Chaos Control in Shuttle Bus Schedule Matthew Avetian Lynette Guzman Ran Kumada Robert Soimaru Salim Zerkoune Math 485 4/1/20102 Introduction Chaos, it is a term that defines randomness and unpredictability. This feature appears often in everyday activities, crowd movements, inner city driving, even brushing your teeth is a process of random strokes of the toothbrush. There has become an increasing interest in determining what portions of these chaotic systems is predictable or even controllable, in order to make certain processes more efficient. In this report we attempt to reproduce Takashi Nagatani’s results on shuttle bus schedules. The system of the shuttle buses is a rather simple one. There are only two shuttle stops, and two buses. This system may represent an amusement park that has a rather large parking lot. To assist the visitors there is a shuttle system in place to move them from their cars to the park, or vice versa. The program we have developed is meant to simulate this process and watch the headway of the buses with time. Background The article “Chaos Control and Schedule of Shuttle Buses” by Takashi Nagatani analyzes two shuttle buses as they pass each other and pick up and drop off passengers. The buses will make up for any time delay caused by dropping off passengers by speeding up, indicated by speedup parameter. There are four different cases of speedup parameter to be considered. We will translate the equations in the article to a Matlab code which will show the buses have periodic behavior and thus predictability in the shuttle bus system. In our goal to measure chaos of shuttle buses we must first define what chaos means to better understand the issue at hand. We will use the definition by Strogatz from Nonlinear Dynamics and Chaos that states, “Chaos is aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions.” (Strogatz, p. 323) A3 deterministic system is one that has no random or noisy parameters and sensitive dependence allows the nearby trajectories to separate exponentially. Mathematician and physicist Jules Henri Poincaré was the first to discover a chaotic deterministic system which paved the way for chaos theory. However, the invention of the modern computer in the late 1950’s led to far more advanced work that Poincaré could not have imagined. Mathematician and meteorologist Edward Lorenz took advantage of the advances in technology to study weather patterns and noticed the dependence on initial conditions. He noticed the solutions to his equations had structure within the chaos and were patterned in the shape of a butterfly, coining the term “butterfly effect.” In another famous example, Mathematician Kevin Cuomo used synchronized chaos to encode hidden messages, namely pop music. Cuomo set up an experiment using resistors, capacitors, and amplifiers to mask singer Mariah Carey’s song “Emotions” so that outsiders could only hear the chaos or static. However, when the song was sent to the receiver, the output was synchronized almost exactly to the original chaos. After some electronic subtraction, the static disappeared and the song emerged albeit fuzzy (Strogatz, 337). As chaos pertains to our problem, the chaotic motion for shuttle buses depends on both the loading and speedup parameter. Furthermore, chaos is closely related to nonlinear dynamics and our shuttle bus system can be modeled similarly to traffic flow. Map model The arrival time, ,ti(m 1) ti(m 1) of bus i at the origin for trip (m+1) is given by, ti(m 1) ti(m) ( )Bi(m)2LVi(m) for i=1,2…,M. New passengers arrive at rate μ and the new passengers that have arrived since the bus ahead iʹ leaves the origin is (ti(m) ti'(m')),. This gives an equation for the motion of the bus,4 ti(m 1) ti(m) ( )(ti(m) ti'(m'))2LV0si( )(ti(m) ti'(m')). Given this equation we want to obtain an equation for dimensionless arrival time. This can be done by dividing the equation by 2L /V0 for bus i at the origin. Completing this step we are left with Ti(m 1) Ti(m) (Ti(m) Ti'(m'))11 Si(Ti(m) Ti'(m')), where Ti(m) ti(m)V0/2L, ( ), and Sisi( )2L/V02.. Thus, we have created a system controlled by two parameters, Γ the loading parameter and Si the speedup parameter. This equation states that as the number of perspective passengers increases, the value of the loading parameter becomes high. The article by Nagatani compares four different cases of speedup parameters. First, both Bus 1 and Bus 2 have a speedup parameter of 0, thus no speedup by either bus. The second case looks at both buses having the same parameter of 0.2 and the last case takes into account two different parameters. In the first case Bus 1 is 0.3 and in the second case Bus 1 is 0.5 while Bus 2 maintains a speedup parameter of 0.2. The goal is to show the chaotic motion of buses is suppressed by the speedup parameter and thus we are able to control the chaos. The first step is to translate the arrival time equation given in the article to work in MATLAB. MATLAB code We have two codes to translate the given equation into MATLAB. The first code is the main programming that calculates the headway of the two busses as a function of the loading parameter. Another code is to plot the result and create graphs with four different cases of speedup parameters.5 The main code, busMap, calculates each arrival time of bus i at the origin, using the equation Ti(m 1) Ti(m) (Ti(m) Ti'(m'))11 Si(Ti(m) Ti'(m')). The article we refer to does not give a specific initial condition, and it only plots the arrival time of 901-1000th trip on the graph. Therefore, we choose the equation t(i) ii 1i, where i is the bus number, to calculate the initial conditions. One reason is to let bus 1 has a simple initial value, but to let bus 2 has non-integer initial condition with greater value than bus 1. Thus, t =1 and t = 2.5 are initial values for bus 1 and bus 2, respectively. However, we have to be aware that the different initial conditions will results in a different graph shape since the system is chaotic. The structure of busMap is as follows. Suppose that bus 1 arrives at the origin. Then, the arrival time and trip number is stored in t[] and arrivalTimes[]. Here, t[] stores the bus number and the time at which trip is completed. The arrivalTimes[] stores four different information; 1) arrival time, 2) bus number, 3) trip number for either bus 1 or bus 2,