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In mathematics, a phase portrait is a geometric representation of the orbits of a dynamical system in the phase plane. Each set of initial conditions is represented by a different point or curve. Phase portraits are an invaluable tool in studying dynamical systems. They consist of a plot of typical trajectories in the phase space.
Phase portrait showing saddle-node bifurcation. Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations.
In mathematics, a phase portrait is a geometric representation of the orbits of a dynamical system in the phase plane. Each set of initial conditions is represented by a different point or curve. Phase portraits are an invaluable tool in studying dynamical systems. They consist of a plot of typical trajectories in the phase space.
Phase portrait of the Van der Pol oscillator, a one-dimensional system. Phase space was the original object of study in symplectic geometry.. Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form.
In applied mathematics, in particular the context of nonlinear system analysis, a phase plane is a visual display of certain characteristics of certain kinds of differential equations; a coordinate plane with axes being the values of the two state variables, say (x, y), or (q, p) etc. (any pair of variables).
The parameters in the above equation are: controls the amount of damping,; controls the linear stiffness,; controls the amount of non-linearity in the restoring force; if =, the Duffing equation describes a damped and driven simple harmonic oscillator,
As can be seen by the animation obtained by plotting phase portraits by varying the parameter , When α {\displaystyle \alpha } is negative, there are no equilibrium points. When α = 0 {\displaystyle \alpha =0} , there is a saddle-node point.
Phase portraits (p vs. x) of the classical kicked rotor at different kicking strengths. The top row shows, from left to right, K = 0.5, 0.971635, 1.3. The bottom row shows, from left to right, K = 2.1, 5.0, 10.0. The phase portrait at the chaotic boundary is the upper middle plot, with K C = 0.971635.