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Asymptotic stability of fixed points of a non-linear system can often be established using the Hartman–Grobman theorem. Suppose that v is a C 1 - vector field in R n which vanishes at a point p , v ( p ) = 0 .
A fixed-point theorem is a result saying that at least one fixed point exists, under some general condition. [ 1 ] For example, the Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, fixed-point iteration will always converge to a fixed point.
More accurately, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues—of the linearization around the fixed point—crosses the complex plane imaginary axis as a parameter crosses a threshold value.
The stability of a given fixed point can be determined by the direction of flow on the x-axis; for instance, in Figure 2, the green point is unstable (divergent flow), and the red one is stable (convergent flow). At first, when r is greater than 0, the system has
A ball located at this point, ball 3, is in equilibrium but unstable; the slightest disturbance will cause it to move to one of the stable points. Light switch, a bistable mechanism. In a dynamical system, bistability means the system has two stable equilibrium states. [1] A bistable structure can be resting in either of two states.
Let : be a smooth map with hyperbolic fixed point at .We denote by () the stable set and by () the unstable set of .. The theorem [2] [3] [4] states that is a smooth manifold and its tangent space has the same dimension as the stable space of the linearization of at .
The definition for discrete-time systems is almost identical to that for continuous-time systems. The definition below provides this, using an alternate language commonly used in more mathematical texts. Let (X, d) be a metric space and f : X → X a continuous function. A point x in X is said to be Lyapunov stable, if,
For r < 1, exists outside [0, 1] as an unstable fixed point, but for r = 1, the two fixed points collide, and for r > 1, appears between [0, 1] as a stable fixed point. When the parameter r = 1, the trajectory of the logistic map converges to 0 as before , but the convergence speed is slower at r = 1 .