>> par=m. o c I n s t a n t . ParameterValue ;>> opt=ode s e t ( ’ AbsTol ’ ,1 e−5, ’ R e l t o l ’ ,1 e−8) ;>> s o l=ode45 ( @momCanonicalSystem , [ 0 2 ] , [ 1 1 ] , opt , par , 1 )s o l =
s o l v e r : ’ ode45 ’x : [ 0 0 .2000 0 .3802 0 .5802 0 .7802 0 .9802 1 .1802
>> opt=d e f a u l t o c o p t i o n s ;>> ocTr j=od e s o l v e (m, 2 , occu rve ( [ 1 1 ] ’ , 1 ) , opt )ocTr j =
o c t r a j e c t o r y o b j e c t :dynVar : [ 2 x12 doub le ]t : [ 1 x12 doub le ]a r c i d : 1
t i m e i n t e r v a l s : 2
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
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Phase space
Native MATLAB
Overloaded OCMat
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Native MATLAB
Overloaded OCMat
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OCMat functions
2.48
Solving boundary value problems (BVPs)
We consider the following ODE
x(t) = f (x(t)), t ∈ [0,T ]
b(x(0), x(T )) = 0 ∈ Rn
Remark
To numerically solve a BVP there exist different solvers forMATLAB (bvp4c,bvp5c).
Remark
For the numeric computation an initial solution has to be provided.
Remark
For an IVP the solution (at T = 0) is already given by the initialcondition.
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
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Graphics
Phase space
Native MATLAB
Overloaded OCMat
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Native MATLAB
Overloaded OCMat
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OCMat functions
2.49
A boundary value problem (BVP)
Nerve impulse model
x1(t) = 3(
x1 + x2 − ax31 − b)
x2(t) = −c(x1(t) + d − ex2(t) t ∈ [0,T ]
x1(0) = x1(T ), x2(0) = x2(T )
The problem is now to find a periodic solution and the period T .
Time scaling and phase condition
x ′1(τ ) = 3T(
x1(τ ) + x2(τ ) + ax31 + b)
x ′2(τ ) = cT (x1(τ ) + d + ex2(τ ) t ∈ [0, 1]
x1(0) = x1(1), x2(0) = x2(1)
x ′2(0) = T (x1(0) + d + ex2(0)) = −1. (1)
Phase condition (1) necessary to specify a single point.
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
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OCMat functions
2.50
Solving the BVP using MATLAB
MATLAB Syntax
f u n c t i o n dxdt=func ( t , x ,T, par )% ODE fun c i o ndxdt =[3∗T∗( x (1 )+x (2 )+par (1 ) ∗( x (1 ) ˆ3)+par (2 ) ) ;
par (3 )∗T∗( x (1 )+par (4 )+par (5 )∗x (2 ) ) ] ;
f u n c t i o n r e s=bc ( xa , xb ,T, par )% Boundary c o n d i t i o n sr e s =[xa (1 )−xb (1 )
xa (2 )−xb (2 )T∗par (3 ) ∗( xa (1 )+par (4 )+par (5 )∗xa (2 ) ) +1] ;
f u n c t i o n v=gues s ( t )% Guess f u n c t i o nv=[ s i n (2∗ p i ∗ t ) ; . . .
co s (2∗ p i ∗ t ) ] ;
>> s o l i n i t=b v p i n i t ( l i n s p a c e (0 , 1 , 5 ) , @guess ,2∗ p i ) ;>> opt=bvpse t ( ’ AbsTol ’ ,1 e−5) ;>> par=[−1/3 −1.3 −1/3 −0.7 0 . 8 ] ;>> s o l=bvp4c ( @func , @bc , s o l i n i t , opt , par ) ;
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
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Native MATLAB
Overloaded OCMat
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Native MATLAB
Overloaded OCMat
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OCMat functions
2.51
Nerve impulse model
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.52
Initialization of a boundary value solver
0 2 4 6 8 10−2
−1
0
1
2Solution detected
t
x1
0 2 4 6 8 10−1
−0.5
0
0.5
1
1.5
2
2.5Solution detected
t
x2
0 2 4 6 8 10−2
−1
0
1
2No solution detected
t
x1
0 2 4 6 8 10−0.5
0
0.5
1
1.5
2
2.5No solution detected
t
x2
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.53
Implicit functions: Lemniscate I
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.54
Implicit functions: Lemniscate II
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.55
Theoretical background
Definition (Stable Manifolds)
Let x be an equilibrium, then the set
Wsloc (x) = {x ∈ Bε(x) : lim
t→∞x(t) = x , and x(0) = x ∈ Bε(x)}
is called the stable local manifold of x .The set
Ws(x) =⋃
t≥0
{x(−t) : x(0) ∈ Wsloc (x), t ≥ 0}
is called the global stable manifold.A trajectory x(·) with x(0) ∈ Ws(x) is called a stable path.
Theorem (Manifold Theorem)
Let J = fx (x), and the corresponding stable eigenspace
Then the local stable manifold Wsloc (x) exists with
TWsloc (x) = Es(x)
and
dimWsloc(x) = n−.
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
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Graphics
Phase space
Native MATLAB
Overloaded OCMat
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Native MATLAB
Overloaded OCMat
References
OCMat functions
2.56
Calculating a stable path by the IVP approach
Remark
For the initial value approach we use:
1 An initial point x0 ∈ Es(x) ∩ Bε is chosen.
2 The trajectory x(−t), 0 ≤ t ≤ T is calculated.
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
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Graphics
Phase space
Native MATLAB
Overloaded OCMat
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Native MATLAB
Overloaded OCMat
References
OCMat functions
2.57
Calculating a stable path by an IVP approach (2)
Native MATLAB Syntax
>> [ e i g v e c e i g v a l ]= e i g ( ocEP{1})e i g v e c =
0.6290 0.2952−0.7774 0.9554
e i g v a l =1.6180 0
0 −0.6180>> x0=1e−6∗e i g v e c ( : , 2 )+ocEP{1}. dynVar ( : , 1 ) ;>> opt=ode s e t ( ’ AbsTol ’ ,1 e−5, ’ Re lTo l ’ ,1 e−8, ’ Even t s ’ , ’ momEvents ’ ) ;>> s o l=ode45 ( @momCanonicalSystem , [ 0 −50] ,x0 , opt , par , 1 )s o l =
s o l v e r : ’ ode45 ’x : [ 1 x37 doub l e ]y : [ 2 x37 doub l e ]
xe : −26.7580
OCMat Syntax
>> opt=s e t o c o p t i o n s ( ’OC ’ , ’ L i n e a r i z a t i o nD i s t a n c e ’ ,1 e−6) ;>> opt=s e t o c o p t i o n s ( opt , ’ODE ’ , ’ AbsTol ’ ,1 e−5, ’ Re lTo l ’ ,1 e−8, ’ Even t s ’ , ’ on ’ ) ;>> ocAsym1=s t a b l e p a t h (m, ocEP{1} ,50 , opt )ocAsym1 =
oca s ymp to t i c o b j e c t :l i m i t s e t : [ d y n p r im i t i v e o b j e c t ]dynVar : [ 2 x37 doub l e ]t : [ 1 x37 doub l e ]a r c i d : 1t i m e i n t e r v a l s : 26 .758
>> opt=s e t o c o p t i o n s ( opt , ’OCCONT ’ , ’ Backward ’ , 1 ) ;>> ocAsym2=s t a b l e p a t h (m, ocEP{1} ,50 , opt ) ;>> m=s t o r e (m,{ocAsym1 , ocAsym2} , ’ Fie ldName ’ , ’ Stab lePath IVP ’ ) ;
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.58
Calculating a stable path using the BVP approach
Remark
For the boundary value approach with x ∈ R2 we use.
