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![Page 1: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/1.jpg)
Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660.Beckmann, M. J. 1953, "The partial equilibrium of a continuous space economy", Weltwirtschaftliches Archiv 71:73-89
![Page 2: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/2.jpg)
Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660.Beckmann, M. J. 1953, "The partial equilibrium of a continuous space economy", Weltwirtschaftliches Archiv 71:73-89
Beckmann, M. J., 1976, "Equilibrium and stability in a continuous space market", Operations Research Verfahren 14:48-63
![Page 3: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/3.jpg)
Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660.Beckmann, M. J. 1953, "The partial equilibrium of a continuous space economy", Weltwirtschaftliches Archiv 71:73-89
Beckmann, M. J., 1976, "Equilibrium and stability in a continuous space market", Operations Research Verfahren 14:48-63
T. Puu, 1979, "Regional modelling and structural stability", Environment and Planning A 11:1431-1438T. Puu, 1981, 2004, "Catastrophic structural change in a continuous regional model", Regional Science and Urban Economics 11:317-333
![Page 4: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/4.jpg)
Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660.Beckmann, M. J. 1953, "The partial equilibrium of a continuous space economy", Weltwirtschaftliches Archiv 71:73-89
Beckmann, M. J., 1976, "Equilibrium and stability in a continuous space market", Operations Research Verfahren 14:48-63
T. Puu, 1979, "Regional modelling and structural stability", Environment and Planning A 11:1431-1438T. Puu, 1981, 2004, "Catastrophic structural change in a continuous regional model", Regional Science and Urban Economics 11:317-333
M.J. Beckmann and T. Puu, 1985, Spatial Economics: Potential, Density, and Flow, (North-Holland Publishing Company, ISBN 0-444-87771-1)
![Page 5: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/5.jpg)
Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660.Beckmann, M. J. 1953, "The partial equilibrium of a continuous space economy", Weltwirtschaftliches Archiv 71:73-89
Beckmann, M. J., 1976, "Equilibrium and stability in a continuous space market", Operations Research Verfahren 14:48-63
T. Puu, 1979, "Regional modelling and structural stability", Environment and Planning A 11:1431-1438T. Puu, 1981, 2004, "Catastrophic structural change in a continuous regional model", Regional Science and Urban Economics 11:317-333
M.J. Beckmann and T. Puu, 1985, Spatial Economics: Potential, Density, and Flow, (North-Holland Publishing Company, ISBN 0-444-87771-1)
T. Puu, 2003, Mathematical Location and Land Use Theory; An Introduction, Second Revised and Enlarged Edition, (Springer-Verlag ISBN 3-540-00931-0)
![Page 6: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/6.jpg)
Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660.Beckmann, M. J. 1953, "The partial equilibrium of a continuous space economy", Weltwirtschaftliches Archiv 71:73-89
Beckmann, M. J., 1976, "Equilibrium and stability in a continuous space market", Operations Research Verfahren 14:48-63
T. Puu, 1979, "Regional modelling and structural stability", Environment and Planning A 11:1431-1438T. Puu, 1981, 2004, "Catastrophic structural change in a continuous regional model", Regional Science and Urban Economics 11:317-333
M.J. Beckmann and T. Puu, 1985, Spatial Economics: Potential, Density, and Flow, (North-Holland Publishing Company, ISBN 0-444-87771-1)
T. Puu, 2003, Mathematical Location and Land Use Theory; An Introduction, Second Revised and Enlarged Edition, (Springer-Verlag ISBN 3-540-00931-0)
T. Puu and M.J. Beckmann, 2003, "Continuous Space Modelling", in R. Hall (Ed.), Handbook of Transportation Science, Second Edition 279-320 (Kluwer Academic Publishers, ISBN 1-4020-7246-5)
![Page 7: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/7.jpg)
Definitions
1
1 1 2 2 1 2x x x x, , ,b gb gc h
2
Flow Vector:
![Page 8: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/8.jpg)
Definitions
1
1 1 2 2 1 2x x x x, , ,b gb gc h
2
Flow Vector:
12
22
FHG
IKJ
1
12
22
2
12
22
,
Flow Volume:
Unit Direction Vector:
![Page 9: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/9.jpg)
Definitions
1
1 1 2 2 1 2x x x x, , ,b gb gc h
2
Flow Vector:
12
22
FHG
IKJ
1
12
