Methods of construction of phase portraits

Acquaintance with features of two methods of construction of phase portraits: a method of isoclinic lines and a method of Lenara is the lecture purpose.

Method of isoclinic lines. It is the general method, allowing to build phase paths at any nonlinearity of the equation presenting oscillations in system. As isoclinic lines are called curves on a phase plane, being geometrical places of points in which tangents to phase paths have the same inclination. For example, a geometrical place of points in which tangents to phase paths are vertical, the axis х is. Isoclinic lines find, setting values of slopes of tangents and substituting them in the differential equation :.

Method of isoclinic lines

 

Method of Lenara

Method of Lenara. A method of construction of phase portraits for oscillatory systems in which nonlinearity is connected only with a dissipation. For such systems the differential equation presenting free oscillations, looks like:

. (1)

Let's inject a dimensionless time and we gain

(2)

Where .

In these designations the differential equation for a phase plane looks like:

. (3)

From it follows that the function built in co-ordinates of a phase plane, is an isoclinic line of horizontal tangents to phase paths.

Lecture 5

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