An equilibrium point, also known as a fixed point or steady state, represents a condition within a dynamic system where all rates of change are exactly zero. If a system begins at an equilibrium point, it will remain there indefinitely unless an external force acts upon it. Dynamic systems are often described by differential equations, which model how a variable changes over time. Analyzing these points is a fundamental practice across fields like physics, engineering, ecology, and economics, as they reveal the long-term behavior of the modeled system.
Finding Equilibrium in Single-Variable Equations
The simplest dynamic systems involve only one variable, modeled by a single ordinary differential equation (ODE) in the form of \(dx/dt = f(x)\). The mathematical procedure to locate an equilibrium point in this scenario is straightforward and relies on the definition that the rate of change must be zero.
To begin the calculation, the derivative term, \(dx/dt\), is formally set to zero. This action transforms the dynamic equation into a purely algebraic equation: \(f(x) = 0\). The next step involves solving this new algebraic equation for the values of \(x\). These resulting values are the system’s equilibrium points.
The algebraic solutions for \(x\) may yield zero, one, or multiple distinct equilibrium points. This technique fundamentally converts a calculus problem concerning rates into a more familiar algebra problem concerning roots.
Calculating Equilibrium Points for Systems
Finding equilibrium points becomes more complex when dealing with coupled systems involving multiple interacting variables. Such systems are modeled by a set of differential equations, where the rate of change of each variable depends on the values of all the other variables. For a two-variable system, this structure typically takes the form of \(dx/dt = f(x, y)\) and \(dy/dt = g(x, y)\).
For the entire system to be at rest, the rate of change for every variable must be zero simultaneously. This requires setting both \(dx/dt\) and \(dy/dt\) equal to zero. The calculation thus yields a system of simultaneous algebraic equations: \(f(x, y) = 0\) and \(g(x, y) = 0\). The equilibrium points are the \((x, y)\) coordinates that satisfy both equations.
Solving this simultaneous system often involves algebraic manipulation, such as substitution or elimination. For example, one might solve the second equation for \(y\) in terms of \(x\), and then substitute that expression into the first equation. This substitution step reduces the problem to a single-variable algebraic equation, which can be solved for \(x\). Once the values for \(x\) are found, they are plugged back into the substitution expression to determine the corresponding values for \(y\). Because the original differential equations are often non-linear, solving the simultaneous algebraic equations can be challenging and may lead to multiple solutions. Each resulting coordinate pair \((x, y)\) represents a distinct equilibrium point.
Determining the Stability of Equilibrium Points
Once all equilibrium points have been located, the next step is to determine their stability. Stability refers to the system’s behavior when it experiences a small disturbance away from the equilibrium point. A point is considered stable if the system returns to it after a small perturbation, and unstable if the system moves further away.
Single-Variable Stability
For single-variable equations, stability can be determined using a simple derivative test, which involves examining the sign of the function’s derivative \(f'(x)\) at the equilibrium point. If \(f'(x)\) is negative, the point is stable. Conversely, if \(f'(x)\) is positive, the point is unstable, causing the system to move away from the equilibrium. A derivative of zero at the point indicates a more complex stability scenario that requires additional analysis.
Multi-Variable Stability
For multi-variable systems, stability analysis requires a more advanced technique called linearization, which involves constructing the Jacobian matrix. The Jacobian matrix is a square matrix composed of the first-order partial derivatives of all the rate functions evaluated at the specific equilibrium point.
The stability of the equilibrium point is then determined by calculating the eigenvalues of the Jacobian matrix. If the real part of every eigenvalue is negative, the equilibrium point is stable. If at least one eigenvalue has a positive real part, the equilibrium point is unstable. This procedure allows the classification of fixed points as attractors, repellers, or saddle points.