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The stability of fixed points of a system of constant coefficient linear differential equations of first order can be analyzed using the eigenvalues of the corresponding matrix. An autonomous system ′ =, where x(t) ∈ R n and A is an n×n matrix with real entries, has a constant solution =
The appearance or the disappearance of a periodic orbit through a local change in the stability properties of a fixed point is known as the Hopf bifurcation. The following theorem works for fixed points with one pair of conjugate nonzero purely imaginary eigenvalues. It tells the conditions under which this bifurcation phenomenon occurs.
In mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by small perturbations (to be exact C 1-small perturbations). Examples of such qualitative properties are numbers of fixed points and periodic orbits (but not their periods
In continuous systems, this corresponds to the real part of an eigenvalue of an equilibrium passing through zero. In discrete systems (described by maps), this corresponds to a fixed point having a Floquet multiplier with modulus equal to one. In both cases, the equilibrium is non-hyperbolic at the bifurcation point. The topological changes in ...
The system's fixed points are represented by where the phase portrait curve crosses the x-axis. 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).
In many fields, equilibria or stability are fundamental concepts that can be described in terms of fixed points. Some examples follow. In projective geometry, a fixed point of a projectivity has been called a double point. [8] [9] In economics, a Nash equilibrium of a game is a fixed point of the game's best response correspondence.
On the Lémeray diagram, a stable fixed point corresponds to the segment of the staircase with progressively decreasing stair lengths or to an inward spiral, while an unstable fixed point is the segment of the staircase with growing stairs or an outward spiral.
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 .