1 The first coordinate of x0 is fixed.
2 The trajectory has to satisfy limt→∞ x(t) = x .
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
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Native MATLAB
Overloaded OCMat
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OCMat functions
2.59
Problems of the BVP approach
Remark
Two principle problems have to be resolved for a numericalcomputation
1 The convergence property limt→∞ x(t) = x has to bereplaced by some “finite” approximation.
2 An initial solution has to be provided.
Convergence condition
limt→∞
x(t) = x
Using continuation
First step:x (0)(·) ≡ x , t ∈ [0, T ]
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
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Native MATLAB
Overloaded OCMat
References
OCMat functions
2.59
Problems of the BVP approach
Remark
Two principle problems have to be resolved for a numericalcomputation
1 The convergence property limt→∞ x(t) = x has to bereplaced by some “finite” approximation.
2 An initial solution has to be provided.
Asymptotic transversality condition
x(T ) ∈ TWsloc(x) = Es(x), T > 0 fixed
Using continuation
First step:x (0)(·) ≡ x , t ∈ [0, T ]
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.59
Problems of the BVP approach
Remark
Two principle problems have to be resolved for a numericalcomputation
1 The convergence property limt→∞ x(t) = x has to bereplaced by some “finite” approximation.
2 An initial solution has to be provided.
Asymptotic transversality condition
Π (x(T )− x) = 0, Π ∈ R1×2
Using continuation
First step:x (0)(·) ≡ x , t ∈ [0, T ]
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.59
Problems of the BVP approach
Remark
Two principle problems have to be resolved for a numericalcomputation
1 The convergence property limt→∞ x(t) = x has to bereplaced by some “finite” approximation.
2 An initial solution has to be provided.
Asymptotic transversality condition
Π (x(T )− x) = 0, Π ∈ R1×2
Using continuation
Step n: x (n)(·) solves
x(n)1 (0) = x1 + n
x0,1 − x1N
Π(
x (n)(T )− x)
= 0
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.59
Problems of the BVP approach
Remark
Two principle problems have to be resolved for a numericalcomputation
1 The convergence property limt→∞ x(t) = x has to bereplaced by some “finite” approximation.
2 An initial solution has to be provided.
Asymptotic transversality condition
Π (x(T )− x) = 0, Π ∈ R1×2
Using continuation
Last step: x (N)(·) solves
x(N)1 (0) = x0
Π(
x (N)(T )− x)
= 0
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.60
Calculating a stable path with OCMat
Main tool: occont
>> opt=s e t o c o p t i o n s ( ’OCCONT ’ , ’ I n i t S t epWid th ’ , 0 . 002 , ’ MaxStepWidth ’ , 0 . 0 1 ) ;>> i n i t S t r u c t=i n i t o c c o n t ( ’ e x t r ema l ’ ,m, ’ i n i t p o i n t ’ , 1 , 5 , ocEP{1}, ’ I n t e g r a t i o nT ime ’ , 500) ;>> s o l=occon t (m, i n i t S t r u c t , opt )
Con t i n ua t i on s t ep No . : 0s t epw id th : 0 . 002Mesh s i z e : 10I t e r a t i o n number : 12Residuum norm : 0R e l a t i v e d i s t a n c e : 1...Con t i n ua t i on s t ep No . : 104s t epw id th : 0 . 002Mesh s i z e : 32I t e r a t i o n number : 9Residuum norm : 3.97047 e−023R e l a t i v e d i s t a n c e : 0
s o l =oca s ymp to t i c o b j e c t :
l i m i t s e t : [ d y n p r im i t i v e o b j e c t ]o c t r a j e c t o r y :
dynVar : [ 2 x32 doub l e ]t : [ 1 x32 doub l e ]t i m e i n t e r v a l s : 500
>> m=s t o r e (m) ;>> m. o cRe s u l t sans =
Ex t r ema l S o l u t i o n : {[1 x1 oca s ymp to t i c ]}>> ocEx=e x t r ema l s o l (m)ocEx =
[1 x1 oca s ymp to t i c ]
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.61
Calculating a stable path with OCMat (2)
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.62
Implementation
Initialization
>> i n i t S t r u c t=i n i t o c c o n t ( ’ e x t r ema l ’ ,m, ’ i n i t p o i n t ’ , 1 , 5 , ocEP{1}, ’ I n t e g r a t i o nT ime ’ , 500)i n i t S t r u c t =
Asymptot icBCMatr ix : [ 0 . 9952 −0.0978]BVPFunction : [ 1 x1 s t r u c t ]
Cont inuat ionArgument : ’ e x t r ema l ’Con t inuat i onType : ’ f ’
Con t i n u a t i o nVa r i a b l e : ’ i n i t p o i n t ’I n i tBVSo l u t i o n : [ 1 x1 s t r u c t ]
I n t e g r a t i o nT ime : 500Mode lParameterValue : [ 0 . 5000 3 I n f ]
ODEFunction : [ 1 x1 s t r u c t ]PathType : ’ s ’
Targe tVa lue : 5>> i n i t S t r u c t . ODEFunctionans =
Dynamics : ’mom4Cont ’Gen e r a l : ’ d yn4 i 2 ep s ’
>> i n i t S t r u c t . BVPFunctionans =
Con t i nua t i on : ’ momBVP4Continuation ’A s ymp to t i cT ran s v e r s a l i t yCond : ’momBVP4AsymTransCond ’
Gene r a l : ’ b c4 i2ep ’>> i n i t S t r u c t . I n i tBVSo l u t i o nans =
y : [ 2 x10 doub l e ]x : [ 0 0 .1111 0.2222 0.3333 0.4444 0.5556 0.6667 0.7778 0.8889 1 ]
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.63
Implementation: Files describing dynamics
General dynamics
f u n c t i o n out = dyn4 i 2 ep s ( t , dynVar , v a r a r g i n )%% DYN4I2EPS dynamics f o r BVP%i n t e r n a l p a r a r g = [ ] ;i f n a r g i n==3
i n t e r n a l p a r a r g=v a r a r g i n {1};end
% g l o b a l v a r i a b l eg l o b a l OcBVPVar
out=f e v a l (OcBVPVar . ODEFunction . Dynamics , t , dynVar , OcBVPVar . ModelParameterValue ,#,#,#,OcBVPVar . I n t eg r a t i onT ime ,#) ;
Model specific dynamics
f u n c t i o n out = mom4Cont ( t , dynVar , pararg , a r ca rg , sw i t c h i d , sw i t chcoord , t r un c t ,#)%% the dynamics f i l e f o r model mom f o r the BVP
% Dynamical sys tem
sw i t ch l e n g t h ( sw i t c h i d )case 0
out ( 1 : 2 , : )=t r u n c t∗momCanonicalSystem( t , dynVar ( 1 : 2 , : ) , pararg , a r c a r g (1 ) ) ;%ADDCASE
o t h e rw i s ee r r o r ( [ num2str ( l e n g t h ( sw i t c h i d ) ) ’ %s not d e f i n e d ’ ] , s w i t c h i d )
f u n c t i o n out = bc4 i2ep ( dynVara , dynVarb , v a r a r g i n )%% BC4I2EP boundary c o n d i t i o n s f o r s o l u t i o n conv e r g i n g to an e q u i l i b r i um%
i n t e r n a l p a r a r g = [ ] ;i f n a r g i n==3
i n t e r n a l p a r a r g=v a r a r g i n {1};end
% g l o b a l v a r i a b l eg l o b a l OcBVPVarout=[ f e v a l (OcBVPVar . BVPFunction . Cont inuat i on , dynVara , OcBVPVar . Con t i n u a t i o nVa r i a b l e ,# ,
OcBVPVar . ActCont inuat ionType , OcBVPVar . Targe tCoord inate ,# ,OcBVPVar . I n i t i a l V a l u e ,#); . . .
f e v a l (OcBVPVar . BVPFunction . WeierstrassErdmannCond ,# , . . . ) ; . . .f e v a l (OcBVPVar . BVPFunction . Im p l i c i t C o n t r o l , # , . . . ) ; . . .f e v a l (OcBVPVar . BVPFunction . A s ymp to t i cT ran s v e r s a l i t yCond ,# , ’EP ’ ,OcBVPVar .