22
2
12
22
,
k x x1 2,b g x x1 2,b g
Flow Volume:
Unit Direction Vector:
Transportation Cost:
Commodity Price:
![Page 10: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/10.jpg)
OperatorsGradient: (direction of steepest ascent)
Divergence: (source density)
FHG
IKJ
x x1 2
,
1
1
2
2x x
![Page 11: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/11.jpg)
1
11
11
x
dx
OperatorsGradient: (direction of steepest ascent)
Divergence: (source density)
FHG
IKJ
x x1 2
,
1
1
2
2x x
dx1
![Page 12: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/12.jpg)
FHG
IKJ
x x1 2
,
1
1
2
2x x
OperatorsGradient: (direction of steepest ascent)
Divergence: (source density)
1
11
11
x
dx
2
2
2
22
x
dx
dx2
dx1
![Page 13: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/13.jpg)
OperatorsGradient: (direction of steepest ascent)
Divergence: (source density)
FHG
IKJ
x x1 2
,
1
1
2
2x x
1
11
11
x
dx
2
2
2
22
x
dx
dx2
dx2
zzz dx dx dsRR 1 2 nGauss’s Divergence Theorem:
![Page 14: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/14.jpg)
Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660.Beckmann, M. J. 1953, "The partial equilibrium of a continuous space economy", Weltwirtschaftliches Archiv 71:73-89
![Page 15: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/15.jpg)
k
Gradient Law:
Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660.Beckmann, M. J. 1953, "The partial equilibrium of a continuous space economy", Weltwirtschaftliches Archiv 71:73-89
![Page 16: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/16.jpg)
k
Gradient Law:
1) prices increase with transportation cost along the flow
2) commodities flow in the direction of the price gradient
Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660.Beckmann, M. J. 1953, "The partial equilibrium of a continuous space economy", Weltwirtschaftliches Archiv 71:73-89
![Page 17: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/17.jpg)
k
Gradient Law:
1) prices increase with transportation cost along the flow
2) commodities flow in the direction of the price gradient
zbg0Divergence Law:
Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660.Beckmann, M. J. 1953, "The partial equilibrium of a continuous space economy", Weltwirtschaftliches Archiv 71:73-89
![Page 18: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/18.jpg)
3) excess demand/supply is withdrawn from/added to flow
k
zbg0
Gradient Law:
1) prices increase with transportation cost along the flow
2) commodities flow in the direction of the price gradient
Divergence Law:
Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660.Beckmann, M. J. 1953, "The partial equilibrium of a continuous space economy", Weltwirtschaftliches Archiv 71:73-89
![Page 19: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/19.jpg)
k 2
2
2
FHGIKJ b gTake square of gradient law:
![Page 20: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/20.jpg)
k 2
2
2
FHGIKJ b gTake square of gradient law:
As and (unit vector squared)
FHGIKJ
2
1 FHGIKJ
FHGIKJ b g2
1
2
2
2
x x
![Page 21: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/21.jpg)
As and (unit vector squared)
FHGIKJ
2
1
k 2
2
2
FHGIKJ b g
FHGIKJ
FHGIKJ b g2
1
2
2
2
x x
FHGIKJ
FHGIKJ
x x
k x x1
2
2
2
1 2
2,b g
Take square of gradient law:
we have
![Page 22: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/22.jpg)
Constructive solution for , disk radius 1/k
As and (unit vector squared)
FHGIKJ
2
1
k 2
2
2
FHGIKJ b g
FHGIKJ
FHGIKJ b g2
1
2
2
2
x x
FHGIKJ
FHGIKJ
x x
k x x1
2
2
2
1 2
2,b g
Take square of gradient law:
we have
![Page 23: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/23.jpg)
Constructive solution for , disk radius 1/k Orthogonal trajectories
As and (unit vector squared)
FHGIKJ
2
1
k 2
2
2
FHGIKJ b g
FHGIKJ
FHGIKJ b g2
1
2
2
2
x x
FHGIKJ
FHGIKJ
x x
k x x1
2
2
2
1 2
2,b g
Take square of gradient law:
we have
![Page 24: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/24.jpg)
Assume Radial flow or hyperbolic depends on boundary conditions. k x x2
12
22
EXAMPLES
![Page 25: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/25.jpg)
Assume Radial flow or hyperbolic depends on boundary conditions. k x x2
12
22
FHG
IKJ
12 1
222
1 2
1 2 12
22
12
22 1
12
22
2
12
22 1 2
2
x x x x
x x x x
x xx
x x
x
x xx x
c h b gb g
b g
, ,
, , ,
, ,
EXAMPLES
![Page 26: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/26.jpg)
Assume Radial flow or hyperbolic depends on boundary conditions.
