On the n-Dimensional Phase Portraits

Abstract

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.

Nonlinear System Stability and Behavioral Analysis for Effective Implementation of Artificial Lower Limb

“…Using the Gauss elimination method, the obtained simplified equation is 19. The produced graphical phase portrait [10] plot is given below in Figure 11.…”

“…The phase plot lines are presented through red lines in Figure 12. The produced graphical phase portrait [10] plot is given below in Figure 11. This output shows asymptotic stability because for Eigen vectors the first phase plot is going outward or away from the origin of the plane and the second phase plot is coming towards the origin of the plane as shown in Figure 12.…”

“…Basic theoretical knowledge and stability-related methodology is discussed for the nonlinear systems [9]. Linearization of differential equations and phase portrait analysis of nonlinear systems has been shown in the nonlinear domain [10]. Exponential stability analysis and design of sampled-data nonlinear systems has been shown here [11].…”

Automatic Generation of Interactive Multidimensional Phase Portraits

“…From an n -dimensional system, a 2-dimensional phase plot can be generated representing a planar slice showing the dynamics of the system with respect to only two dependent variables and setting the rest of dependent variables as constants. Crucially, the location of a n -dimensional point or line in Ω can be unequivocally determined from all the combinations of 2-dimensional plots of x i , x j out of the set ( x 1 , …, x n ) [12]. Hence, a number v of 2-dimensional views equal to the binomial coefficient is sufficient to determine unambiguously the location of critical points, nullclines, and the numerical solution of a trajectory in an n -dimensional system.…”

“…Complete phase portraits are generally limited to present information in two dimensions. However, dynamical systems with more than two dependent variables can be visualized without loss of information with state combinations [12]. In general, constructing phase portrait diagrams requires expert ability and the use of specialized mathematical software tools.…”