Asymptot icBCMatr ix , OcBVPVar . Sadd l ePo i n t ) ] ;
File for asymptotic boundary condition
f u n c t i o n out = momBVP4AsymTransCond ( dynVar , L im i t s e tType , Asymptot icBCMatr ix , Sadd l ePo i n t )%% the BC f i l e f o r the a s ymp to t i c t r a n s v e r s a l i t y c o n d i t i o ng l o b a l OcBVPVar
out = [ ] ;sw i t ch L im i t s e tType
case ’EP ’out=Asymptot icBCMatr ix ∗(dynVar−Sadd l ePo i n t ) ;
Model specific boundary condition for continuation
f u n c t i o n out = momBVP4Continuation ( dynVara , Cont inuat ionArgument , ActCont inuat ionType ,Targe tCoord inate ,#,#,#,#, I n i t i a l V a l u e ,#)
%% the BC f i l e r e t u r n i n g the c o n t i n u a t i o n res iduum
% g l o b a l v a r i a b l eg l o b a l OcBVPVar
out = [ ] ;
sw i t ch Cont inuat ionArgument (1 )case ’ i ’
c o n t r e s = [ ] ; % con t i n u a t i o n res iduump o s r e s = [ ] ; % po s i t i o n res iduumi f Ac tCont inuat i onType==0
po s r e s=dynVara ( Ta rge tCoo rd i n a t e )−I n i t i a l V a l u e ;end
endout=[ p o s r e s ; c o n t r e s ] ;
ContinuationArgument: continuation of the initial point, otherpossibilities are the time horizon, parameter, etc.
ActContinuationType: Fixed step width (0), linear adaptive (1),quadratic adaptive (2).
InitialValue: This value returns the actual initial state value and isgenerated during continuation in the function occont.
TargetCoordinate: Actual coordinate of the canonical system forcontinuation.
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.66
Implementation: Continuation
Continuation command
f u n c t i o n v a r a r g ou t=occon t ( ocObj , I n i tBVSt ru c t , v a r a r g i n )% OCCONT con t i n u e an ex t r ema l s o l u t i o n o f an oc model% g l o b a l v a r i a b l eg l o b a l OcBVPVarOcBVPVar=I n i tBVS t r u c t ;b v s o l{3}= i n i t b v s o l ; % bv s o l{3} denotes the a c t u a l s o l u t i o nwh i l e c o n t f l a g
i f contnum > 1t r y
b v s o l p r e d i c t . x=b v s o l{3}. x ;b v s o l p r e d i c t . y = b v s o l{3}. y∗(1+a l p h a f a c ) − a l p h a f a c∗d e v a l ( b v s o l{2}, b v s o l{3}. x )
; % p r e d i c t e d s o l u t i o ne l s e
b v s o l p r e d i c t = b v s o l{3};s t epw id th=i n i t s t e p w i d t h ;
endt r y
s o l v e r% VIOLATION CHECK
endi f ˜ e r r o r f l a g % s o l u t i o n found
% SETTING STEP WIDTH, DETERMINE RELATIVE DISTANCE% STORING PREVIOUS RESULTS , HANDLE LAST STEPi n i t i a l p o i n t s =[ i n i t i a l p o i n t s [ b v s o l a c t u a l . y ( : , 1 ) ; a c t u a l r e l d i s t a n c e ] ] ;e ndpo i n t s =[ endpo i n t s [ b v s o l a c t u a l . y ( : , end ) ; a c t u a l r e l d i s t a n c e ] ] ;% PLOT ACTUAL SOLUTIONOcBVPVar . I n i t i a l V a l u e=OcBVPVar . S t a r tVa l u e+a c t u a l r e l d i s t a n c e∗OcBVPVar .
Con t i n ua t i onVec t o r ;% update c e l l o f s o l u t i o n sb v s o l ( 1 ) = [ ] ;b v s o l{3}=b v s o l a c t u a l ;
endcontnum=contnum+1;
end
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.67
Implementation: Solver
Solver section
f u n c t i o n s o l v e r
bc func=s t r 2 f u n c (OcBVPVar . BVPFunction . Gen e r a l ) ;ode func=s t r 2 f u n c (OcBVPVar . ODEFunction . Gen e r a l ) ;
sw i t ch b v p s o l v e rcase ’ bvp4c ’
b v s o l a c t u a l= bvp4c ( odefunc , bcfunc , b v s o l p r e d i c t , opt .BVP) ;case ’ bvp5c ’
b v s o l a c t u a l= bvp5c ( odefunc , bcfunc , b v s o l p r e d i c t , opt .BVP) ;o t h e rw i s e
e r r o r ( ’BV s o l v e ’ ’%s ’ ’ not implemented . ’ , b v p s o l v e r )endi f ˜OcBVPVar . ActCont inuat i onType
b v s o l a c t u a l . con tparamete r=a c t u a l r e l d i s t a n c e ;end% r e t u r n s the res iduum o f the a c t u a l s o l u t i o nb c r e s=f e v a l ( bcfunc , b v s o l a c t u a l . y ( : , 1 ) , b v s o l a c t u a l . y ( : , end ) ) ;
f p r i n t f (1 , ’ Mesh s i z e : %d\n ’ , s i z e ( b v s o l a c t u a l . y , 2 ) ) ;f p r i n t f (1 , ’ I t e r a t i o n number : %d\n ’ , b v s o l a c t u a l . b c e v a l s ) ;i t e rnum=b v s o l a c t u a l . b c e v a l s ;f p r i n t f (1 , ’ Residuum norm : %g\n ’ , norm ( b c r e s ( 1 : end ) ) )i f norm ( b c r e s ( 1 : end ) ) > maxresnorm
e r r o r f l a g = 1 ;e l s e
e r r o r f l a g = 0 ;end
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.68
Turning point of a stable manifold
Default continuation: fixed continuation direction
>> m=changeparameter (m, ’ r , c ’ , [ 0 . 5 3 ] ) ;>> ocEP=ca l c e p (m) ; b=i s a dm i s s i b l e (m, ocEP) ; ocEP(˜ b ) = [ ] ; i s s a d d l e ( ocEP{:})ans =
1 0 0 1 1>> opt=s e t o c o p t i o n s ( ’OCCONT ’ , ’ I n i t S t epWid th ’ , 0 . 002 , ’ MaxStepWidth ’ , 0 . 0 1 ) ;>> i n i t S t r u c t=i n i t o c c o n t ( ’ e x t r ema l ’ ,m, ’ i n i t p o i n t ’ , 1 , 0 , ocEP{4}, ’ I n t e g r a t i o nT ime ’ , 500) ;>> s o l=occon t (m, i n i t S t r u c t , opt ) ;
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.69
Adaptive continuation: Initialization
>> opt=s e t o c o p t i o n s ( opt , ’OCCONT ’ , ’ I n i t S t epWid th ’ , 0 . 01 , ’ StepWidth ’ , 0 . 01 , ’MaxStepWidth ’, 0 . 0 1 ) ;
>> i n i t S t r u c t=i n i t o c c o n t ( ’ e x t r ema l ’ ,m, ’ i n i t p o i n t ’ , 1 , 0 , ocEP{4}, ’ I n t e g r a t i o nT ime ’ ,500 , ’Con t inuat i onType ’ , ’ a l ’ ) ;
>> s o l=occon t (m, i n i t S t r u c t , opt ) ;
Model specific boundary condition for adaptive continuation
f u n c t i o n out = momBVP4Continuation ( dynVara , Cont inuat ionArgument , ActCont inuat ionType ,Targe tCoord inate ,#,#,#,#, I n i t i a l V a l u e , i n t e r n a l p a r a r g )
% the BC f i l e r e t u r n i n g the c o n t i n u a t i o n res iduum