FHG
IKJ
12 1
222
1 2
1 2 12
22
12
22 1
12
22
2
12
22 1 2
2
x x x x
x x x x
x xx
x x
x
x xx x
c h b gb g
b g
, ,
, , ,
, ,
k x x212
22
FHG
IKJ
12 1
222
1 2
1 2 12
22
12
22 1
12
22
2
12
22 1 2
0
x x x x
x x x x
x xx
x x
x
x xx x
c h b gb g
b g
, ,
, , ,
, ,
EXAMPLES
![Page 27: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/27.jpg)
Beckmann, M. J., 1976, "Equilibrium and stability in a continuous space market", Operations Research Verfahren 14:48-63
![Page 28: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/28.jpg)
Beckmann, M. J., 1976, "Equilibrium and stability in a continuous space market", Operations Research Verfahren 14:48-63
Given excess demand decreasing function of price, given transportation cost, and boundary conditions (for instance outflow on boundary increasing function of of price), unique price/flow solution.
![Page 29: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/29.jpg)
Beckmann, M. J., 1976, "Equilibrium and stability in a continuous space market", Operations Research Verfahren 14:48-63
Given excess demand decreasing function of price, given transportation cost, and boundary conditions (for instance outflow on boundary increasing function of of price), unique price/flow solution.
z
kFurther, dynamization,
![Page 30: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/30.jpg)
Beckmann, M. J., 1976, "Equilibrium and stability in a continuous space market", Operations Research Verfahren 14:48-63
Given excess demand decreasing function of price, given transportation cost, and boundary conditions (for instance outflow on boundary increasing function of of price), unique price/flow solution.
z
kFurther, dynamization,
equilibrium pattern globally asymptotically stable.
![Page 31: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/31.jpg)
Beckmann, M. J., 1976, "Equilibrium and stability in a continuous space market", Operations Research Verfahren 14:48-63
z
k
Given excess demand decreasing function of price, given transportation cost, and boundary conditions (for instance outflow on boundary increasing function of of price), unique price/flow solution.
Further, dynamization,
The Beckmann model is compatible with any spatial pattern, so
How can we obatain more information?
equilibrium pattern globally asymptotically stable.
![Page 32: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/32.jpg)
SINGULARITIES (Stagnation Points in Flow)
1) Linear Systems – just One Point
Stable Node
![Page 33: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/33.jpg)
SINGULARITIES (Stagnation Points in Flow)
1) Linear Systems – just One Point
Stable Node Unstable Node
![Page 34: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/34.jpg)
SINGULARITIES (Stagnation Points in Flow)
1) Linear Systems – just One Point
Stable Node Unstable Node
Stable Focus
![Page 35: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/35.jpg)
SINGULARITIES (Stagnation Points in Flow)
1) Linear Systems – just One Point
Stable Node Unstable Node
Stable Focus Unstable Focus
No Foci inGradient Flow
![Page 36: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/36.jpg)
SINGULARITIES (Stagnation Points in Flow)
1) Linear Systems – just One Point
Stable Node Unstable Node
Stable Focus
Saddle Point
Unstable Focus
NOTHING ELSE
![Page 37: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/37.jpg)
2) Nonlinear Systems
Everything is Possible - Unless Structural Stability is Assumed
![Page 38: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/38.jpg)
2) Nonlinear Systems
Everything is Possible - Unless Structural Stability is Assumed
Topological Equivalence Defined:
A) Each singularity cna be mapped onto a singularity of the same kindB) Each trajectory can be mapped onto another orientation being preserved
![Page 39: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/39.jpg)
2) Nonlinear Systems
Everything is Possible - Unless Structural Stability is Assumed
Topological Equivalence Defined:
A) Each singularity cna be mapped onto a singularity of the same kindB) Each trajectory can be mapped onto another orientation being preserved
![Page 40: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/40.jpg)
Assume solved for Flow lines determined by
dx
dtx xx
11 21
( , )dx
dtx xx
