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Linear Systems of Autonomous First-Order DE's Part A: Examples The system of DEs = = ) , ( ) , (  y  x G dt dy  y  x  F dt dx  where F and G are two given functions of  x and y is called a system of first-order autonomous DEs. t is the independent variable. There are two dependent variables,  x and y. The system of  DEs describes the simultaneous evolution of  x and y in time. The plane (  x,  y) is called the phase plane. We represent the behavior of the system in the phase plane by plotting a few trajectories  (  x(t ), y(t )), each starting at a different point (  x(0), y(0)), with t running over a suitable interval. The independent variable t does not appear explicitly in the phase plane. We draw the velocity field (dx/dt , dy/dt ) = (  F , G) = (  F (  x, y), G(  x, y)) at regularly spaced  points (  x, y) in the phase plane. This velocity field is sometimes called a vector field or direction field. The slope dy/dx of the trajectory through point (  x,  y) is  y'/  x' = G(  x,  y)/  F (  x,  y). As usual, x' is short for dx/dt . Alternatively, we will sometimes use the notation  x for dx/dt . The loci (generally curves, or lines) of co nstant slope in the phase plane a re called isoclines. Of particular interest are the loci where one of the two derivatives is 0. These loci are called nullclines . The locus of the points (  x, y) such that dx/dt = 0, i.e., F (  x,  y) = 0, is called the vertical nullcline, or V-nullcline , because the trajectories there all have vertical slope. Similarly, the locus of the points (  x, y) such that dy/dt = 0, i.e., G(  x,  y) = 0, is called the horizontal nullcline, or H-nullcline , because the trajectories there all have horizontal slope. The trajectories, velocity field and nullclines make up a phase portrait of the system. If both F and G are linear functions of  x and y, the system is called a linear system of first-order autonomous DEs. The following are examples of such systems. The phase  portraits were created by Linear_System.m.

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