% g l o b a l v a r i a b l eg l o b a l OcBVPVarsw i t ch Cont inuat ionArgument (1 )
case ’ i ’c o n t r e s = [ ] ; % con t i n u a t i o n res iduump o s r e s = [ ] ; % po s i t i o n res iduumi f Ac tCont inuat i onType==0
po s r e s=dynVara ( Ta rge tCoo rd i n a t e )−I n i t i a l V a l u e ;e l s e i f ActCont inuat i onType==2
nv=d i f f ( Prev iousBVSol ( [ 1 : 2 ] , : ) , 1 , 2 ) /norm ( d i f f ( Prev iousBVSol ( [ 1 : 2 ] , : ) , 1 , 2 ) ) ;c o n t r e s=sum ( [ Prev iousBVSol ( [ 1 : 2 ] , 2 )−dynVara ( 1 : 2 ) ] . ’∗ nv )+StepWidth ;p o s r e s=dynVara ( Ta rge tCoo rd i n a t e )−Star tVa lue−i n t e r n a l p a r a r g ( end )∗
Con t i nua t i onVec t o r ;end
endout=[ p o s r e s ; c o n t r e s ] ;
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.71
Plotting the calculated solutions
Load stored model into the MATLAB workspace
>> l o ad ( ’ F i r s tMode l . mat ’ ) ;m=ocObj>> ocEx=e x t r ema l s o l (m) ;>> ocEP=e q u i l i b r i um (m) ;>> m. o cRe su l t s . Stab lePath IVP ; ocAsym=ans ;
Native MATLAB syntax
>> p l o t ( ocEx {1} . t , ocEx {1} . dynVar ( 1 , : ) )>> s e t ( gca , ’ Xlim ’ , [ 0 0 . 0 5 ] , ’YLim ’ , [−0.05 5 ] )
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.71
Plotting the calculated solutions
Load stored model into the MATLAB workspace
>> l o ad ( ’ F i r s tMode l . mat ’ ) ;m=ocObj>> ocEx=e x t r ema l s o l (m) ;>> ocEP=e q u i l i b r i um (m) ;>> m. o cRe su l t s . Stab lePath IVP ; ocAsym=ans ;
Native MATLAB syntax
>> p l o t ( ocEx {1} . t , ocEx {1} . dynVar ( 1 , : ) )>> s e t ( gca , ’ Xlim ’ , [ 0 0 . 0 5 ] , ’YLim ’ , [−0.05 5 ] )
>> ho ld on , p l o t ( ocEx {1} . dynVar ( 1 , : ) , ocEx {1} . dynVar ( 2 , : ) )
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
1.2
1.4
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.71
Plotting the calculated solutions
Load stored model into the MATLAB workspace
>> l o ad ( ’ F i r s tMode l . mat ’ ) ;m=ocObj>> ocEx=e x t r ema l s o l (m) ;>> ocEP=e q u i l i b r i um (m) ;>> m. o cRe su l t s . Stab lePath IVP ; ocAsym=ans ;
Native MATLAB syntax
>> p l o t ( ocEx {1} . t , ocEx {1} . dynVar ( 1 , : ) )>> s e t ( gca , ’ Xlim ’ , [ 0 0 . 0 5 ] , ’YLim ’ , [−0.05 5 ] )
>> ho ld on , p l o t ( ocEx {1} . dynVar ( 1 , : ) , ocEx {1} . dynVar ( 2 , : ) )
>> p l o t ( ocAsym{2} . dynVar ( 1 , : ) , ocAsym {2} . dynVar ( 2 , : ) , ’ Co lo r ’ , [ 00 .5 0 ] )
−6 −4 −2 0 2 4 6−1.5
−1
−0.5
0
0.5
1
1.5
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.72
Plotting the calculated solutions (2)
Overloaded OCMat syntax
>> p l o t p h a s e (m, ocEP{1} , ’ c o s t a t e ’ , 1 , 1 , ’ Co lo r ’ , [ 1 0 0 ] )>> ho ld on , p l o t p h a s e (m, ocEx{1} , ’ c o s t a t e ’ , 1 , 1 )>> p l o t p h a s e (m, ocAsym{2} , ’ c o s t a t e ’ , 1 , 1 , ’ Co lo r ’ , [ 0 0 .5 0 ] )
−6 −4 −2 0 2 4 6−1.5
−1
−0.5
0
0.5
1
1.5
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.72
Plotting the calculated solutions (2)
Overloaded OCMat syntax
>> p l o t p h a s e o c r e s u l t (m, ’ c o s t a t e ’ , 1 , 1 )
−6 −4 −2 0 2 4 6−1.5
−1
−0.5
0
0.5
1
1.5
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.72
Plotting the calculated solutions (2)
Overloaded OCMat syntax
>> p l o t p h a s e o c r e s u l t (m, ’ c o s t a t e ’ , 1 , 1 , ’ e x c l u d e ’ ,{ ’E x t r ema l S o l u t i o n ’ , ’ S l i c eMan i f o l d ’ })
−6 −4 −2 0 2 4 6−1.5
−1
−0.5
0
0.5
1
1.5
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.72
Plotting the calculated solutions (2)
Overloaded OCMat syntax
>> p l o t p h a s e o c r e s u l t (m, ’ c o s t a t e ’ , 1 , 1 , ’ o n l y ’ ,{ ’ E q u i l i b r i um ’ } , ’Marker ’ , ’ . ’ , ’ Marke rS ize ’ ,16)
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.72
Plotting the calculated solutions (2)
Overloaded OCMat syntax
>> p l o t p h a s e o c r e s u l t (m, ’ c o s t a t e ’ , 1 , 1 , ’ o n l y ’ ,{ ’ S tab lePath IVP ’ } ,’ l i m i t s e t ’ , ’ on ’ , ’ l i m i t s e tMa r k e r ’ , ’ . ’ , ’ l i m i t s e tMa r k e r S i z e ’,16)
−6 −4 −2 0 2 4 6−1.5
−1
−0.5
0
0.5
1
1.5
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.72
Plotting the calculated solutions (2)
Overloaded OCMat syntax
>> p l o t p h a s e o c r e s u l t (m, ’ c o s t a t e ’ , 1 , 1 , ’ o n l y ’ ,{ ’ S tab lePath IVP ’ } ,’ l i m i t s e t ’ , ’ o f f ’ )
−6 −4 −2 0 2 4 6−1.5
−1
−0.5
0
0.5
1
1.5
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.72
Plotting the calculated solutions (2)
Overloaded OCMat syntax
>> p l o t p h a s e o c r e s u l t (m, ’ c o s t a t e ’ , 1 , 1 , ’ o n l y ’ ,{ ’ S tab lePath IVP ’ } ,’ o n l y i n d e x ’ ,{2})
−6 −5 −4 −3 −2 −1 0−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.72
Plotting the calculated solutions (2)
Overloaded OCMat syntax
>> p l o t p h a s e o c r e s u l t (m, ’ c o n t r o l ’ , 1 , 1 , ’ o n l y ’ ,{ ’ S tab lePath IVP ’ })
−6 −4 −2 0 2 4 6−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.72
Plotting the calculated solutions (2)
Overloaded OCMat syntax
>> p l o t p h a s e o c r e s u l t (m, ’ h am i l t o n i a n ’ , 1 , 1 , ’ o n l y ’ ,{ ’S tab lePath IVP ’ })
−6 −4 −2 0 2 4 60
0.5
1
1.5
2
2.5
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.73
Plotting in the time domain
Load stored model into the MATLAB workspace
>> l o ad ( ’ F i r s tMode l . mat ’ ) ;m=ocObj>> ocEx=e x t r ema l s o l (m) ;>> ocEP=e q u i l i b r i um (m) ;>> m. o cRe su l t s . Stab lePath IVP ; ocAsym=ans ;
Native MATLAB syntax
>> p l o t ( ocEx {1} . t , ocEx {1} . dynVar ( 1 , : ) )>> s e t ( gca , ’ Xlim ’ , [ 0 0 . 0 5 ] , ’YLim ’ , [−0.05 5 ] )
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.73
Plotting in the time domain
Load stored model into the MATLAB workspace
