Nullcline

In mathematical analysis, nullclines, sometimes called zero-growth isoclines, are encountered in a system of ordinary differential equations : x_1'=f_1(x_1, \ldots, x_n) : x_2'=f_2(x_1, \ldots, x_n) :: \vdots : x_n'=f_n(x_1, \ldots, x_n) where x' here represents a derivative of x with respect to another parameter, such as time t . The j 'th nullcline is the geometric shape for which 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. Nullclines are useful for visualization in phase plane plot analysis. Nullclines split the plot into regions of potentially similar dynamics .

In mathematical analysis, nullclines, sometimes called zero-growth isoclines, are encountered in a system of ordinary differential equations x_1'=f_1(x_1, \ldots, x_n) x_2'=f_2(x_1, \ldots, x_n) \vdots x_n'=f_n(x_1, \ldots, x_n) where x' here represents a derivative of x with respect to another parameter, such as time t . The j 'th nullcline is the geometric shape for which 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. Nullclines are useful for visualization in phase plane plot analysis. Nullclines split the plot into regions of potentially similar dynamics . \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. Simonyi developed a new stability test method from these new definitions, and with it he studied differential equations. This method, beyond the usual stability examinations, provided semi-quantitative results. ==References== ==Notes== E. Simonyi – M. Kaszás: Method for the Dynamic Analysis of Nonlinear Systems, Periodica Polytechnica Chemical Engineering – Chemisches Ingenieurwesen, Polytechnical University Budapest, 1969 ==External links== SOS Mathematics: Qualitative Analysis

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