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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., 1952, "A continuous model of transportation", Econometrica 20:642-660.

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slide1

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

slide2

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

slide3

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 11:1431-1438

T. Puu, 1981, 2004, "Catastrophic structural change in a continuous regional model", Regional Science and Urban Economics 11:317-333

slide4

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 11:1431-1438

T. 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)

slide5

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 11:1431-1438

T. 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)

slide6

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 11:1431-1438

T. 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)

slide7

Definitions

Flow Vector:

slide8

Definitions

Flow Vector:

Flow Volume:

Unit Direction Vector:

definitions
Definitions

Flow Vector:

Flow Volume:

Unit Direction Vector:

Transportation Cost:

Commodity Price:

slide10

Operators

Gradient: (direction of steepest ascent)

Divergence: (source density)

slide11

Operators

Gradient: (direction of steepest ascent)

Divergence: (source density)

slide12

Operators

Gradient: (direction of steepest ascent)

Divergence: (source density)

operators
Operators

Gradient: (direction of steepest ascent)

Divergence: (source density)

Gauss’s Divergence Theorem:

slide14

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

slide15

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

Gradient Law:

slide16

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

Gradient Law:

  • prices increase with transportation cost along the flow
  • commodities flow in the direction of the price gradient
slide17

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

Gradient Law:

  • prices increase with transportation cost along the flow
  • commodities flow in the direction of the price gradient

Divergence Law:

3 excess demand supply is withdrawn from added to flow

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

Gradient Law:

  • prices increase with transportation cost along the flow
  • commodities flow in the direction of the price gradient

3) excess demand/supply is withdrawn from/added to flow

Divergence Law:

slide20

Take square of gradient law:

As and (unit vector squared)

slide21

Take square of gradient law:

As and (unit vector squared)

we have

slide22

Take square of gradient law:

As and (unit vector squared)

we have

Constructive solution for l, disk radius 1/k

slide23

Take square of gradient law:

As and (unit vector squared)

we have

Constructive solution for l, disk radius 1/k

Orthogonal trajectories

slide24

EXAMPLES

Assume Radial flow or hyperbolic depends on boundary conditions.

slide25

EXAMPLES

Assume Radial flow or hyperbolic depends on boundary conditions.

assume radial flow or hyperbolic depends on boundary conditions

EXAMPLES

Assume Radial flow or hyperbolic depends on boundary conditions.
slide27

Beckmann, M. J., 1976, "Equilibrium and stability in a continuous space market", Operations Research Verfahren 14:48-63

slide28

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.

slide29

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.

Further, dynamization,

slide30

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.

Further, dynamization,

equilibrium pattern globally asymptotically stable.

slide31

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.

Further, dynamization,

equilibrium pattern globally asymptotically stable.

The Beckmann model is compatible with any spatial pattern, so

How can we obatain more information?

slide32

SINGULARITIES (Stagnation Points in Flow)

  • Linear Systems – just One Point

Stable Node

slide33

SINGULARITIES (Stagnation Points in Flow)

  • Linear Systems – just One Point

Stable Node

Unstable Node

slide34

SINGULARITIES (Stagnation Points in Flow)

  • Linear Systems – just One Point

Stable Node

Unstable Node

Stable Focus

slide35

SINGULARITIES (Stagnation Points in Flow)

  • Linear Systems – just One Point

Stable Node

Unstable Node

Stable Focus

Unstable Focus

No Foci in

Gradient Flow

slide36

SINGULARITIES (Stagnation Points in Flow)

  • Linear Systems – just One Point

Stable Node

Unstable Node

Stable Focus

Unstable Focus

Saddle Point

NOTHING ELSE

slide37

2) Nonlinear Systems

Everything is Possible - Unless Structural Stability is Assumed

slide38

2) Nonlinear Systems

Everything is Possible - Unless Structural Stability is Assumed

  • Topological Equivalence Defined:
  • Each singularity cna be mapped onto a singularity of the same kind
  • Each trajectory can be mapped onto another orientation being preserved
slide39

2) Nonlinear Systems

Everything is Possible - Unless Structural Stability is Assumed

  • Topological Equivalence Defined:
  • Each singularity cna be mapped onto a singularity of the same kind
  • Each trajectory can be mapped onto another orientation being preserved
slide41