>> l o ad ( ’ F i r s tMode l . mat ’ ) ;m=ocObj>> ocEx=e x t r ema l s o l (m) ;>> ocEP=e q u i l i b r i um (m) ;>> m. o cRe su l t s . Stab lePath IVP ; ocAsym=ans ;
Native MATLAB syntax
>> p l o t ( ocEx {1} . t , ocEx {1} . dynVar ( 1 , : ) )>> s e t ( gca , ’ Xlim ’ , [ 0 0 . 0 5 ] , ’YLim ’ , [−0.05 5 ] )
>> p l o t t imep a t h (m, ocEx{1} , ’ c o s t a t e ’ , 1 )>> s e t ( gca , ’ Xlim ’ , [ 0 50 ] , ’YLim ’ , [−0.05 1 . 5 ] )
0 5 10 15 20 25 30 35 40 45 50
0
0.5
1
1.5
Dieter Graß
Numerical basics
Solving IVPs
Solving BVPs
Continuation
Stable paths
Definitions
IVP approach
BVP approach
BVP Implementation
Initialization
Dynamics
Boundary conditions
Continuation
Solver
Turning points
Adaptive continuation
Graphics
Phase space
Native MATLAB
Overloaded OCMat
Time path
Native MATLAB
Overloaded OCMat
References
OCMat functions
2.75
References
P. Kunkel and O. von dem Hagen. Numerical solution of infinite-horizonoptimal-control problems. Computational Economics, 16(3):189–205,2000
T. Pampel. Numerical approximation of generalized connecting orbits.PhD thesis, University of Bielefeld, 2000
R. Winkler. Path-following for two-point boundary value problems.Seminarbericht 78, Department of Mathematics, Humboldt-UniversityBerlin, Germany, 1985
D. Grass. Numerical computation of the optimal vector field in a fisherymodel. The Beijer Discussion Paper 224, The Beijer Institute, TheRoyal Swedish Academy of Sciences, 2010
Y.A. Kuznetsov. Elements of Applied Bifurcation Theory. SpringerVerlag, New York, 2nd edition, 1998
A.S. Dontchev and R.T. Rockafellar. Implicit Functions and Solution
Mappings: A View from Variational Analysis. Springer Verlag, 2009
B. Krauskopf, H.M. Osinga, and J. Galan-Vioque, editors. Numerical
Continuation Methods for Dynamical Systems: Path following and
boundary value problems. Springer Verlag, Dordrecht, 2007
Dieter Graß
IVPs
Solve ODEs
Stable path
Manage results
BVPs
Stable path
Store paths
Plotting
Phase space
Time path
2.76
Solving IVPs
odesolve
f u n c t i o n [ s o l , s o l n ]= od e s o l v e ( ocObj , t ime l eng th , ocTr j , v a r a r g i n )%% ODESOLVE s o l v e s an i n i t i a l v a l u e problem o f ocmodel .%% SOL=ODESOLVE(OCOBJ,TIMELENGTH,OCELEMENT) OCOBJ op t ima l c o n t r o l model TIMELENGTH% denotes % the l e n g t h o f the i n t e g r a t i o n i n t e r v a l . With OCELEMENT the i n i t i a l% s t a t e and a c t u a l a r c % i d e n t i f i e r i s p r ov i d ed . OCELEMENT can be e i t h e r an% o c t r a j e c t o r y o r an occu r v e . By d e f a u l t the f i r s t column o f the dynVar f i e l d i s% taken as the i n t i a l p o i n t and the c o r r e s p on d i n g a r c i d as the a r c i d e n t i f i e r .%% SOL=ODESOLVE(OCOBJ,TIMELENGTH, X0 , ARCID) Here X0 i s the i n i t i a l p o i n t and ARCID% the v a l u e o f the a r c i d e n t i f i e r . I f ARCID i s omi t t ed i t i s assumed as one .%% SOL=ODESOLVE(OCOBJ,TIMESPAN , . . . , OPT) op t i on s t r u c t u r e OPT ( s e e d e f a u l t o c o p t i o n s ) .% For the u s u a l ode op t i o n s s e e MATLAB r e f e r e n c e f o r od e s e t . I n the o c op t i o n s% s t r u c t u r e t h e s e can be ad r e s s e d by :% opt=s e t o c o p t i o n s ( ’ODE’ , o p t i o n f i e l d , o p t i o n v a l u e )% Only f o r the ’ Even t s and Jacob ian f i e l d you do must not p r o v i d e the f i l e n ame% ( as i n od e s e t ) , but on l y s e t i t to 1 ’ Even t s/ Jaco iban ’ on or 0 ’ Even t s/ Jaco iban ’% o f f . Depending on the OCOBJ the , e . g . , the even t f i l e i s assumed as% ’ modelnameEvents ’ .%% Other o p t i o n s impor tan t f o r the o d e s o l v e r a r e s e t i n the OC f i e l d . There you can% change the d e f a u l t o d e s o l v e r ’ ode45 ’ to some o th e r s o l v e r s e t t i n g :% opt=s e t o c o p t i o n s ( ’OC’ , ’ ODESolver ’ , o d e s o l v e r )% I f you want to chage the d e f a u l t i n d ex f o r the i n i t i a l p o i n t from 1 to some o th e r% s t a r t i n g i nd ex you can s e t :% opt=s e t o c o p t i o n s ( ’OC’ , ’ S t a r t I n d ex ’ , i d x )
Dieter Graß
IVPs
Solve ODEs
Stable path
Manage results
BVPs
Stable path
Store paths
Plotting
Phase space
Time path
2.77
Calculating manifold paths
f u n c t i o n ocAsy = s t a b l e p a t h ( ocObj , dynPrim , t ime l eng th , v a r a r g i n )%% STABLEPATH c a l c u l a t e s the s t a b l e path u s i n g the IVP approach .%% For the IVP a po i n t l y i n g on the s t a b l e e i g e n s p a c e near% the e q u i l i b r i um i s used as an i n i t i a l p o i n t f o r s o l v i n g the% c a n o n i c a l sys tem backwards i n t ime .%% STABLEPATH(OCOBJ,DYNPRIM,T) c a l c u l a t e s the s t a b l e path o f DYNPRIM wi th% t ime l e n g t h abs (T) .%% STABLEPATH(OCOBJ,DYNPRIM,T,OPT) d i s t a n c e o f the i n i t i a l p o i n t from the% equ i l b r i um i s de te rm ined by the op t i on en t r y ’ L i n e a r i z a t i o nD i s t a n c e ’ o f the OC% f i e l d . Thus to change the d e f u a l t d i s t a n c e 1e−3 you can s e t :% opt=s e t o c o p t i o n s ( ’OC’ , ’ L i n e a r i z a t i o nD i s t a n c e ’ , e i g t o l )% To c a l c u l a t e the path i n the op p o s i t e d i r e c t i o n one can s e t% opt=s e t o c o p t i o n s ( ’OC’ , ’ Backward ’ , 1 )%% STABLEPATH(OCOBJ,DYNPRIM,T,OPT,FLAG) i f FLAG i s s e t to ’ s t rong ’ than the% s t r o n g s t a b l e path i s c a l c u l a t e d ( e i g e n v e c t o r c o r r e s p on d i n g to the% e i g e n v a l u e w i th maximum ab s o l u t r e a l v a l u e ) .%% STABLEPATH(OCOBJ,DYNPRIM,T,OPT,FLAG ,W) For s t a b l e man i f o l d s w i th d imens ion% dim>1 the f a c t o r W can be p rov ided , we i gh t i n g the d i f f e r e n t e i g e n v e c t o r s .
