mathematics//dynamical systems//equilibrium and stability

Where can the system stay, and what happens when it is perturbed there? An **equilibrium** of \(\dot x=f(x)\) satisfies \(f(x^*)=0\): starting exactly there produces a constant trajectory. That says nothing yet about the surroundings. With a constant input \(u^*\) the condition is \(f(x^*,u^*)=0\): a hovering drone needs thrust to cancel its weight, so equilibrium does not mean absence of input.

Where can the system stay, and what happens when it is perturbed there? An equilibrium of x˙=f(x)\dot x=f(x)x˙=f(x) satisfies f(x∗)=0f(x^)=0f(x∗)=0: starting exactly there produces a constant trajectory. That says nothing yet about the surroundings. With a constant input u∗u^u∗ the condition is f(x∗,u∗)=0f(x^,u^)=0f(x∗,u∗)=0: a hovering drone needs thrust to cancel its weight, so equilibrium does not mean absence of input.

Stability answers another question. An equilibrium is stable in the sense of Lyapunov if sufficiently close initial conditions stay close. It is asymptotically stable if they also converge to it, and exponentially stable if the distance is bounded by a decaying exponential. It is unstable if arbitrarily small perturbations end up moving away. A saddle mixes attracting and repelling directions. "Neutral" is often used for stability without return, but it is better to state the exact property than to read it off a zero eigenvalue.

Local, regional and global. "Local" restricts initial conditions to a neighborhood; "global" refers to the whole declared domain. For x˙=rx−γx2\dot x=rx-\gamma x^2x˙=rx−γx2, with r,γ>0r,\gamma>0r,γ>0, the positive equilibrium attracts every x(0)>0x(0)>0x(0)>0 but not x(0)=0x(0)=0x(0)=0, which stays at another equilibrium. In a closed diffusive network (diffusion and consensus) there is a whole family of constant equilibria: the dynamics removes disagreement but does not return to the same common value after a perturbation that changes the total amount.

Attractor and periodic regime. An attractor is an invariant set that attracts a region of initial conditions, under the adopted definition. It can be an equilibrium or a periodic orbit. A limit cycle is an isolated periodic orbit of an autonomous system; not every limit cycle is attracting. A response with x(t+T)=x(t)x(t+T)=x(t)x(t+T)=x(t) forced by a periodic input is neither a fixed equilibrium nor automatically a limit cycle: holding a point and following a trajectory are different control goals (feedback control).

Chaos is not any instability. Deterministic chaos combines nontrivial evolution and sensitivity to initial conditions; in the usual reading, bounded aperiodic trajectories with exponential separation over some interval, plus specific formal conditions. A Lyapunov exponent of a trajectory is not an instantaneous eigenvalue of the Jacobian. x˙=x\dot x=xx˙=x is unstable without being chaotic; x˙=−x−x3\dot x=-x-x^3x˙=−x−x3 is nonlinear and convergent. Nonlinearity, instability and chaos are different properties.

The linearization supplies local tests; a Lyapunov function can widen the conclusion; discretization separates the behavior of the model from errors of the algorithm.