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Thursday, July 30, 2020 | History

2 edition of Phase-plane methods found in the catalog.

Phase-plane methods

by Yun-lan Hsiung

  • 93 Want to read
  • 26 Currently reading

Published by Naval Postgraduate School in Monterey, California .
Written in English


ID Numbers
Open LibraryOL25128779M

Phase-plane analysis of nonlinear systems provides an understanding of which steady-state solution that a particular set of initial conditions will converge to • The local behavior (close to one of the steady-state solutions) can be understood from a linear phase-plane analysis of the particular steady-state solution (equilibrium point). The phase plane is introduced to discuss nonlinear phenomena. Discrete models for population growth are also presented, and when teaching in recent years I have supplemented the book with a discussion of iterations of the logistic map and the period doubling route to chaos.

Choudhury(), andMooney() for descriptions of uses in various fields. Since the method is extremely algorithmic in nature, software tools would clearly provide a powerful system by which to execute the techniques. Despite this, few programmes are available for implementing phase plane methods.   We discuss the use of Phase Plane Plots for understanding the qualitative behavior of systems of differential equations.

refer to numerical methods especially developed for such purpose, which have appeared in the last few years. The brief Chapter 5 is concerned with dynamic analysis of non-linear systems. The phase plane method to evaluate graphically the response of single degree-of- freedom systems, and a general numerical method. An Overview of the Book 12 Notes and References 13 Part I: Nonlinear Systems Analysis 14 Introduction to Part I 14 2. Phase Plane Analysis 17 Concepts of Phase Plane Analysis 18 Phase Portraits 18 Singular Points 20 Symmetry in Phase Plane Portraits 22 Constructing Phase Portraits


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Phase-plane methods by Yun-lan Hsiung Download PDF EPUB FB2

CHAPTER 4 Systems of ODEs. Phase Plane. Qualitative Methods. Tying in with Chap. 3, we present another method of solving higher order ODEs in Sec.

This converts any nth-order ODE into a system of n first-order ODEs. We also show some applications. Moreover, in the same section we solve systems of first-order ODEs that occur directly in applications, that is, not derived from an nth-order.

9 - Nonlinear Ordinary Differential Equations: Phase Plane Methods. King, University of Birmingham, J. Billingham, University of Birmingham, S. Otto, University of Birmingham; Publisher: Cambridge University Press Email your librarian or administrator to recommend adding this book to your organisation's collection.

Differential Author: A. King, J. Billingham, S. Otto. The system Phase-plane methods book can be represented graphically by the locus of x (t) versus x (t), that is, parametrized in t. The pair {x (t), x (t)} corresponds to the coordinates of a point in the so-called phase plane (PP).

As time varies in the interval t ∈ [0, ∞ [, this point describes a PP trajectory. A family of PP trajectories is called a phase : J. Tenreiro Machado, António M. Lopes, Duarte Valério, Alexandra M. Galhano. An illustration of an open book. Books. An illustration of two cells of a film strip.

Video. An illustration of an audio speaker. Audio. An illustration of a " floppy disk. Software. An illustration of two photographs. Phase-plane methods. Phase-plane methods book Preview remove-circle Share or Embed This Item. Practical application of phase-plane method is given in the end of this article.

Discover the world's research. In his popular book Chaos on the history and science of chaos theory, James Author: Mark Polyak. Phase Plane Methods EquilibriaPlot in the xy-plane all equilibria of (1).

See Figure 3. WindowSelect an x-range and a y-range for the graph window which includes all signi cant equilibria (Figure 3). GridPlot a uniform grid of Ngrid points (Nˇ50 for hand work) within the graph window, to populate the graph- ical white space (Figure 4).

Phase Plane Methods Population biology. Planar autonomous systems have been applied to two-species populations like two species of trout, who compete for food from the same supply, and foxes and rabbits, who compete in a predator-prey situation. Trout system.