f u n c t i o n ocAsy = man i f o l d pa th ( ocObj , dynPrim , t ime l eng th , v a r a r g i n )%% MANIFOLDPATH c a l c u l a t e s the s t a b l e path%% MANIFOLDPATH(OCOBJ,DYNPRIM,T,OPT,FLAG) FLAG can be one o f the e n t r i e s% ’ s ’ : s t a b l e o r ’ s s ’ : s t r o n g s t a b l e o r ’ws ’ : weak s t a b l e% ’u ’ : u n s t a b l e o r ’ su ’ : s t r o n g u n s t ab l e o r ’wu ’ : weak u n s t ab l e% ’ c ’ : c e n t e r% For the o th e r e n t r i e s and op t i o n s s e e ’ s t a b l e p a t h ’
Dieter Graß
IVPs
Solve ODEs
Stable path
Manage results
BVPs
Stable path
Store paths
Plotting
Phase space
Time path
2.78
Storing results
f u n c t i o n [ newocObj , v a r a r g ou t ]= s t o r e ( ocObj , v a r a r g i n )%% STORE r e s u l t s%% OCOBJ=STORE(OCOBJ, ’CONTINUATION ’ ) s t o r e r e s u l t s o f the l a s t% c o n t i n u a t i o n p r o c e s s i n the r e s u l t s o f the oc model OCOBJ . The i d e n t i f i e r% ’CONTINUATION’ can a l s o be omi t t ed .%% OCOBJ=STORE(OCOBJ, ’ DATAFILE .MAT’ ) s t o r e data g i v en by the MAT f i l e DATAFILE .MAT as% a r e s u l t o f a c o n t i n u a t i o n p r o c e s s .%% OCOBJ=STORE(OCOBJ,OCELEMENT) s t o r e s the ob j e c t OCELEMENT i n the r e s u l t s o f the oc% ob j e c t OCOBJ (OCELEMENT can be a c e l l a ray o f o b j e c t s ) . P o s s i b l e c l a s s e s f o r% OCELEMENT are% dy n p r im i t i v e : e q u i l i b r i um or l i m i t c y c l e% oca s ymp to t i c : a s ymp to t i c s o l u t i o n s%% The d e f a u l t f i e l d n ame s o f the o cRe s u l t s a r e% ocRe s u l t s . E q u i l i b r i um : f o r a d y n p r im i t i v e b e i n g an e q u i l i b r i um% ocRe s u l t s . L im i tCy c l e : f o r a d y n p r i m i t i v e b e i n g a l i m i t c y c l e% ocRe s u l t s . E x t r ema l S o l u t i o n : f o r an oca s ymp to t i c% ocRe s u l t s . NonAdmi s s i b l eSo l u t i on : f o r an oca s ymp to t i c v i o l a t i n g the% models c o n s t r a i n t s%% OCOBJ=STORE ( . . . , ’ REPLACE’ ,VAL) f o r c e s to r e p l a c e an a l r e a d y c a l c u l a t e d% ob j e c t by the new one i f VAL=’on ’ ( d e f a u l t i t i s s e t ’ o f f ’ ) . I f t h e% ob j e c t does not a l r e a d y e x i s t i t i s appended .%% OCOBJ=STORE ( . . . , ’ APPEND’ ,VAL) f o r c e s to append the ob j e c t to an a l r e a d y% e x i s t i n g c e l l a r r a y o f o b j e c t s o r c r e a t e s a new c e l l a r r a y .%% OCOBJ=STORE ( . . . , ’ FIELDNAME’ ,VAL) f o r c e s to s t o r e the ob j e c t a t the% ocRe s u l t s under the f i e l d n ame VAL, i f the ob j e c t i s a c e l l a r ray , VAL i s% e i t h e r a s t r i n g o r a c e l l a r r a y o f s t r i n g s o f the same l e n g t h as VAL .%% [OCOBJ,OCELEMENT]=STORE(OCOBJ , . . . ) r e t u r n s the s t o r e d e l ement s OCELEMENT.%% [OCOBJ,OCELEMENT,FIELDNAMES]=STORE(OCOBJ , . . . ) r e t u r n s c o r r e s p on d i n g% f i e l d n ame s .
Dieter Graß
IVPs
Solve ODEs
Stable path
Manage results
BVPs
Stable path
Store paths
Plotting
Phase space
Time path
2.79
Removing results
Remove specific results
f u n c t i o n ocObj=d e l e t e ( ocObj , v a r a r g i n )%% DELETE d e l e t e s s p e c i f i c r e s u l t s s t o r e d i n OCOBJ .%% OCOBJ=DELETE(OCOBJ,FIELDNAME, INDEX) d e l e t e s the r e s u l t s s t o r e d under FIELDNAME i n the% ocResu l t s , where INDEX i s an a r r a y s p e c i f y i n g the i nd ex o f the r e s u l t s to be d e l e t e d .%% OCOBJ=DELETE(OCOBJ, FIELDNAME1, INDEX1 , FIELDNAME2, INDEX2 , . . . )
Remove entire field
f u n c t i o n newocObj=remove ( ocObj , f i e l dname , v a r a r g i n )%% REMOVE removes a l l r e s u l t s from a o cRe s u l t s f i e l d .%% OCOBJ=REMOVE(OCOBJ,FIELDNAME) removes the f i e l d FIELDNAME from the% ocRe s u l t s .%% OCOBJ=REMOVE(OCOBJ,FIELDNAME, ’FORCE’ ) f o r c e s to remove the f i e l d w i thou t% a warn ing .%% OCOBJ=REMOVE(OCOBJ,FIELDNAME, INDEX) remove o b j e c t s w i th i n d ex INDEX from% the f i e l d FIELDNAME o f the o cRe s u l t s .
Dieter Graß
IVPs
Solve ODEs
Stable path
Manage results
BVPs
Stable path
Store paths
Plotting
Phase space
Time path
2.80
Retrieving stored results
Remark
To retrieve the elements of the fields Equilibrium andExtremalSolution in the ocResults specific commands aredefined, see below.
