Chaos in transit systems
Author: Villalobos Durán, Jorge
Director: Zarama Urdaneta, Roberto
Publication date: 2010
Content type: doctoralThesis
We present two traffic models based on the same idea: that of a single vehicle interacting with traffic lights. The vehicle is capable of accelerating, traveling at constant speed, and braking. Braking takes place when the vehicle is interacting with a traffic light. The first model (Car model) will mimic the expected dynamics of a vehicle on a long road that contains several traffic lights. This car will interact with the lights depending on their color (either red or green) in the usual way that drivers do i.e. stop if a red light is present at a given point in space, continue if green. The second model (Bus model) will obey the same dynamics as the Car model but with the possibility of a forced stop between traffic lights, this in order to copy the action of picking up or leaving passengers. Two variants of the bus model will be developed and studied. All models are highly minimalistic and avoid the issue of human reactions (the vehicles will never run a red light, or try to emulate other human behavior like traveling at different speeds or such). The principal objective of this document is to characterize chaotic behavior in these simple traffic models. The characterization we propose is based on the following program: First, based on the dynamic equations of the system, write a discrete non linear map. Then use bifurcation diagrams to explore the parameters, find symbolic solutions to period 1 and period 2 orbits, as well as some interesting values for the diagrams. Since there are many bifurcation parameters, represent combinations of parameter values that have resulted in chaotic behavior in 2D plots where positive Lyapunov exponents are marked; these 2D Lyapunov plots give a broad picture of the combinations of parameters for which chaos is present on the systems. We will aim our efforts in order to achieva a goal and answer two simple questions: 1.To characterize the systems asymptotic behavior. 2.Is non trivial dynamics and chaos (understood as high sensitivity to initial conditions) a fundamental part of traffic systems? 3.Is this sensitivity consequence of human behavior or not?
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