathbf {u} _{0},\lambda _{0})}

path of solutions



(

u

(
s
)
,
λ
(
s
)
)


{\displaystyle (\mathbf {u} (s),\lambda (s))}





(

u

(
0
)
,
λ
(
0
)
)
=
(


u


0


,

λ

0


)
,

(

u

(
1
)
,
λ
(
1
)
)
=
(

u

,
λ
)


{\displaystyle (\mathbf {u} (0),\lambda (0))=(\mathbf {u} _{0},\lambda _{0}),\,(\mathbf {u} (1),\lambda (1))=(\mathbf {u} ,\lambda )}

and



f
(

u

(
s
)
,
λ
(
s
)
)
=
0


{\displaystyle f(\mathbf {u} (s),\lambda (s))=0}

.

this figure shows 2 solution components, 1 red , other blue. note these 2 components may connected outside region of interest.

numerical continuation

a numerical continuation algorithm takes input system of parametrized nonlinear equations , initial solution



(


u


0


,

λ

0


)


{\displaystyle (\mathbf {u} _{0},\lambda _{0})}

,



f
(


u


0


,

λ

0


)
=
0


{\displaystyle f(\mathbf {u} _{0},\lambda _{0})=0}

, , produces set of points on solution component



Γ
(


u


0


,

λ

0


)


{\displaystyle \gamma (\mathbf {u} _{0},\lambda _{0})}

.

regular point

a regular point of



f


{\displaystyle f}

point



(

u

,
λ
)


{\displaystyle (\mathbf {u} ,\lambda )}

@ jacobian of



f


{\displaystyle f}

full rank



(
n
)


{\displaystyle (n)}

.

near regular point solution component isolated curve passing through regular point (the implicit function theorem). in figure above point



(


u


0


,

λ

0


)


{\displaystyle (\mathbf {u} _{0},\lambda _{0})}

regular point.

singular point

a singular point of



f


{\displaystyle f}

point



(

u

,
λ
)


{\displaystyle (\mathbf {u} ,\lambda )}

@ jacobian of f not full rank.

near singular point solution component may not isolated curve passing through regular point. local structure determined higher derivatives of



f


{\displaystyle f}

. in figure above point 2 blue curves cross singular point.

in general solution components



Γ


{\displaystyle \gamma }

branched curves. branch points singular points. finding solution curves leaving singular point called branch switching, , uses techniques bifurcation theory (singularity theory, catastrophe theory).

for finite-dimensional systems (as defined above) lyapunov-schmidt decomposition may used produce 2 systems implicit function theorem applies. lyapunov-schmidt decomposition uses restriction of system complement of null space of jacobian , range of jacobian.

if columns of matrix



Φ


{\displaystyle \phi }

orthonormal basis null space of



j
=

[





f

x





f

λ






]



{\displaystyle j=\left[{\begin{array}{cc}f_{x}&f_{\lambda }\\\end{array}}\right]}

, columns of matrix



Ψ


{\displaystyle \psi }

orthonormal basis left null space of



j


{\displaystyle j}

, system



f
(
x
,
λ
)
=
0


{\displaystyle f(x,\lambda )=0}

can rewritten




[




(
i
−
Ψ

Ψ

t


)
f
(
x
+
Φ
ξ
+
η
)





Ψ

t


f
(
x
+
Φ
ξ
+
η
)




]

=
0
,


{\displaystyle \left[{\begin{array}{l}(i-\psi \psi ^{t})f(x+\phi \xi +\eta )\\\psi ^{t}f(x+\phi \xi +\eta )\\\end{array}}\right]=0,}





η


{\displaystyle \eta }

in complement of null space of



j


{\displaystyle j}





(

Φ

t



η
=
0
)


{\displaystyle (\phi ^{t}\,\eta =0)}

.

in first equation, parametrized null space of jacobian (



ξ


{\displaystyle \xi }

), jacobian respect



η


{\displaystyle \eta }

non-singular. implicit function theorem states there mapping



η
(
ξ
)


{\displaystyle \eta (\xi )}

such



η
(
0
)
=
0


{\displaystyle \eta (0)=0}

,



(
i
−
Ψ

Ψ

t


)
f
(
x
+
Φ
ξ
+
η
(
ξ
)
)
=
0
)


{\displaystyle (i-\psi \psi ^{t})f(x+\phi \xi +\eta (\xi ))=0)}

. second equation (with



η
(
ξ
)


{\displaystyle \eta (\xi )}

substituted) called bifurcation equation (though may system of equations).

the bifurcation equation has taylor expansion lacks constant , linear terms. scaling equations , null space of jacobian of original system system can found non-singular jacobian. constant term in taylor series of scaled bifurcation equation called algebraic bifurcation equation, , implicit function theorem applied bifurcation equations states each isolated solution of algebraic bifurcation equation there branch of solutions of original problem passes through singular point.

another type of singular point turning point bifurcation, or saddle-node bifurcation, direction of parameter



λ


{\displaystyle \lambda }

reverses curve followed. red curve in figure above illustrates turning point.