Assume solved for l. Flow lines determined by

Structural stability for flows in the plane (M.M. Peixoto), consider two systems

slide42

Assume solved for l. Flow lines determined by

Structural stability for flows in the plane (M.M. Peixoto), consider two systems

Such that

and

slide43

Assume solved for l. Flow lines determined by

Structural stability for flows in the plane (M.M. Peixoto), consider two systems

Such that

and

Structurally stable if flows topologically equivalent after e-perturbation

slide44

Assume solved for l. Flow lines determined by

Structural stability for flows in the plane (M.M. Peixoto), consider two systems

Such that

and

Structurally stable if flows topologically equivalent after e-perturbation

Equivalent, structurally stable

slide45

Assume solved for l. Flow lines determined by

Structural stability for flows in the plane (M.M. Peixoto), consider two systems

Such that

and

Structurally stable if flows topologically equivalent after e-perturbation

Nonequivalent, unstable

(singularity splits)

Equivalent, structurally stable

slide46

Assume solved for l. Flow lines determined by

Structural stability for flows in the plane (M.M. Peixoto), consider two systems

Such that

and

Structurally stable if flows topologically equivalent after e-perturbation

Nonequivalent, unstable

(singularity splits)

Nonequivalent, unstable (trajectory splits)

Equivalent, structurally stable

slide47

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.

slide48

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.

slide49

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.

slide50

Structurally stable flow in 2-D:

  • finite number of same type of singularities as in linear systems, i.e.
  • Stable node or sink
  • Unstable node or source
  • Saddle points
slide51

Structurally stable flow in 2-D:

  • finite number of same type of singularities as in linear systems, i.e.
  • Stable node or sink
  • Unstable node or source
  • Saddle points

No foci or centres in gradient flow

slide52

Structurally stable flow in 2-D:

  • finite number of same type of singularities as in linear systems, i.e.
  • Stable node or sink
  • Unstable node or source
  • Saddle points

No foci or centres in gradient flow

Global result:

4) No heteroclinic/homoclinic saddle connections

slide53

Structurally stable flow in 2-D:

  • finite number of same type of singularities as in linear systems, i.e.
  • Stable node or sink
  • Unstable node or source
  • Saddle points

No foci or centres in gradient flow

Global result:

4) No heteroclinic/homoclinic saddle connections

Stable grid

slide54

Structurally stable flow in 2-D:

  • finite number of same type of singularities as in linear systems, i.e.
  • Stable node or sink
  • Unstable node or source
  • Saddle points

No foci or centres in gradient flow

Global result:

4) No heteroclinic/homoclinic saddle connections

Stable grid

Corresponding flow

slide55

Results of Structural Stability depend on what it is applied to, above it was to flow:

Then pure hexagonal patterns are ruled out.

slide56

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 areas

come together in each corner (Transverse Intersection), never four or more.

slide57

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 areas

come 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 better

argument, in tessellations three 6-gons, four 4-gons, and six 3-gons meet. - the

Last two nontransverse.

slide58

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 areas

come 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 better

argument, in tessellations three 6-gons, four 4-gons, and six 3-gons meet. - the

Last two nontransverse.

Euclidean Metric

slide59

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 areas

come 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 better

argument, in tessellations three 6-gons, four 4-gons, and six 3-gons meet. - the

Last two nontransverse.

Euclidean Metric

Manhattan Metric

slide60

Deleting pairs of sinks and saddles in a square grid produces new structurally stable grid,

which can be deformed into a hexagon

slide61

Deleting pairs of sinks and saddles in a square grid produces new structurally stable grid,

which can be deformed into a hexagon

slide62

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 connected
  • Not two sinks, nor two sources (otherwise impossible to orient flow)
  • Not two saddles (heteroclinic connection forbidden in stable flow)
slide63

Three ways of organizing triangles (one source, one sink, one saddle)

cyclically to produce tessellation elements

slide64

Three ways of organizing triangles (one source, one sink, one saddle)

cyclically to produce tessellation elements

Equal numbers of

sources and sinks

slide65

Three ways of organizing triangles (one source, one sink, one saddle)

cyclically to produce tessellation elements

Twice as many

sources as sinks

Equal numbers of

sources and sinks

slide66

Three ways of organizing triangles (one source, one sink, one saddle)

cyclically to produce tessellation elements

Twice as many

sources as sinks

Equal numbers of

sources and sinks

Twice as many

sinks as sources

local change of structure
Local Change of Structure

Elliptic Umblic Catastrophe

global change of structure
Global Change of Structure

Periodic Monkey Saddle, unfolding added