Phase Plane Analysis Phase Plane Analysis is a graphical method for studying second-order systems respect to initial conditions by: providing motion trajectories corresponding to various initial conditions.

examining the qualitative features of the trajectories obtaining information regarding the stability of the equilibrium points. In this section we will give a brief introduction to the phase plane and phase portraits. We define the equilibrium solution/point for a homogeneous system of differential equations and how phase portraits can be used to determine the stability of the equilibrium solution.

We also show the formal method of how phase portraits are constructed. phase plane approach to study oscillations in electrical circuits.] West, J. Analytical Techniques for Nonlinear Control Systems. London: EUP [First book published in the UK on nonlinear control systems. Gives an excellent introduction to phase plane and describing function methods.] West, J.

C., Douce, J. and Naylor, R. Phase Plane Analysis Phase plane analysis is a graphical method for studying second-order systems.

This chapter’s objective is to gain familiarity of the nonlinear systems through the simple graphical method. Concepts of Phase Plane Analysis. In applied mathematics, in particular the context of nonlinear system analysis, a phase plane is a visual display of certain characteristics of certain kinds of differential equations; a coordinate plane with axes being the values of the two state variables, say, or etc.

It is a two-dimensional case of the general n-dimensional phase space. The phase plane method refers to graphically determining the existence.

Sketching phase trajectories: The sketching of the phase trajectory is the basis of phase plane analysis. Analytical method and graphical method are two main methods for plotting the phase trajectory.

The analytical method leads to a functional relationship between x and ẋ by solving the differential equation, then the phase trajectory can. Amalia Torre, in Linear Ray and Wave Optics in Phase Space, The ellipse of second-order moments. According to the results of §as far as the linear approximation is concerned, the phase-plane dynamics of the Wigner chart mirrors that of the ray-density distribution in the geometrical optical means that one may choose any closed domain in the Wigner plane to.

CONCEPT OF PHASE PLANE ANALYSIS phase portraits:The phase plane method is concerned with the graphical study of second-order autonomous systems described by ˙x1 = f1(x1, x2) (1) ˙x2 = f2(x1, x2) (2) where x1, x2: states of the system f1, f2 nonlinear functions of the states Geometrically, the state space of this system is a plane having x1, x2 as plane is.

Chapter 14 - Introduction to Phase Plane Analysis Pages - The goal of this chapter is to examine the cone and horizontal cell system using a qualitative visualization technique called phase plane analysis.

In this chapter you will be studying a retinal feedback model. second-order systems, and apply phase plane analysis. I Disadvantage of Phase Plane Method: It is restricted to at most second-order and graphical study of higher-order is computationally and geometrically complex.

Farzaneh Abdollahi Nonlinear Control Lecture 2 3/ Differential Equations with Boundary-Value Problems (9th Edition) Edit edition. Problem 32E from Chapter Use the phase-plane method to show that the solution to the.

This book presents a collection of exercises on dynamical systems, modelling and control. Each topic covered includes a summary of the theoretical background, problems. the phase plane. The parametric curves traced by the solutions are sometimes also called their trajectories.

Remark: It is quite labor-intensive, but it is possible to sketch the phase portrait by hand without first having to solve the system of equations that it represents. Just like a direction field, a phase portrait can be a tool to predict. About this book The increase of consumer electronics and communications applications using Radio Frequency (RF) and microwave circuits has implications for oscillator design.Phase-Plane Methods and Qualitative Solutions.

This Chapter Appears in. Title Information. Published: ISBN: eISBN: Book Code: CL Series: Classics in Applied Mathematics. Pages: Buy the Print Edition. Nothing is. Performs a qualitative analysis of one- and two-dimensional autonomous ordinary differential equation systems, using phase plane methods.

Programs are available to identify and classify equilibrium points, plot the direction field, and plot trajectories for multiple initial conditions.

In the one-dimensional case, a program is also available to plot the phase portrait.