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Self-Organisation in Spatial Systems-From Fractal Chaos to Regular Patterns and Vice Versa.

Banaszak M, Dziecielski M, Nijkamp P, Ratajczak W - PLoS ONE (2015)

Bottom Line: This study offers a new perspective on the evolutionary patterns of cities or urban agglomerations.Such developments can range from chaotic to fully ordered.Our approach is dynamic in nature and forms a generalisation of hierarchical principles in geographic space.

View Article: PubMed Central - PubMed

Affiliation: Faculty of Physics, A. Mickiewicz University, ul. Umultowska 85, 61-614 Poznan, Poland.

ABSTRACT
This study offers a new perspective on the evolutionary patterns of cities or urban agglomerations. Such developments can range from chaotic to fully ordered. We demonstrate that in a dynamic space of interactive human behaviour cities produce a wealth of gravitational attractors whose size and shape depend on the resistance of space emerging inter alia from transport friction costs. This finding offers original insights into the complex evolution of spatial systems and appears to be consistent with the principles of central place theory known from the spatial sciences and geography. Our approach is dynamic in nature and forms a generalisation of hierarchical principles in geographic space.

No MeSH data available.


Related in: MedlinePlus

Dissipation of energy as a function of distance travelled, s, by an agent starting at point (1,1).Fit: 3.2 exp(−0.24s2). Data for the red line were obtained from the numerical solution of the equation of motion Eq 10.
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pone.0136248.g005: Dissipation of energy as a function of distance travelled, s, by an agent starting at point (1,1).Fit: 3.2 exp(−0.24s2). Data for the red line were obtained from the numerical solution of the equation of motion Eq 10.

Mentions: In Fig 4 we show the potential for a symmetric tri-city system. It is evident that every trajectory has to end in one of the metropolitan centres. We will next confirm this prediction by performing extensive numerical simulations for a wide variety of parameters characterizing the metropolitan system concerned. The structure of Model (10) allows investigating the effect of the resistance of space on the organisational level of a spatial system by changing the value of the parameter μ. This can be interpreted as the unit transport cost prevailing in this system. The system of attractors presented in Fig 1a stabilises when the value of μ changes from 0.09 to 0.7. This is corroborated by Fig 1c. The pattern it depicts accords with the effect foreseen by CPT. In other words, it is possible for a spatial system to pass from chaotic impacts to fully ordered ones, and vice versa. Fig 1c inspires us to ask an important cognitive question: what is the form of the model describing the dependence between energy loss by the agent and the distance it has travelled? Apparently, complex spatial systems can move to ordered systems under conditions of low distance costs. This intriguing question will be addressed shortly. Fig 5 presents total energy as a function of the distance (red line) of the agent. We can see that, with increasing time, the agent’s total energy decreases. This is a result of the action of friction forces (μ = 0.7). This behaviour differs significantly from the situation presented in Fig 3, which ignores friction (μ = 0).


Self-Organisation in Spatial Systems-From Fractal Chaos to Regular Patterns and Vice Versa.

Banaszak M, Dziecielski M, Nijkamp P, Ratajczak W - PLoS ONE (2015)

Dissipation of energy as a function of distance travelled, s, by an agent starting at point (1,1).Fit: 3.2 exp(−0.24s2). Data for the red line were obtained from the numerical solution of the equation of motion Eq 10.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4581867&req=5

pone.0136248.g005: Dissipation of energy as a function of distance travelled, s, by an agent starting at point (1,1).Fit: 3.2 exp(−0.24s2). Data for the red line were obtained from the numerical solution of the equation of motion Eq 10.
Mentions: In Fig 4 we show the potential for a symmetric tri-city system. It is evident that every trajectory has to end in one of the metropolitan centres. We will next confirm this prediction by performing extensive numerical simulations for a wide variety of parameters characterizing the metropolitan system concerned. The structure of Model (10) allows investigating the effect of the resistance of space on the organisational level of a spatial system by changing the value of the parameter μ. This can be interpreted as the unit transport cost prevailing in this system. The system of attractors presented in Fig 1a stabilises when the value of μ changes from 0.09 to 0.7. This is corroborated by Fig 1c. The pattern it depicts accords with the effect foreseen by CPT. In other words, it is possible for a spatial system to pass from chaotic impacts to fully ordered ones, and vice versa. Fig 1c inspires us to ask an important cognitive question: what is the form of the model describing the dependence between energy loss by the agent and the distance it has travelled? Apparently, complex spatial systems can move to ordered systems under conditions of low distance costs. This intriguing question will be addressed shortly. Fig 5 presents total energy as a function of the distance (red line) of the agent. We can see that, with increasing time, the agent’s total energy decreases. This is a result of the action of friction forces (μ = 0.7). This behaviour differs significantly from the situation presented in Fig 3, which ignores friction (μ = 0).

Bottom Line: This study offers a new perspective on the evolutionary patterns of cities or urban agglomerations.Such developments can range from chaotic to fully ordered.Our approach is dynamic in nature and forms a generalisation of hierarchical principles in geographic space.

View Article: PubMed Central - PubMed

Affiliation: Faculty of Physics, A. Mickiewicz University, ul. Umultowska 85, 61-614 Poznan, Poland.

ABSTRACT
This study offers a new perspective on the evolutionary patterns of cities or urban agglomerations. Such developments can range from chaotic to fully ordered. We demonstrate that in a dynamic space of interactive human behaviour cities produce a wealth of gravitational attractors whose size and shape depend on the resistance of space emerging inter alia from transport friction costs. This finding offers original insights into the complex evolution of spatial systems and appears to be consistent with the principles of central place theory known from the spatial sciences and geography. Our approach is dynamic in nature and forms a generalisation of hierarchical principles in geographic space.

No MeSH data available.


Related in: MedlinePlus