What are the methods available for constructing phase trajectory?
2.2 Constructing Phase Portraits There are a number of methods for constructing phase plane trajectories for linear or nonlinear system, such that so-called analytical method, the method of isoclines, the delta method, Lienard’s method, and Pell’s method.
What is the method of isoclines?
Isoclines are often used as a graphical method of solving ordinary differential equations. In an equation of the form y’ = f(x, y), the isoclines are lines in the (x, y) plane obtained by setting f(x, y) equal to a constant.
What is phase plane technique in control system?
The phase plane method refers to graphically determining the existence of limit cycles in the solutions of the differential equation. The solutions to the differential equation are a family of functions. Graphically, this can be plotted in the phase plane like a two-dimensional vector field.
What is the phase trajectory?
The trajectory of a point in a phase space, representing how the state of a dynamical system changes with time. A point w of a non-closed phase trajectory divides it into two parts — the positive and negative semi-trajectories.
What do Isoclines represent?
An isocline is a curve through points at which the parent function’s slope will always be the same, regardless of initial conditions. The word comes from the Greek words Isos meaning “same” and Klisi meaning “slope”. It is often used as a graphical method of solving ordinary differential equations.
What are Isoclines in economics?
An isocline is a curve that starts from the origin and passes through the isoquant map of the firm, and along which the marginal rate of technical substitution of input X for input Y, i.e., the numerical slope of the isoquants is constant.
What is trajectory control system?
In dynamical systems, a trajectory is the set of points in state space that are the future states resulting from a given initial state. In a discrete dynamical system, a trajectory is a set of isolated points in state space. In a continuous dynamical system, a trajectory is a curve in state space.
How do you draw a trajectory?
One way to get a sketch of trajectories is to do something similar to what we did the first time we looked at equilibrium solutions. We can choose values of →x (note that these will be points in the phase plane) and compute A→x A x → . This will give a vector that represents →x′ at that particular solution.
How to create a phase portrait by isocline?
Isocline Method The algorithm of constructing the phase portrait by isocline method: Plot the curveS(x) =in state-space (phase plane) Draw small line with slope. Note that the direction of the line depends onthe sign off1andf2at that point.
How are solution curves constructed in the isocline method?
By using Eqn (1.94), the trajectories (solution curves) can be constructed in the (α1, α2) plane. The isocline method turns out to be straightforward and simple to employ. The pattern of the trajectories is strongly influenced by the so-called singular points. In these points the motion finds an equilibrium.
How is the pattern of the trajectories influenced by isoclines?
The isocline method turns out to be straightforward and simple to employ. The pattern of the trajectories is strongly influenced by the so-called singular points. In these points the motion finds an equilibrium. In the singular points, the motion is stationary and consequently the differentials of the state variables vanish.
How are th and IL2 = 0 isoclines related?
(A) The Th’ = 0 and IL2’ = 0 isoclines for various values of S (i.e. S = 1, 2, 4, 8, 16, 32). The straight diagonal IL2’ = 0 isocline is independent of the S (influx) parameter. The bent Th′ = 0 isocline moves upwards if S increases. The region in which both Th and IL2 increase is shaded.