Inmathematics, specifically indifferential equations, anequilibrium point is a constant solution to a differential equation.

The point${\displaystyle \tilde{\mathbf{x}}\in {\mathbb{R}}^n}$ is anequilibrium point for thedifferential equation

${\displaystyle \frac{d\mathbf{x}}{dt}=\mathbf{f}(t,\mathbf{x})}$

if${\displaystyle \mathbf{f}(t,\tilde{\mathbf{x}})=\mathbf{0}}$ for all${\displaystyle t}$.

Similarly, the point${\displaystyle \tilde{\mathbf{x}}\in {\mathbb{R}}^n}$ is anequilibrium point (orfixed point) for thedifference equation

${\mathbf{x}}_{k+1}=\mathbf{f}(k,{\mathbf{x}}_k)$

if${\displaystyle \mathbf{f}(k,\tilde{\mathbf{x}})=\tilde{\mathbf{x}}}$ for${\displaystyle k=0,1,2,\dots }$.

Equilibria can be classified by looking at the signs of theeigenvalues of the linearization of the equations about the equilibria. That is to say, by evaluating theJacobian matrix at each of the equilibrium points of the system, and then finding the resulting eigenvalues, the equilibria can be categorized. Then the behavior of the system in the neighborhood of each equilibrium point can be qualitatively determined, (or even quantitatively determined, in some instances), by finding the eigenvector(s) associated with each eigenvalue.

An equilibrium point ishyperbolic if none of the eigenvalues have zero real part. If all eigenvalues have negative real parts, the point isstable. If at least one has a positive real part, the point isunstable. If at least one eigenvalue has negative real part and at least one has positive real part, the equilibrium is asaddle point and it is unstable. If all the eigenvalues are real and have the same sign the point is called anode.

• Autonomous equation
• Critical point
• Steady state

• Boyce, William E.; DiPrima, Richard C. (2012).Elementary Differential Equations and Boundary Value Problems (10th ed.). Wiley.ISBN 978-0-470-45831-0.
• Perko, Lawrence (2001).Differential Equations and Dynamical Systems (3rd ed.). Springer. pp. 102–104.ISBN 1-4613-0003-7.

• Stability theory
• Dynamical systems

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