Nullcline

In mathematical analysis, nullclines, sometimes called zero-growth isoclines, are encountered in a system of ordinary differential equations In mathematical analysis, nullclines, sometimes called zero-growth isoclines, are encountered in a system of ordinary differential equations where x ′ {displaystyle x'} here represents a derivative of x {displaystyle x} with respect to another parameter, such as time t {displaystyle t} . The j {displaystyle j} 'th nullcline is the geometric shape for which x j ′ = 0 {displaystyle x_{j}'=0} . The equilibrium points of the system are located where all of the nullclines intersect.In a two-dimensional linear system, the nullclines can be represented by two lines on a two-dimensional plot; in a general two-dimensional system they are arbitrary curves. The definition, though with the name ’directivity curve’, was used in a 1967 article by Endre Simonyi. This article also defined 'directivity vector' as w = s i g n ( P ) i + s i g n ( Q ) j {displaystyle mathbf {w} =mathrm {sign} (P)mathbf {i} +mathrm {sign} (Q)mathbf {j} } ,where P and Q are the dx/dt and dy/dt differential equations, and i and j are the x and y direction unit vectors.