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

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 economy", Weltwirtschaftliches Archiv 71:73-89

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 economy", Weltwirtschaftliches Archiv 71:73-89 Beckmann, M. J., 1976, "Equilibrium and stability in a continuous space market", Operations Research Verfahren 14:48-63

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 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

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 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)

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 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)

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 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)

7. Definitions Flow Vector:

8. Definitions Flow Vector: Flow Volume: Unit Direction Vector:

9. Definitions Flow Vector: Flow Volume: Unit Direction Vector: Transportation Cost: Commodity Price:

10. Operators Gradient: (direction of steepest ascent) Divergence: (source density)

11. Operators Gradient: (direction of steepest ascent) Divergence: (source density)

12. Operators Gradient: (direction of steepest ascent) Divergence: (source density)

13. Operators Gradient: (direction of steepest ascent) Divergence: (source density) Gauss’s Divergence Theorem:

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 economy", Weltwirtschaftliches Archiv 71:73-89

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 economy", Weltwirtschaftliches Archiv 71:73-89 Gradient Law:

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 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

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 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:

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 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:

19. Take square of gradient law:

20. Take square of gradient law: As and (unit vector squared)

21. Take square of gradient law: As and (unit vector squared) we have

22. Take square of gradient law: As and (unit vector squared) we have Constructive solution for l, disk radius 1/k

23. Take square of gradient law: As and (unit vector squared) we have Constructive solution for l, disk radius 1/k Orthogonal trajectories

24. EXAMPLES Assume Radial flow or hyperbolic depends on boundary conditions.

25. EXAMPLES Assume Radial flow or hyperbolic depends on boundary conditions.

26. EXAMPLES Assume Radial flow or hyperbolic depends on boundary conditions.

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

28. 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.

29. 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,

30. 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.

31. 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?

32. SINGULARITIES (Stagnation Points in Flow) • Linear Systems – just One Point Stable Node

33. SINGULARITIES (Stagnation Points in Flow) • Linear Systems – just One Point Stable Node Unstable Node

34. SINGULARITIES (Stagnation Points in Flow) • Linear Systems – just One Point Stable Node Unstable Node Stable Focus

35. SINGULARITIES (Stagnation Points in Flow) • Linear Systems – just One Point Stable Node Unstable Node Stable Focus Unstable Focus No Foci in Gradient Flow

36. SINGULARITIES (Stagnation Points in Flow) • Linear Systems – just One Point Stable Node Unstable Node Stable Focus Unstable Focus Saddle Point NOTHING ELSE

37. 2) Nonlinear Systems Everything is Possible - Unless Structural Stability is Assumed

38. 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

39. 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

40. Assume solved for l. Flow lines determined by

41. Assume solved for l. Flow lines determined by Structural stability for flows in the plane (M.M. Peixoto), consider two systems

42. Assume solved for l. Flow lines determined by Structural stability for flows in the plane (M.M. Peixoto), consider two systems Such that and

43. 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

44. 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

45. 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

46. 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

47. 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.

48. 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.

49. 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.

50. 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