By Prof. Edward R. Scheinerman, Mathematics
This article permits readers from a variety of backgrounds and with restricted technical necessities to discover the dynamical platforms and arithmetic in most cases. The ebook is designed for readers who are looking to proceed exploring arithmetic past linear algebra, yet should not prepared for hugely summary fabric.
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Additional resources for Invitation to Dynamical Systems
Suppose a person has three accounts: checking, savings and retirement. Each month, the checking account is credited with a pay check. Each month the person pays rent, utilities, and other expenses from the checking account and makes a deposit into savings and into retirement (assume all these amounts are the same from month to month). The checking account has a monthly fee and earns no interest. The savings and retirement accounts earn interest. Furthermore, the person has a car loan (paid from checking).
6) we notice that if x0 = x ˜ = −b/a, then x(t) explodes as t → ∞. However, if x0 = −b/a, then the system is stuck at −b/a. The value x ˜ = −b/a is an unstable When a is negative, the system tends to a stable fixed point. When a is positive, the system explodes. 34 CHAPTER 2. 2: The possible behaviors of one-dimensional continuous time linear systems. 7: The dynamical system x = f (x) = ax + b with a < 0. fixed point of the system. It is unstable because even if we start very near (but not at) x ˜, the system moves far away from x ˜.
Analysis We begin our study of multidimensional continuous time linear systems with our usual simplification, namely, that b = 0. The system becomes x = Ax. Now, instead of having a single differential equation in one unknown quantity, we have n differential equations in n variables! For example, x1 x2 2 1 = 1 2 x1 x2 which can be written out as dx1 dt dx2 dt = 2x1 + x2 , = x1 + 2x2 . , 42 Understanding “e raised to a matrix”. CHAPTER 2. LINEAR SYSTEMS These equations are difficult to solve because each is dependent on the other.