By Hüseyin Kocak
Phaser is a cosmopolitan software for IBM own com- puters, constructed atBrown collage through the writer and a few of his scholars, which allows usersto test with differential and distinction equations and dynamical structures in an interactive surroundings utilizing pix. This publication starts with a quick dialogue of the geometric inter- pretation of differential equations and numerical equipment, and proceeds to lead the scholar by utilizing this system. To run Phaser, you would like an IBM notebook, XT, AT, or PS/2 with an IBM colour images Board (CGB), superior photos Adapter (VGA). A math coprocessor is supported; although, one isn't really required for Phaser to run at the above hardware.
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Additional resources for Differential and Difference Equations through Computer Experiments: With Diskettes Containing PHASER: An Animator/Simulator for Dynamical Systems for IBM Personal Computers
4. 3. COMMAND: RESPONSE/EXPLANATION: u UTILITIES: To bring up the UTILITIES menu. b Big View: To select a view for the entire viewing area. A submenu containing the names of nine possible views will appear, and you will be prompted to select one of them to enlarge. p PhasePort: To blow up the Phase portrait view. The submenu of views will be erased, and the previous main menu, the UTILITIES menu, will be redisplayed. d DirField: To draw the direction field; see Lesson 3. 6
P -1, with f (p l(x O) = xO is called a periodic orbit of period p. 5) Therefore, periodic orbits of period p can be found geometrically by intersecting the graph of f (p 1 with the 45° line. Observe that a fixed point is a periodic orbit of period 1. 5), respectively. This is usually a formidable task. Certain fixed points and periodic orbits, however, can be located numerically. A periodic orbit is called asymptotically stable if all orbits starting "near" it converge to the periodic orbit as k -+ 00.
We will prefer to write the equation above as xk+l = f for (xk) k = 0,1,2,3, ... 2) xO . 2), is called an initial-value problem for a first-order difference equation. 2). The elements of this sequence are also called the iterates of xO. 3) What is a Difference Equation? 21 Notation: We will use the notation f (k l to denote the composition of the function f with itself k times, and will call it the k til iterate or the k til power of f . Consequently, the k til iterate of an initial point x will be denoted by f (k l(xO).
Differential and Difference Equations through Computer Experiments: With Diskettes Containing PHASER: An Animator/Simulator for Dynamical Systems for IBM Personal Computers by Hüseyin Kocak