21 22
( , )
![Page 41: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/41.jpg)
Assume solved for Flow lines determined by
dx
dtx xx
11 21
( , )dx
dtx xx
21 22
( , )
dx
dtf x x
dx
dtg x x
11 2
21 2
RS|T|
( , )
( , )
dx
dtF x x
dx
dtG x x
11 2
21 2
RS|T|
( , )
( , )
Structural stability for flows in the plane (M.M. Peixoto), consider two systems
![Page 42: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/42.jpg)
Structural stability for flows in the plane (M.M. Peixoto), consider two systems
Assume solved for Flow lines determined by
dx
dtx xx
11 21
( , )dx
dtx xx
21 22
( , )
dx
dtf x x
dx
dtg x x
11 2
21 2
RS|T|
( , )
( , )
dx
dtF x x
dx
dtG x x
11 2
21 2
RS|T|
( , )
( , )
G
x
g
x
G
x
g
x1 1 2 2
,
F f G g ,
F
x
f
x
F
x
f
x1 1 2 2
, ,and
Such that
![Page 43: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/43.jpg)
Structural stability for flows in the plane (M.M. Peixoto), consider two systems
Assume solved for Flow lines determined by
dx
dtx xx
11 21
( , )dx
dtx xx
21 22
( , )
dx
dtf x x
dx
dtg x x
11 2
21 2
RS|T|
( , )
( , )
dx
dtF x x
dx
dtG x x
11 2
21 2
RS|T|
( , )
( , )
G
x
g
x
G
x
g
x1 1 2 2
,
F f G g ,
F
x
f
x
F
x
f
x1 1 2 2
, ,and
Such that
Structurally stable if flows topologically equivalent after -perturbation
![Page 44: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/44.jpg)
Structural stability for flows in the plane (M.M. Peixoto), consider two systems
Assume solved for Flow lines determined by
dx
dtx xx
11 21
( , )dx
dtx xx
21 22
( , )
dx
dtf x x
dx
dtg x x
11 2
21 2
RS|T|
( , )
( , )
dx
dtF x x
dx
dtG x x
11 2
21 2
RS|T|
( , )
( , )
G
x
g
x
G
x
g
x1 1 2 2
,
F f G g ,
F
x
f
x
F
x
f
x1 1 2 2
, ,and
Such that
Structurally stable if flows topologically equivalent after -perturbation
Equivalent, structurally stable
![Page 45: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/45.jpg)
Structural stability for flows in the plane (M.M. Peixoto), consider two systems
Assume solved for Flow lines determined by
dx
dtx xx
11 21
( , )dx
dtx xx
21 22
( , )
dx
dtf x x
dx
dtg x x
11 2
21 2
RS|T|
( , )
( , )
dx
dtF x x
dx
dtG x x
11 2
21 2
RS|T|
( , )
( , )
G
x
g
x
G
x
g
x1 1 2 2
,
F f G g ,
F
x
f
x
F
x
f
x1 1 2 2
, ,and
Such that
Structurally stable if flows topologically equivalent after -perturbation
Equivalent, structurally stable
Nonequivalent, unstable (singularity splits)
![Page 46: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/46.jpg)
dx
dtx xx
11 21
( , )dx
dtx xx
21 22
( , )
G
x
g
x
G
x
g
x1 1 2 2
,
dx
dtf x x
dx
dtg x x
11 2
21 2
RS|T|
( , )
( , )
F f G g ,
F
x
f
x
F
x
f
x1 1 2 2
, ,
Assume solved for Flow lines determined by Structural stability for flows in the plane (M.M. Peixoto), consider two systems
dx
dtF x x
dx
dtG x x
11 2
21 2
RS|T|
( , )
( , )
and
Structurally stable if flows topologically equivalent after -perturbation
Nonequivalent, unstable (trajectory splits)
Such that
Equivalent, structurally stable
Nonequivalent, unstable (singularity splits)
![Page 47: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/47.jpg)
As we will see, only nodes (sinks and sources) and saddle points are possible stagnation points in a structurally stable flow, it can be topologically organized around such stagnation points.
![Page 48: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/48.jpg)
As we will see, only nodes (sinks and sources) and saddle points are possible stagnation points in a structurally stable flow, it can be topologically organized around such stagnation points.
To nodes infinitely many flow lines are incindent – to saddles only two pairs, ingoing and outgoing. These can be taken as organizing elements for the ”skeleton” of a flow, along with the stagnation points.
![Page 49: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/49.jpg)
As we will see, only nodes (sinks and sources) and saddle points are possible stagnation points in a structurally stable flow, it can be topologically organized around such stagnation points.
To nodes infinitely many flow lines are incindent – to saddles only two pairs, ingoing and outgoing. These can be taken as organizing elements for the ”skeleton” of a flow, along with the stagnation points.
Everywhere else the flow is topologically equivalent to a set of parallel staright lines.