Predefined commands
>> ocEP=e q u i l i b r i um (m)ocEP =
[1 x1 d y n p r im i t i v e ] [ 1 x1 d y n p r im i t i v e ] [ 1 x1 d y n p r im i t i v e ]>> ocEx=e x t r ema l s o l (m)ocEx =
[1 x1 o c t r a j e c t o r y ] [ 1 x1 o c t r a j e c t o r y ] [ 1 x1 o c t r a j e c t o r y ]
Retrieve elements of other (user defined) fields
>> ocAsym=m. o cRe s u l t s . S tab l ePathocAsym =
[1 x1 oca s ymp to t i c ] [ 1 x1 oca s ymp to t i c ] [ 1 x1 oca s ymp to t i c ]>> ocEP=m. o cRe s u l t s . E q u i l i b r i umocEP =
[1 x1 d y n p r im i t i v e ] [ 1 x1 d y n p r im i t i v e ] [ 1 x1 d y n p r im i t i v e ]
Dieter Graß
IVPs
Solve ODEs
Stable path
Manage results
BVPs
Stable path
Store paths
Plotting
Phase space
Time path
2.81
Initialization file for the BVP ansatz
f u n c t i o n i n i t S t r u c t=i n i t o c c o n t ( contarg , ocObj , va r t ype , t a r g e t c o o r d i n a t e , t a r g e t v a l u e ,i n i t t r j , v a r a r g i n )
%% INITOCCONT r e t u r n s the i n i t i a l i z i n g s t r u c t u r e f o r the c o n t i n u a t i o n% p r o c e s s%% INITSTRUCT=INITOCCONT(CONTARG,OCOBJ,VARTYPE,TCOORD,TVAL, INITTRJ , . . . )% r e t u r n s the i n i t i a l i z i n g s t r u c t u r e INITSTRUCT f o r the c o n t i n u a t i o n type% s p e c i f i e d i n CONTARG, where CONTARG i s% ’ ex t r ema l ’ . . . i f an ex t r ema l s o l u t i o n i s con t i n u ed% ’ boundary ’ . . . i f t h e boundary o f a c o n s t r a i n t i s con t i n u ed% ’ dnss ’ . . . i f a DNSS po i n t i s con t i n u ed% INITTRJ i s an i n i t i a l s o l u t i o n f o r the ana l y z ed BVP and i s o f c l a s s% ’ d y n p r im i t i v e ’ , ’ ocasymptot i c ’ o r ’ o c t r a j e c t o r y ’ .% The arguments VARTYPE, TCOORD and TVAL a r e e x p l a i n e d nex t :%% INITSTRUCT=INITOCCONT(CONTARG,OCOBJ, ’ INITPOINT ’ , INITPTIDX , INITPTVAL , . . . )% INITPTIDX denotes the c o o r d i n a t e ( s ) o f the i n i t i a l s t a t e s to which the% s o l u t i o n i s con t i n u ed . I f INITPTIDX i s empty the e n t i r e c o o r d i n a t e s o f% the s t a t e space a r e assumed .%% INITSTRUCT=INITOCCONT(CONTARG,OCOBJ, ’PARAMERTER’ , PARIDX ,PARVAL , . . . ) w i th% PARIDX the i nd ex o f the v a r i e d paramete r i s s p e c i f i e d and PARVAL denotes% the paramete r v a l u e to which the s o l u t i o n i s con t i n u ed .%% INITSTRUCT=INITOCCONT ( . . . , ’ I n t eg r a t i onT ime ’ ,T) f o r an i n f i n i t e t ime% ho r i z o n model t h i s denotes the t r u n c a t i o n time , o t h e rw i s e i t i s s e t to% the t ime h o r i z o n .%% INITSTRUCT=INITOCCONT ( . . . , ’ Cont inuat ionType ’ ,CT) p o s s i b l e v a l u e s f o r CT% are% ’ f ’ . . . f i x s t ep method% ’ a l ’ . . . a d ap t i v e s t ep method wi th l i n e a r e x t r a p o l a t i o n% ’ aq ’ . . . a d ap t i v e s t ep method wi th q u ad r a t i c e x t r a p o l a t i o n%% INITSTRUCT=INITOCCONT ( . . . , ’ PathType ’ ,PT) f o r an i n f i n i t e t ime h o r i z o n% model the path type PT i s e i t h e r ’ s ’ f o r a s t a b l e , ’ u ’ f o r an un s t ab l e ,% ’ ss ’ f o r the s t r o n g s t a b l e , and ’ uu ’ f o r the s t r o n g u n s t ab l e path .
Dieter Graß
IVPs
Solve ODEs
Stable path
Manage results
BVPs
Stable path
Store paths
Plotting
Phase space
Time path
2.82
Calculating manifold paths using the BVP ansatz
f u n c t i o n v a r a r g ou t=occon t ( ocObj , I n i tBVSt ru c t , v a r a r g i n )%% OCCONT s o l v e s and con t i n u e s a BVP%% OCXP=OCCONT(OCOBJ, INITSTRUCT) c o n t i n u a t i o n o f a s o l u t i o n s p e c i f i e d i n% INITSTRUCT o f the oc model OCOBJ . OCXP i s the l a s t a dm i s s i b l e s o l u t i o n o f% the c o n t i n u a t i o n p r o c c e s s . INITSTRUCT i s a s t r u c t u r r e t u r n e d by INITOCCONT ( . . . ) .%% OCXP=OCCONT(CONTARG,OCOBJ, INITSTRUCT ,OPT) OPT p r o v i d e s o c op t i o n s f o r the% c o n t i n u a t i o n p r o c e s s . Impoerant o p t i o n s f o r the c o n t i n u a t i o n p r o c e s s a r e :% MaxCont inuat i onStep s : maximum number o f c o n t i n u a t i o n s t e p s [ I n f ]% Check : ch eck i n g a d m i s s i b i l i t y o f the s o l u t i o n ’ on ’% Backward : con t i n u e fo rward [ 0 ] o r backward [ 1 ]% In i t S t epWid th : i n i t i a l s t ep width [ 1 e−4]% MaxStepWidth : maximum s t ep width [ 0 . 5 ]% Mean I t e r a t i on : used as a ‘ ‘ measure ’ ’ f o r i n c r e a s i n g the s t ep width [ 1 5 ]% MinStepWidth : min imal s t ep width [ 1 e−8] , belwo tha t the c o n t i n u a t i o n% p r o c e s s i s i n t e r r u p t e d% Admi s s i b l eTo l : t o l e r a n c e to dete rm ine s a dm i s s i b l i t y o f a s o l u t i o n [ 1 e−6]% Sav e I n t e rmed i a t e : s t o r i n g a l s o i n t e rm e d i a t e s o l u t i o n s du r i n g c o n t i n a u t i o n% [ ’ o f f ’ ] . I f s e t to ’ on ’ the i n t e rm e d i a t e s o l u t i n o s
can% be added to the o cRe s u l t s u s i n g m=s t o r e (m)% StepFacto r : f a c t o r f o r i n c r e a s i n g the s t ep width [ 5 ]% MaxGr idPoints : p r o v i d e s an upper l i m i t f o r the t ime g r i d o f the s o l u t i o n .% I f t h i s i s exceeed the s t ep width i s r educed [ 5 0 0 ] .%% These op t i o n s can be s e t u s i n g : opt=s e t o c o p t i o n s ( ’OCCONT’ , optionname , o p t i o n v a l u e )%% To change the d e f a u l t BVP s o l v e r ’ bvp4c ’ use :% opt=s e t o c o p t i o n s ( ’OC’ , ’ BVPSolver ’ , b v p s o l v e r )%% The op t i o n s f o r the BVP s o l v e r can be s e t u s i n g :% opt=s e t o c o p t i o n s ( ’BVP’ , optionname , o p t i o n v a l u e )% f o r t h e s e op t i o n s s e e MATLAB r e f e r e n c e f o r ’ bvp4c ’ .
Dieter Graß
IVPs
Solve ODEs
Stable path
Manage results
BVPs
Stable path
Store paths
Plotting
Phase space
Time path
2.83
Usage of the store command after continuation
Remark
During continuation the results are stored in the mat-file
ContinuationData.mat
in the data directory of the model. Using the store command thisfile is loaded and the data is added to the ocResults.
Example
>> m=ocmodel ( ’mom’ )>> m. o cRe s u l t sans =1x1 s t r u c t a r r a y w i th no f i e l d s .>> ocEP=ca l c e p (m) ; b=i s a dm i s s i b l e (m, ocEP) ; ocEP(˜ b ) = [ ] ;>> i n i t S t r u c t=i n i t o c c o n t ( ’ e x t r ema l ’ ,m, ’ i n i t p o i n t ’ , 1 , 5 , ocEP{1}, ’ I n t e g r a t i o nT ime ’ , 500)>> s o l=occon t (m, i n i t S t r u c t , opt ) ;...
>> m=s t o r e (m) ;>> m. o cRe s u l t sans =
Ex t r ema l S o l u t i o n : {[1 x1 oca s ymp to t i c ]}
generates or adds results to m.ocResults.ExtremalSolution.
