Lyapunov–Malkin theorem
The Lyapunov–Malkin theorem (named for Aleksandr Lyapunov and Ioel Malkin) is a mathematical theorem detailing stability of nonlinear systems. == Theorem == In the system of differential equations, x ˙ = A x + X ( x , y ) , y ˙ = Y ( x , y ) {\displaystyle {\dot {x}}=Ax+X(x,y),\quad {\dot {y}}=Y(x,y)} where x ∈ R m {\displaystyle x\in \mathbb {R} ^{m}} and y ∈ R n {\displaystyle y\in \mathbb {R} ^{n}} are components of the system state, A ∈ R m × m {\displaystyle A\in \mathbb {R} ^{m\times m}} is a matrix that represents the linear dynamics of x {\displaystyle x} , and X : R m × R n → R m {\displaystyle X:\mathbb {R} ^{m}\times \mathbb {R} ^{n}\to \mathbb {R} ^{m}} and Y : R m × R n → R n {\displaystyle Y:\mathbb {R} ^{m}\times \mathbb {R} ^{n}\to \mathbb {R} ^{n}} represent higher-order nonlinear terms.