Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.

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, 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)

Definitions

1

1 1 2 2 1 2x x x x, , ,b gb gc h

2

Flow Vector:

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:

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:

OperatorsGradient: (direction of steepest ascent)

Divergence: (source density)

FHG

IKJ

x x1 2

,

1

1

2

2x x

1

11

11

x

dx



FHG

IKJ

x x1 2

,

1

1

2

2x x

dx1

FHG

IKJ

x x1 2

,

1

1

2

2x x



1

11

11

x

dx

2

2

2

22

x

dx

dx2

dx1



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:


k

Gradient Law:


k

Gradient Law:

1) prices increase with transportation cost along the flow

2) commodities flow in the direction of the price gradient


k

Gradient Law:



zbg0Divergence Law:


3) excess demand/supply is withdrawn from/added to flow

k

zbg0

Gradient Law:



Divergence Law:


k 2

2

2

FHGIKJ b gTake square of gradient law:

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


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

Constructive solution for , disk radius 1/k


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


we have

Constructive solution for , disk radius 1/k Orthogonal trajectories


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


we have

Assume Radial flow or hyperbolic depends on boundary conditions. k x x2

12

22

EXAMPLES

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

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



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,



z

kFurther, dynamization,

equilibrium pattern globally asymptotically stable.


z

k


Further, dynamization,

The Beckmann model is compatible with any spatial pattern, so

How can we obatain more information?

equilibrium pattern globally asymptotically stable.

SINGULARITIES (Stagnation Points in Flow)

1) Linear Systems – just One Point

Stable Node



Stable Node Unstable Node




Stable Focus




Stable Focus Unstable Focus

No Foci inGradient Flow




Stable Focus

Saddle Point

Unstable Focus

NOTHING ELSE

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



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

Assume solved for Flow lines determined by

dx

dtx xx

11 21

( , )dx

dtx xx

21 22

( , )


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



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



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



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


Equivalent, structurally stable



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



Nonequivalent, unstable (singularity splits)

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


Nonequivalent, unstable (trajectory splits)

Such that


Nonequivalent, unstable (singularity splits)

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.


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.

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



Stable grid Corresponding flow


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


Euclidean Metric Manhattan Metric



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)

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


Twice as manysources as sinks

Twice as manysinks as sources


Local Change of Structure

Elliptic Umblic Catastrophe x x x a x x bx cx13

1 22

12

22

1 23 c h

Global Change of Structure

sin( ) sin( ) sin( )x y x y x3 3 2 Periodic Monkey Saddle, unfolding added

Beckmann, M.J., 1952, "A continuous model of transportation", Econometrica 20:642-660. Beckmann, M. J. 1953, "The partial equilibrium of a continuous space.

Documents

continuous regional

continuous space economy

continuous space market

structural stability

partial equilibrium

regional science

regional modelling

weltwirtschaftliches