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Beckmann, M. J. 1953, "The partial equilibrium of a continuous space economy", Weltwirtschaftliches Archiv 71:73-89

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

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

Beckmann, M. J. 1953, "The partial equilibrium of a continuous space economy", Weltwirtschaftliches Archiv 71:73-89

Beckmann, M. J. 1953, "The partial equilibrium of a continuous space economy", Weltwirtschaftliches Archiv 71:73-89

Beckmann, M. J. 1953, "The partial equilibrium of a continuous space economy", Weltwirtschaftliches Archiv 71:73-89

Beckmann, M. J. 1953, "The partial equilibrium of a continuous space economy", Weltwirtschaftliches Archiv 71:73-89

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

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

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

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

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.

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.

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

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

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

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)

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

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)

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

T. Puu, 1979, "Regional modelling and structural stability", Environment and Planning A 11:1431-1438

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)

Flow Vector:

Operators

Gradient: (direction of steepest ascent)

Divergence: (source density)

Gauss’s Divergence Theorem:

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

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

Gradient Law:

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

Gradient Law:

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

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

Gradient Law:

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

Divergence Law:

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

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:

As and (unit vector squared)

As and (unit vector squared)

we have

Constructive solution for l, disk radius 1/k

As and (unit vector squared)

we have

Constructive solution for l, disk radius 1/k

Orthogonal trajectories

Assume Radial flow or hyperbolic depends on boundary conditions.

Assume Radial flow or hyperbolic depends on boundary conditions.

Assume Radial flow or hyperbolic depends on boundary conditions.

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.

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,

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.

Further, dynamization,

equilibrium pattern globally asymptotically stable.

The Beckmann model is compatible with any spatial pattern, so

How can we obatain more information?

SINGULARITIES (Stagnation Points in Flow)

- Linear Systems – just One Point

Stable Node

Unstable Node

Stable Focus

SINGULARITIES (Stagnation Points in Flow)

- Linear Systems – just One Point

Stable Node

Unstable Node

Stable Focus

Unstable Focus

No Foci in

Gradient Flow

SINGULARITIES (Stagnation Points in Flow)

- Linear Systems – just One Point

Stable Node

Unstable Node

Stable Focus

Unstable Focus

Saddle Point

NOTHING ELSE

Everything is Possible - Unless Structural Stability is Assumed

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

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

Assume solved for l. Flow lines determined by

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

Assume solved for l. Flow lines determined by

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

Such that

and

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

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

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

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

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.

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.

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.

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

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

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

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

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

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

Then pure hexagonal patterns are ruled out.

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.

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.

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

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

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

which can be deformed into a hexagon

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

which can be deformed into a hexagon

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)

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

cyclically to produce tessellation elements

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

cyclically to produce tessellation elements

Equal numbers of

sources and sinks

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

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

Elliptic Umblic Catastrophe

Global Change of Structure

Periodic Monkey Saddle, unfolding added

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