The phase portrait for dynamic systems is a tool used to graphically determine the instantaneous behavior of its trajectories for a set of initial conditions. Classic phase portraits are limited to two dimensions and occasionally snapshots of 3D phase portraits are presented; unfortunately, a single point of view of a third or higher order system usually implies information losses. To solve that limitation, some authors used an additional degree of freedom to represent phase portraits in three dimensions, for example color graphics. Other authors perform states combinations, empirically, to represent higher dimensions, but the question remains whether it is possible to extend the two-dimensional phase portraits to higher order and their mathematical basis. In this paper, it is reported that the combinations of states to generate a set of phase portraits is enough to determine without loss of information the complete behavior of the immediate system dynamics for a set of initial conditions in an n-dimensional state space. Further, new graphical tools are provided capable to represent methodically the phase portrait for higher order systems.

https://www.mdpi.com/2076-3417/9/5/872/pdf

On the n-Dimensional Phase Portraits

The class of systems considered in this investigation is a cascade combination of a linear memory system and a non-linear no-memory system in the forward path of a unity feedback control system. The output of the non-linear no-memory system is assumed to be a polynomial function of the input. Regardless of the exact nature of the non-linearity, the objective of this method of analysis is to predict the behaviour of higher-order non-linear systems with different initial conditions for step inputs. Two different cascade combinations of linear and non-linear blocks in the forward path are considered. For both configurations a similar non-linear differential equation is obtained for some variable in the system. The non-linear differential equation is further reduced to a first-order equation, explicitly independent of the independent variable, time t. Treating all other coefficients as parameters and eliminating each in turn, finally the required phase-plane trajectory is obtained.

Graphical analysis of a class of non-linear feedback control systems with step input†

By considering simultaneously the N-x (or the N-\dot{x} ), where N=d\dot{x}/dx and the \dot{x}-x planes, second order non-linear autonomous systems (not easily amendable to the existing methods) described by the differential equation \ddot{x}=F(x, \dot{x}) can be studied. Changes in system behavior due to changes in initial conditions, nonlinear dampling and restoring forces can easily be studied. Simple and general as it is, the method presented here is believed to be novel.

Generalized isocline method of plotting phase-plane trajectories

In the method of analysis presented, phase plane techniques have been successfully modified to cater to a class of third order non-linear systems This method is applicable to control systems having non-linearities in the feedback loopsThe technique involves plotting an (x1, x3) master plane having all possible trajectories for any non-zero value of x2 The variation and initial conditions of variable x2 are incorporated by adding a set of x3-scales, and an origin shift scale, for various values of x2, to the (x1, x3) master plane Using this master plane, the required (x1, x2 and (x1, x3) trajectories can be obtained for any set of initial conditions, by following the procedure illustrated by typical examples

Analysis of a Class of Third Order Non-linear Systems Using Phase Plane Techniques

The following method is proposed for graphical analysis of various third order non-linear systems, defined by The technique involves treating ẋ as a parameter and introduces dummy variables, such as N = ẍ/ẋ = dẋ/dx and M = dN/dx, to obtain a single plane, to be known as the (N, x) master plane, which would contain all the data about the variation and initial conditions of all the three variables, N, ẋ and x. The variation of ẋ is incorporated into the (N, x) plane by effecting a change of x-scale, and change of the N-axis for various ẋ values. The given examples illustrate the use of this master plane and a simple procedure, by which the required (x, ẋ) and (N, x) trajectories for a third order system are obtained. It has been shown that this technique is also applicable to system defined by where f4 is a composite non-linear function having unique values for specific ẋ and ẍ.

Graphical Analysis of a General Class of Third Order Non-Linear Systems

This paper suggests the use of simple transformations like ?=kx, kx2 for second-order nonlinear differential equations to effect rapid plotting of the phase-plane trajectories. The method is particularly helpful in determining quickly the trajectory slopes along simple curves in any desired region of the phase plane. New planes such as the t?-x, t?2-x are considered for the study of some groups of nonlinear time-varying systems. Suggestions for solving certain higher-order nonlinear systems are also made.

Techniques for Analysis of Certain Nonlinear Systems

Analysis of certain second-order nonlinear systems, not easily amenable to the phase-plane methods, and described by either of the following differential equations x?n-2?+ f(x)x?2n+g(x)x?n+h(x)=0 ?+f(x)x?n+h(x)=0 n≫0 can be effected easily by drawing the entire portrait of trajectories on a new plane; that is, on one of the x?n?x planes. Simple equations are given to evaluate time from a trajectory on any of these n planes. Poincare's fundamental phase plane x??x is conceived of as the simplest case of the general x?n?x plane.

The x¿ n -x Plane for Analysis of Certain Second-Order Nonlinear Systems

Use of some new planes such as the R-x, R2-x (where R represents in the n-dimensional phase space, the radius vector from the origin to any point on the trajectory described by the system) is suggested for analysis of nonlinear systems of any kind. The stability conditions in these planes are given. For easy understanding of the method, the transformation from the phase plane to the R-x, R2-x planes is brought out for second-order systems. In general, while these planes serve as useful as the phase plane, they have proved to be simpler in determining quickly the general behavior of certain classes of second-order nonlinear systems. A chart and a simple formula are suggested to evaluate time easily from the R-x and R2-x trajectories, respectively. A means of solving higher-order nonlinear systems is also illustrated. Finally, a comparative study of the trajectories near singular points on the phase plane and on the new planes is made.

B. L. Deekshatulu

Papers

Techniques for Analysis of Certain Nonlinear Systems

The x¿ n -x Plane for Analysis of Certain Second-Order Nonlinear Systems

The R-x, R 2 -x Planes; Some New Ideas for the Study of Nonlinear Systems