Interrelated Dynamical Systems with Random Initial Inputs

我們把隨機變數放在動力系統的起始值上，而系統本身並無隨機性。然後提出了一個有意思的問題：若起始值是獨立的隨機變數，在動力系統本身的彼此交互作用下，系統將會如何影響隨機變數的相關性呢？隨機變數之間還是彼此獨立嗎？亦或是變數之間會開始產生某種相關性呢？在這論文中，我們分別在差分方程、具有落遲時間的差分方程與微分方程中，檢驗其隨機變數間相關性的演變過程。結果，在這些方程裡，其相關性都有相似的結果：或具有長期的穩定趨勢；或在一段時間的演變後具有某種循環，有的單純、而有的卻很複雜。最後，在處理具有不確定性或相關性的模擬時，該論文提供了一個新的觀點：方程式中，隨著時間前進，變數的相關性可以是非定值但仍然是可預測的。相信在其他科學領域上，例如統計、計量經濟學，該想法提供了不同的思考可能方向。We consider deterministic dynamical systems with random initial inputs, where we can know that the only uncertainty in the systems is determined at the beginning. An interesting question arises––how would the system in which the variables are interacted push the correlations between variables into change? In this thesis, we examine the evolution of correlations under difference equations, delayed difference equation (a kind of difference equation) and differential equation separately. As a result, the paths of correlations under different equations have a similar consequence––the patterns of the paths may have a long-term stable tendency or a cycle, simple or complicated, after several periods of evolution. Moreover, this thesis provides a new point of view on dealing with uncertainty and correlations in simulation––the correlations of variables in equations, as time progresses, may not be constant but still predictable. We believe that the thought provides different possible perspectives on other scientific areas, such as statistics and econometrics.論文口試委員審定書 ....................................................................................................... i要 ................................................................................................................................. iibstract ............................................................................................................................ iiiontents ........................................................................................................................... ivist of Figures .................................................................................................................. viist of Tables ................................................................................................................... vi1. Introduction ................................................................................................................ 12. Systems and Correlations ........................................................................................... 4.1 Linear Difference Equation ................................................................................ 4.1.1 Linear Difference Equations without Constant Term .............................. 4.1.2 Linear Difference Equations with Nonzero Constant Term .................... 7.1.3 Two-Dimensional Case ............................................................................ 9.2 Linear Differential Equation ............................................................................. 23.2.1 Formulation of Covariance and Correlation .......................................... 23.2.2 Three- and Four-Dimensional Case ....................................................... 25.2.3 Generalization ........................................................................................ 49.3 Delay Difference Equation ............................................................................... 56.3.1 Formulation of Covariance and Correlations ........................................ 563. Simulations ............................................................................................................... 59.1 Linear Difference Equation .............................................................................. 59.1.1 Diagonalizable matrix with real-valued eigenvalues ............................. 59.1.2 Not diagonalizable matrix ..................................................................... 60.1.3 Matrix with complex conjugate eigenvalues ......................................... 62.2 Linear Differential Equation ............................................................................. 64.2.1 3×3 matrix with a real and a pair of complex conjugate eigenvalues ... 64.2.2 4×4 matrix with two pairs of complex conjugate eigenvalues .............. 66.3 Delay Difference Equation ............................................................................... 69.4 Algorithm of Simulations with MATLAB ........................................................ 71.4.1 Codes of Section 3.2.2 (Simplified) ...................................................... 714. Conclusion ................................................................................................................ 72eferences ...................................................................................................................... 7