Nonlinear Transformation of Differential Equations into Phase Space
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Nonlinear Transformation of Differential Equations into Phase Space Leon Cohen Department of Physics and Astronomy, Hunter College, City University of New York, 695 Park Avenue, New York, NY 10021, USA Email: [email protected]
Lorenzo Galleani Dipartimento di Elettronica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy Email: [email protected] Received 7 September 2003; Revised 20 January 2004 Time-frequency representations transform a one-dimensional function into a two-dimensional function in the phase-space of time and frequency. The transformation to accomplish is a nonlinear transformation and there are an infinite number of such transformations. We obtain the governing differential equation for any two-dimensional bilinear phase-space function for the case when the governing equation for the time function is an ordinary differential equation with constant coefficients. This connects the dynamical features of the problem directly to the phase-space function and it has a number of advantages. Keywords and phrases: time-frequency distributions, nonstationary signals, linear systems, differential equations.
1.
INTRODUCTION
Ordinary linear differential equations with constant coefficients are the most venerable and studied differential equations, and many ideas and methods have been developed to obtain exact, approximate, and numerical solutions, and to qualitatively study the nature of the solutions [1]. The subject is over 300 years old, but nonetheless we argue that a totally new perspective is achieved when the differential equation, even a simple ordinary differential equation, is transformed into phase space by a nonlinear transformation. Moreover we further argue that this transformation not only results in greater insight into the nature of the solution, but leads to new approximation methods [2]. To illustrate and motivate our method we start with a simple example. Consider the following harmonic oscillator differential equation (it is the equation of the RLC circuit, or the damped spring-mass system): d2 x(t) dx(t) + 2µ + ω02 x(t) = f (t), dt 2 dt
(1)
where f (t) is a given driving force and x(t) the output signal of the system, that is, the solution to the differential equation (µ and ω0 are real constants). Perhaps there is no more studied equation than this one. In principle, this equation can be solved symbolically by many methods, for example, by obtaining Green’s function. However, doing so does not add any particular insight into the nature of the solution. For
practical reasons and to gain insight, one often transforms this equation into the Fourier domain. Defining
1 X(ω) = √ x(t) e−itω dt, 2π 1 F(ω) = √ f (t) e−itω dt, 2π
(2)
the differential equation transforms into [3]
− ω2 + 2iµω + ω02 X(ω) = F(ω),
(3)
whose exact solution is X(ω) =
F(ω) − ω2 + 2iµω + ω02
.
(4)
The reasons for going into the Fourier domain are many. First, we have a practical way of solution, since now one can find the time solution by way of 1 x(t) = √ 2π
F(ω) − ω2 + 2iµω
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