Dieter Graß
IVPs
Solve ODEs
Stable path
Manage results
BVPs
Stable path
Store paths
Plotting
Phase space
Time path
2.84
Plotting in the phase space
f u n c t i o n v a r a r g ou t = p l o t p h a s e ( v a r a r g i n )%% PLOTPHASE p l o t s the phase p o r t r a i t f o r a s o l u t i o n o f an ocmodel .%% PLOTPHASE(OCOBJ,OCELEMENT) p l o t s the f i r s t two c o o r d i n a t e s o f the OCELEMENT, where% OCELEMENT can be e i t h e r a d y n p r im i t i v e , occurve , o c t r a j e c t o r y o r o ca s ymp to t i c .%% PLOTPHASE(OCOBJ,OCTRJ,VARTYPE) wi th VARTYPE one can s p e c i f y the type% o f v a r i a b l e s to be p l o t t ed , p o s s i b l e v a l u e s a r e :% ’ s t a t e ’ . . . s t a t e v a l u e% ’ co s t a t e ’ . . . c o s t a t e v a l u e% ’ c on t r o l ’ . . . c o n t r o l v a l u e% ’ ham i l t on i an ’ . . . v a l u e o f the Hami l ton ian% ’ l m u l t i p l i e r ’ . . . v a l u e o f the Lagrang ian m u l t i p l i e r% ’ u s e r f un c ’ . . . v a l u e o f u s e r d e f i n e d f u n c t i o n s%% PLOTPHASE(OCOBJ,OCTRJ,VARTYPE,XCOORD) XCOORD s p e c i f i e s the s t a t e c o o r d i n a t e% XCOORD. I f i t i s empty the f i r s t s t a t e c o o r d i n a t e i s p l o t t e d .%% PLOTPHASE(OCOBJ,OCTRJ,VARTYPE,XCOORD,YCOORD) YCOORD s p e c i f i e s the c o o r d i n a t e% o f term g i v en by VARTYPE. I f i t i s empty the f i r s t c o o r d i n a t e o f VARTYPE i s p l o t t e d .%% PLOTPHASE(OCOBJ,OCTRJ,VARTYPE,XCOORD,YCOORD,PROPERTY,PROPERTYVAL) f o r p o s s i b l e% p r o p e r t i e s s e e MATLAB r e f e r e n c e o f p l o t .%% Add i t o n a l l y the f o l l o w i n g p r o p e r t i e s ( depend ing on the type o f OCELEMENT) can be s e t :% o c t r a j e c o t r y :% ’ con t inuous ’ . . . i f t h e o c t r a j e c t o r y c o n s i s t s o f d i f f e r e n t a r c s t h e s e can be% p l o t t e d as a cu r v e ’ c o n t i n u o s l y ’ connected at the sw i t c h i n g p o i n t s ’ on ’ . Or% each a r c i s p l o t t e d s e p a r a t e l y ( u s i n g a l s o d i f f e r e n t c o l o r s ) ’ o f f ’ .% occu r v e :% ’ a s s o c i a t e d s o l ’ . . . i f , e . g . , the occu r v e i s a boundary cu r v e c a l c a u l t e d by a% co n t i n u a t i o n p r o c e s s w i th ’ on ’ the a s s o c i a t e d ( ’ l a s t ’ ) s o l u t i o n o f t h i s% c o n t i n u a t i o n p r o c e s s i s p l o t t ed , d e f a u l t i s i s s e t to ’ o f f ’ .% oca s ymp to t i c :% ’ l i m i t s e t ’ . . . i f s e t to ’ on ’ the c o r r e s p on d i n g l i m i t s e t i s a l s o p l o t t ed , by% d e f a u l t t h i s i s s e t ’ o f f ’ .%% P r o p e r t i e s o f l i m i t s e t ( a s s i c a t e d s o l ) can be s e t u s ing , e . g . , ’ l im i t s e tMa r k e r ’ , ’ o ’
Dieter Graß
IVPs
Solve ODEs
Stable path
Manage results
BVPs
Stable path
Store paths
Plotting
Phase space
Time path
2.85
Plotting ocResults in the phase space
f u n c t i o n v a r a r g ou t = p l o t p h a s e o c r e s u l t ( v a r a r g i n )%% PLOTPHASEOCRESULT p l o t s the o cRe s u l t s o f an oc ob j e c t .%% PLOTPHASEOCRESULT(OCOBJ,VARTYPE,XCOORD,YCOORD) f o r VARTYPE,XCOORD and YCOORD% see the PLOTPHASE command .%% PLOTPHASEOCRESULT(AXH,OCOBJ,VARTYPE,XCOORD,YCOORD) wi th AXH you can s p e c i f y a f i g u r e /
a x i s% hand l e where the r e s u l t s s hou l d be p l o t t e d .%% PLOTPHASEOCRESULT(H,OCOBJ,VARTYPE,XCOORD,YCOORD, ’ONLY’ ,CELLNAME) wi th CELLNAME you
can% p r o v i d e a c e l l a r r a y o f s t r i n g s c o n s i t i n g o f f i l e d names o f the o cRe s u l t s . Then on l y% th e s e r e s u l t s a r e p l o t t e d . Even f o r one f i e l d you have to s p e c i f y i t as a c e l l .%% PLOTPHASEOCRESULT( . . . , ’ ONLY’ ,CELLNAME, CELLIDX) CELLIDX i s a% c e l l a r r a y o f the same s i z e as CELLNAME and p r o v i e d the i nd ex o f s p e c i f i c e l ement s o f% the CELLNAME f i e l d s , e . g .% p l o t p h a s e o c r e s u l t (m, ’ s t a t e ’ , 1 , 2 , ’ on ly ’ ,{ ’ E x t r ema lSo l u t i on ’} , ’ on l y i n d ex ’ ,{ [ 2 3]})% on l y p l o t s the 2 and 3 ob j e c t s t o r e d i n o cRe s u l t s . E x t r ema l S o l u t i o n . With { [0]} no% e lement s i s p l o t t e d and wi th { [ ]} a l l e l emnt s a r e p l o t t e d .%% PLOTPHASEOCRESULT( . . . , ’ ONLY’ ,PROPERTY,PROPERTYVAL) wi th PROPERTY and PROPERTYVAL% p r o p e r t i e s can be p rov ided , s e e PLOTPHASE.%% H=PLOTPHASEOCRESULT ( . . . ) r e t u r n s the l i n e hand l e s
Dieter Graß
IVPs
Solve ODEs
Stable path
Manage results
BVPs
Stable path
Store paths
Plotting
Phase space
Time path
2.86
plottimepath
f u n c t i o n v a r a r g ou t = p l o t t imep a t h ( v a r a r g i n )%% PLOTTIMEPATH p l o t s the t ime path f o r the o c t r a j e c t o r y o f an ocmodel .%% PLOTTIMEPATH(OCOBJ,OCTRJ,VARTYPE) p l o t s the t ime path s o f the OCTRJ v a r i a b l e s ,% where each c oo r d i n a t e i s d i s p l a y e d as a s e p e r a t e l i n e . w i th VARTYPE one can s p e c i f y
the type% o f v a r i a b l e s to be p l o t t ed , p o s s i b l e v a l u e s a r e :% ’ s t a t e ’ . . . s t a t e v a l u e% ’ co s t a t e ’ . . . c o s t a t e v a l u e% ’ c on t r o l ’ . . . c o n t r o l v a l u e% ’ ham i l t on i an ’ . . . v a l u e o f the Hami l ton ian% ’ l m u l t i p l i e r ’ . . . v a l u e o f the Lagrang ian m u l t i p l i e r% ’ u s e r f un c ’ . . . v a l u e o f u s e r d e f i n e d f u n c t i o n s%% PLOTTIMEPATH(OCOBJ,OCTRJ,VARTYPE,COORD) s p e c i f y the c o o r d i n a t e s COORD to% be p l o t t e d . I f i t i s empty a l l c o o r d i n a t e s a r e d i s p l a y e d .%% PLOTTIMEPATH(OCOBJ,OCTRJ,VARTYPE,COORD,PROPERTY,PROPERTYVAL) he r e% PROPERTY can e i t h e r be a u s u a l p r o p e r t y o f the MATLAB p l o t command or% one o f the p r o p e r t i e s ot the ’ p l o t pha s e ’ command .%% PLOTTIMEPATH(AXH , . . . ) the r e s u l t i s p l o t t e d to the axes w i th hand l e AXH.%