![Page 50: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/50.jpg)
Structurally stable flow in 2-D:finite number of same type of singularities as in linear systems, i.e.1) Stable node or sink2) Unstable node or source3) Saddle points
![Page 51: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/51.jpg)
Structurally stable flow in 2-D:finite number of same type of singularities as in linear systems, i.e.1) Stable node or sink2) Unstable node or source3) Saddle points
No foci or centres in gradient flow
![Page 52: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/52.jpg)
Structurally stable flow in 2-D:finite number of same type of singularities as in linear systems, i.e.1) Stable node or sink2) Unstable node or source3) Saddle points
No foci or centres in gradient flow
Global result:4) No heteroclinic/homoclinic saddle connections
![Page 53: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/53.jpg)
Structurally stable flow in 2-D:finite number of same type of singularities as in linear systems, i.e.1) Stable node or sink2) Unstable node or source3) Saddle points
No foci or centres in gradient flow
Global result:4) No heteroclinic/homoclinic saddle connections
Stable grid
![Page 54: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/54.jpg)
Structurally stable flow in 2-D:finite number of same type of singularities as in linear systems, i.e.1) Stable node or sink2) Unstable node or source3) Saddle points
Global result:4) No heteroclinic/homoclinic saddle connections
Stable grid Corresponding flow
No foci or centres in gradient flow
![Page 55: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/55.jpg)
Results of Structural Stability depend on what it is applied to, above it was to flow:Then pure hexagonal patterns are ruled out.
![Page 56: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/56.jpg)
Results of Structural Stability depend on what it is applied to, above it was to flow:Then pure hexagonal patterns are ruled out.
However, it is also possible to apply to for instance market areas, then three areascome together in each corner (Transverse Intersection), never four or more.
![Page 57: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/57.jpg)
Results of Structural Stability depend on what it is applied to, above it was to flow:Then pure hexagonal patterns are ruled out.
However, it is also possible to apply to for instance market areas, then three areascome together in each corner (Transverse Intersection), never four or more.
Launhardt-Lösch structures have been motivated by optimality, but only 1.7%saving of transportation cost compared to square structures. Stability is a betterargument, in tessellations three 6-gons, four 4-gons, and six 3-gons meet. - theLast two nontransverse.
![Page 58: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/58.jpg)
Results of Structural Stability depend on what it is applied to, above it was to flow:Then pure hexagonal patterns are ruled out.
However, it is also possible to apply to for instance market areas, then three areascome together in each corner (Transverse Intersection), never four or more.
Launhardt-Lösch structures have been motivated by optimality, but only 1.7%saving of transportation cost compared to square structures. Stability is a betterargument, in tessellations three 6-gons, four 4-gons, and six 3-gons meet. - theLast two nontransverse.
Euclidean Metric
![Page 59: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/59.jpg)
Launhardt-Lösch structures have been motivated by optimality, but only 1.7%saving of transportation cost compared to square structures. Stability is a betterargument, in tessellations three 6-gons, four 4-gons, and six 3-gons meet. - theLast two nontransverse.
Euclidean Metric Manhattan Metric
Results of Structural Stability depend on what it is applied to, above it was to flow:Then pure hexagonal patterns are ruled out.
However, it is also possible to apply to for instance market areas, then three areascome together in each corner (Transverse Intersection), never four or more.
![Page 60: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/60.jpg)
Deleting pairs of sinks and saddles in a square grid produces new structurally stable grid,which can be deformed into a hexagon
![Page 61: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/61.jpg)
Deleting pairs of sinks and saddles in a square grid produces new structurally stable grid,which can be deformed into a hexagon
![Page 62: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/62.jpg)
Deleting pairs of sinks and saddles in a square grid produces new structurally stable grid,which can be deformed into a hexagon
All tessellations can be triangulated. In basic triangle – all corners connected1) Not two sinks, nor two sources (otherwise impossible to orient flow)2) Not two saddles (heteroclinic connection forbidden in stable flow)
![Page 63: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/63.jpg)
Three ways of organizing triangles (one source, one sink, one saddle)cyclically to produce tessellation elements
![Page 64: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/64.jpg)
Three ways of organizing triangles (one source, one sink, one saddle)cyclically to produce tessellation elements
Equal numbers ofsources and sinks
![Page 65: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/65.jpg)
Three ways of organizing triangles (one source, one sink, one saddle)cyclically to produce tessellation elements
Equal numbers ofsources and sinks
Twice as manysources as sinks
![Page 66: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/66.jpg)
Three ways of organizing triangles (one source, one sink, one saddle)cyclically to produce tessellation elements
Twice as manysources as sinks
Twice as manysinks as sources
Equal numbers ofsources and sinks
![Page 67: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/67.jpg)
Local Change of Structure
Elliptic Umblic Catastrophe x x x a x x bx cx13
1 22
12
22
1 23 c h
![Page 68: Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.](https://reader034.fdocuments.in/reader034/viewer/2022052213/56649f225503460f94c3a43b/html5/thumbnails/68.jpg)
Global Change of Structure
sin( ) sin( ) sin( )x y x y x3 3 2 Periodic Monkey Saddle, unfolding added