陳義裕臺灣大學：物理研究所林彥廷Lin, Yen-TingYen-TingLin2007-11-262018-06-282007-11-262018-06-282006http://ntur.lib.ntu.edu.tw//handle/246246/54516我們的工作是建立在由Bando等人在八零年代所提出所謂的最適速度跟車交通模型上。我們在原本的單線道系統上加上了了一個匝道，並且在空間上採用週期條件。車輛採取一些策略來進入或離開匝道。數值模擬的方式顯現系統的一些特性：系統似乎有一個極為強烈的穩定解，在某些車輛數目的區間，以及不同的起始條件或是系統幾何（主支線道的長度），系統都會保有固定的型態以及流量。分析之後的結果我們認為應該要引入所謂延遲微分方程式(delay differential equation, or retarded functional differential equation, RFDE)來描述這個系統。我們提出一個可能的近似變分原理來解釋固定流量的現象。我們也將在最後提出近似的物理詮釋。This work was based on the so-called optimal velocity car-following model proposed by Bando et al. One more route in some region was added for the spatially periodic traffic system. Cars adopt several strategies to enter and leave the branching region. Numerical simulations were done to verify the evolution of the system. The system seems to have a robust stable solution: Within certain range, it appears to have a fixed flux, irrespective of the car number, the initial condtions and the geometry (length of the main and the branch). We tried to analyze such phenomenum and verify the essentiality of introducing delay di¤erential equations. An approximate variational principle with a delay argument is constructed and used to interpret the…fixed-flux phenomenon. The meaning of the approximate scheme is also discussed.1 Introduction 1 2 Single-Lane Bifurcating to Two Branches 3 3 Observations from Simulations 6 3.1 The Flux-Optimized States . . . . . . . . . . . . . . . . . . . . 6 3.2 The Phase Transition . . . . . . . . . . . . . . . . . . . . . . . 11 3.3 The Wavy Pattern . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4 The Divergent and Merging Strategies . . . . . . . . . . . . . 14 4 Wavy Pattern 15 4.1 The Governing Equations . . . . . . . . . . . . . . . . . . . . 15 4.1.1 Mean Fields and Conservation of Car Number . . . . . 15 4.1.2 Linearized Governing Equations of the Headway and the Velocity Fields . . . . . . . . . . . . . . . . . . . . 16 4.2 Delay Di erential Equations . . . . . . . . . . . . . . . . . . . 17 4.2.1 Direct Taylor’s Expansion . . . . . . . . . . . . . . . . 18 4.2.2 Characteristic Equation of the Delay Di erential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2.3 Asymptotic Behavior of Linearized Governing Equations 19 4.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.4 The Failure of Taylor’s Expansion and the Essentiality of D.D.E. 23 5 Some Details Near the Boundaries (M and D) and Nonlinear Governing Equations 26 5.1 Snapshots of the Time-independent Fields . . . . . . . . . . . 26 5.2 The Geometry of the System . . . . . . . . . . . . . . . . . . . 30 5.3 Nonlinear Governing Equations for the Fields (Time-independent Case) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.4 Realization of the Constant Delay . . . . . . . . . . . . . . . . 31 6 Unique Flux of the System { The Variational Principle of D.D.E. 32 6.1 One Must Go Beyond the Traditional Variational Principle for D.D.E.’s: Non-Existence of Absolutely Variational Principle of D.D.E. in Traditional Form . . . . . . . . . . . . . . . . . . . 33 6.2 Approximate Variational Principles for Delay Di erential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6.3 Estimate the Values of the Fields . . . . . . . . . . . . . . . . 36 6.3.1 Numerical Results . . . . . . . . . . . . . . . . . . . . 38 6.4 The Approximate Scheme and Some Details . . . . . . . . . . 40 7 Conclusion 43 7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2845094 bytesapplication/pdfen-US交通系統變分原理延遲微分方程式穩定性Traffic systemvariational principlesdelay equationsdelay differential equationsretarded functional differential equationsstabilitythesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/54516/1/ntu-95-R93222009-1.pdf