mathematics//dynamical systems//equilibrium and stability

Equilibrium and stability say where a system can stay and what happens when it is perturbed there. An **equilibrium point** of \(\dot x=f(x,u)\) is a state and an input with \(f(x^*,u^*)=0\): started exactly there, the system stays put. A hovering drone needs thrust equal to its weight, so equilibrium does not mean absence of input, and it is the point every linearization starts from. An equilibrium is stable if, after a small push, the system comes back; that is the stability that matters in control.

Equilibrium and stability say where a system can stay and what happens when it is perturbed there. An equilibrium point of x˙=f(x,u)\dot x=f(x,u)x˙=f(x,u) is a state and an input with f(x∗,u∗)=0f(x^*,u^*)=0f(x∗,u∗)=0: started exactly there, the system stays put. A hovering drone needs thrust equal to its weight, so equilibrium does not mean absence of input, and it is the point every linearization starts from. An equilibrium is stable if, after a small push, the system comes back; that is the stability that matters in control.

The precise words. An equilibrium is stable in the sense of Lyapunov if sufficiently close initial conditions stay close, 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 (repeller); a saddle mixes attracting and repelling directions. For a linear or linearized system the test is the eigenvalues: all with negative real part, asymptotically stable; any with positive real part, unstable.

λ−0.30 ± 1.38i verdictStable spiral A unit mass on a spring of stiffness 2.00 with damping 0.60, drawn in the plane of position and velocity. Complex pair in the left half: it oscillates and the swing dies out.

Lower the damping through zero and watch the pair of eigenvalues cross the imaginary axis: the spiral that wound inwards becomes a closed orbit, then a spiral that winds out. Make the stiffness negative for a saddle, and tap the plane anywhere to start a trajectory from that state.

The eigenvalues tell the fate.

Each λ=σ+jω\lambda=\sigma+j\omegaλ=σ+jω is a mode: a negative real part dies away with time constant −1/σ-1/\sigma−1/σ, a positive one explodes, the imaginary part oscillates at ω\omegaω rad/s, and the slowest mode, the one closest to the imaginary axis, dominates what is seen in the long run (eigenvalue, modes).

Feedback moves the eigenvalues, which is its power and its danger. It can stabilize the unstable (a drone in hover does not return to its spot by itself) and destabilize the stable (a shower does not oscillate until someone tries to regulate it). A PD on a drone's altitude gives mλ2+Kdλ+Kp=0m\lambda^2+K_d\lambda+K_p=0mλ2+Kd​λ+Kp​=0, stable for any positive gains; adding integral action raises the degree to mλ3+Kdλ2+Kpλ+Ki=0m\lambda^3+K_d\lambda^2+K_p\lambda+K_i=0mλ3+Kd​λ2+Kp​λ+Ki​=0, stable only while KdKp>mKiK_dK_p>mK_iKd​Kp​>mKi​: too much integral destabilizes, and derivative action is what makes room for it (PID controller).

Local, regional and global. A local property restricts initial conditions to a neighbourhood, a global one covers 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; in a closed diffusive network (diffusion and consensus) there is a whole family of constant equilibria, and the dynamics removes disagreement without returning to the same common value.

An attractor can be an equilibrium or a periodic orbit (attractors are how emergence shows up in a state space). A limit cycle is an isolated periodic orbit of an autonomous system; a response forced by a periodic input is neither a fixed equilibrium nor automatically a limit cycle, and holding a point and following a trajectory are different control goals.

Nonlinearity, instability and chaos are different properties: 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, and a chaotic system stays bounded while it loses predictability.

In discrete time the test becomes the unit circle (discrete-time stability); for two states the trace-determinant plane reads it at a glance; a Lyapunov function widens a local conclusion, and discretization separates the behaviour of the model from errors of the algorithm.