Nonlinear dynamical social and political prediction algorithm for city planning and public participation using the Impulse Pattern Formulation

Abstract

A nonlinear-dynamical algorithm for city planning is proposed as an Impulse Pattern Formulation (IPF) for predicting relevant parameters like health, artistic freedom, or financial developments of different social or political stakeholders over the cause of a planning process. The IPF has already shown high predictive precision at low computational cost in musical instrument simulations, brain dynamics, and human-human interactions. The social and political IPF consists of three basic equations of system state developments, self-adaptation of stakeholders, two adaptive interactions, and external impact terms suitable for respective planning situations. Typical scenarios of stakeholder interactions and developments are modeled by adjusting a set of system parameters. These include stakeholder reaction to external input, enhanced system stability through self-adaptation, stakeholder convergence due to adaptive interaction, as well as complex dynamics in terms of fixed stakeholder impacts. A workflow for implementing the algorithm in real city planning scenarios is outlined. This workflow includes machine learning of a suitable set of parameters suggesting best-practice planning to aim at the desired development of the planning process and its output.

Figures (18)

• Figure 1: Time series of $g^s$ with $N=2$ interacting stakeholders and constant $\alpha=0.369$ when suddenly changing $\beta^{s,i}$ at iteration step 51. $\beta^{s,i}$ always starts at $0.2$ and is suddenly increased to 0.25 (pink) converging to a new stationary limit or decreased to 0.1 or 0 leading to a low-order bifurcation or a chaotic behaviour respectively.
• Figure 2: Time series of $g^s$ with $N=2$ interacting stakeholders and constant $\alpha=0.369$ when gradually changing $\beta^{s,i}$ between iteration step 40 to 60. Start and end values of $\beta^{s,i}$ are the same as in Fig. \ref{['fig:betaSudden']} and the same system states are reached in the end. Still, during transition $g^{s,i}$ is stationary even if the fixed point is not.
• Figure 3: System variable $g^1$ of stakeholder one for $\alpha = 0.56$ and different impact $I_{20} = \{0, 1, 5, -0.5, -1\}$ at iteration point 20. Blue: no impact (varying term I version II), yellow positive impact $I_{20} = 1$ (varying term I version I) leading to a disturbance with returning convergence. Green: even stronger impact of $I_{20} = 5$ where convergence again appears only after a larger time interval. Red: negative impact $I_{20} = -0.5$ showing a strong positive reaction of the system followed by convergence. Purple: negative impact $I_{20} = -1$ leading to a negative $g^1$ and therefore a break down of the system.
• Figure 4: Adaptive version I of varying term II compared to non-adaptive version II concerning reaction of external input at iteration time point 20. Red: Version II time series from Fig. \ref{['fig:ipfstadtplanungmodeli']} with constant $\alpha = 0.56$. Blue: Version I time series with adaptive $\alpha$. Compared to Version II, the Version I time series is bifurcating due to adaptivity; still, it is returning to this state faster compared to Version II with static $\alpha$. Gray: Version I adaptive $\alpha$ corresponding to Version I system parameter $g^s_t$. After the impact on time point 20 $\alpha$ is way above the diverging limit of $\alpha \sim 0.37$, the system remains stable due to adaptation with a maximum $\alpha \sim 0.95$.
• Figure 5: Convergence of adaptive Version I of varying term II for varying $C^s$ like $-3 \leq C^s \leq 7$ and initial $\alpha_0$ like 0.1961 $\leq \alpha_0 \leq$ 10 for three impact strength I. Top: $I_{20}$ = 0, no impact, $\alpha_0$ can be decreased way above the lowest value of non-adapting Version II with suitable $C^s$. Middle: reduced variability of $\alpha_0$; still, again, lower values are possible. Bottom: only negative $C^s$ lead to a stable solution and $\alpha_0$ only leads to conversion for values up to $\alpha_0 \sim 0.370$ as in the non-adaptive case.