Phase Portraits Of System Research Articles

Qualitative analysis of behavior of systems of piecewise linear differential equations with two state variables

The set of all solution curves in the phase space for a system of ordinary differential equations (ODEs) is called the phase portrait. A phase portrait provides global qualitative information about how a given system of ODEs behaves under different initial conditions. We are developing a method called Topological Flow Analysis (TFA) for automating analysis of the topological structure of the phase portrait of systems of ODEs. In this paper, we describe the first version of TFA for systems of piecewise linear ODEs with two state variables. TFA has several novel features that have not been achieved before. Firstly, TFA enables to grasp characteristics of all behaviors of a given system of ODEs. Secondly, TFA allows to symbolically represent behaviors in terms of critical geometric features in the phase space. Finally, TFA integrates qualitative and quantitative analysis. The current version of TFA has been implemented as a program called PSX2PWL using Common Lisp.

Phase portraits of transfer systems--detection of nonlinear transmission properties between respiratory movements and heart rate fluctuations

A new approach to the analysis of non-linear systems is proposed. It enables the reconstruction of the system state movement in a multi-dimensional phase space. The resulting phase portraits of the system behaviour were characterised quantitatively by estimating the correlation dimensions and prediction errors. This new approach was validated using simulated typical transfer systems. The measured respiratory movements and heart rate fluctuations of rabbits were employed for the first time to investigate the non-linear transfer properties between respiratory activity and heart rate in the system phase portraits. Significant non-linear transfer characteristics between respiratory movements and heart rate fluctuations were found--also during the awake phase--which have so far not been quantified.

Algebraic and topological classification of the homogeneous cubic vector fields in the plane

Tsutomu Date and Masao Iri in [DI] gave an algebraic classification of systems I = P(x, y), I; = Q(.x, y), where P and Q are homogeneous polynomials of degree 2. For this, they used the classification of the binary cubic forms and also the simultaneous classification of a linear binary form and a cubic binary form given by the algebraic invariant theory. We begin by doing a similar study for systems .?? = P(x, y), ,’ = Q(x, y), where P and Q are homogeneous polynomials of degree three (i.e., cubic systems). The classification’s theorem of such systems is based on the classification of fourth-order binary forms. Gurevich in [Gu] did the classification of fourth-order binary forms on the field of complex numbers. Since we did not find the classification on the real domain, we adapt Gurevich’s proof to obtain it. In Section 1 we give some definitions and preliminary results, while in Section 2 we give the theorem of classification of fourth-order binary forms on the real domain. The method used in the proof (Caley’s method) let us obtain canonical forms of the fourth-order binary forms and the algebraic characteristics. Section 3 is devoted to obtaining the algebraic classification of systems .? = P(x, y), j = Q(x, y), where P and Q are homogeneous polynomials of degree 3. Given an arbitrary system X= (P, Q) with P and Q homogeneous polynomials of degree 3, we can know the equivalence-class at which it belongs through the algebraic characteristics. In Section 4 we study the phase-portraits of systems 1= P(x, y), j = Q(x, y), where P and Q are homogeneous polynomials of degree n and P and Q have no common factor. Such systems have been studied by J. Argemi in [A]. Here we give a shorter new proof of his results by using

Non-Linear Analysis on the Acceleration Velocity Plane

The non-linear system characterized by the differential equation ẍ+f(ẋ) + w20x = A0u(t)+A1t is solved on the ẍ-ẋ plane. The acceleration velocity portraits of typical systems are drawn for all pertinent initial conditions using the isocline concept and the delta dot method. A simple and easy to apply equation is derived for interpreting the state variable x from the ẍ-ẋ trajectory. A method is indicated for plotting the phase portrait of systems characterized by ẍ+f(ẋ)+g(x)= A0u(t) = A0u(t) on the ẍ-ẋ plane with a suitable illustration. The methods presented in this article open a new avenue for the analysis of certain non-linear systems and have certain definite advantage over the existing technique.