Systems of first-order differential equations149Mathematicians have demonstrated a number of properties for the trajectoriesof autonomous systems. Here we shall simply list them.(1)There is no more than one trajectory through any point in the phase plane(2)A trajectory that starts at a point that is not a fixed point will only reacha fixed point in an infinite time period(3)No trajectory can cross itself unless it is a closed curve. If it is a closedcurve then the solution is a periodic one.4.3Vectors of forces in the phase planeWe established in chapter 2, when considering single autonomous differentialequations, that we could establish the direction ofxwhentvaries from the sign of˙x. In the case of a system of two differential equations we can establish the directionofxfrom the sign of ˙xand the direction ofyfrom the sign of ˙y. Such movementsinxandygive us insight into the dynamics of the system around the equilibrium.
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