DStabilityConditions
DStabilityConditions[eqn,x[t],t]
gives the fixed points and stability conditions for a differential equation.
DStabilityConditions[{eqn_{1},eqn_{2},…},{x_{1}[t],x_{2}[t],…},t]
gives the fixed points and stability conditions for a system of differential equations.
DStabilityConditions[{eqn_{1},eqn_{2},…},{x_{1}[t],x_{2}[t],…},t,{pnt_{1},pnt_{2},…}]
gives the stability conditions for the given fixed points.
Details and Options
 Stability is also known as asymptotic stability, and fixed points are also known as equilibrium points or stationary points.
 DStabilityConditions is typically used to qualitatively analyze longterm behavior near fixed points. If the system is stable, then solutions converge to the fixed point if you are close enough.
 For a system of differential equations , a point is a fixed point iff . In effect, the initial value remains stationary; if you initialize at you stay at .
 A fixed point is asymptotically stable iff for and you have for sufficiently small.
 DStabilityConditions returns a list of the form {{{,,…},cond},…}, where {,,…} is a fixed point.
 DStabilityConditions gives sufficient conditions for local stability of fixed points. For linear systems, these conditions are also conditions for global stability.
 DStabilityConditions works for linear and nonlinear ordinary differential equations.
 The following options can be given:

Assumptions $Assumptions assumptions on parameters
Examples
open allclose allBasic Examples (7)
Find the fixed point and determine its stability for the equation :
Find the fixed point and determine the stability for the equation :
Find the fixed points and conditions for stability for the equation :
Plot several solutions for different values of a:
Stability analysis of a twodimensional system:
Plot the parameter region for which the system is stable:
Stability analysis of a nonlinear differential equation:
Use StreamPlot to demonstrate the stability:
The stability of a linear system with constant coefficients:
Use StreamPlot to visualize the stability:
Find the fixed points for a nonlinear equation:
Scope (21)
Linear Equations (5)
Find the fixed point and determine its stability for the equation :
A firstorder linear inhomogeneous equation:
Plot the unstable solution for :
Plot the stable solution for :
Plot the stability region for parameters and :
Higherorder inhomogeneous ODE:
Solve the ODE using coordinates of the fixed point as initial values:
Nonlinear Equations (2)
The stability of a firstorder nonlinear equation:
Plot the solution for the initial value :
Use StreamPlot to demonstrate the stability at :
Consider a secondorder nonlinear ODE:
DSolve is unable to solve this equation:
Analyze the stability of the equation using DStabilityConditions:
Linear Systems (10)
A stable linear system of uncoupled equations:
An unstable linear system of uncoupled equations:
Unstable system with constant coefficients:
Stable system with constant coefficients:
A firstorder system with imaginary eigenvalues:
Use StreamPlot to visualize the stability:
Inhomogeneous unstable system:
Linear system with symbolic coefficients:
Use Assumptions to simplify the stability conditions:
Analyze the stability of a 10×10 linear system with random constant coefficients:
Nonlinear Systems (4)
A nonlinear firstorder system:
Use StreamPlot to visualize the stability:
A nonlinear system with periodic fixed points:
A nonlinear system with unstable fixed point at origin:
Options (2)
Assumptions (2)
Without Assumptions, there are conditions on parameters for stability:
Using Assumptions can often result in simplified conditions:
A system of two nonlinear equations has an infinite number of periodic fixed points:
Use Assumptions to specify the range of a dependent variable:
Applications (11)
Physics (5)
Do stability analysis for the springmass system with damping:
Use assumptions to simplify the stability conditions:
Solve the springmass system equation:
Plot the solution for given values of parameters:
Do stability analysis for the electric circuit equation:
Solve the electric circuit equation:
Plot the solution for given values of parameters:
Stability analysis for the damped pendulum equation:
Plot the phase portrait of the system:
Plot the solution for the initial conditions , :
Stable system of Lorenz equations:
Use StreamPlot3D to visualize the Lorenz attractors:
Biology and Ecology (3)
Stability analysis for the predatorprey model (Lotka–Volterra equation):
Plot the phase portrait of the system:
Solve the system for the initial conditions , :
The Rosenzweig–MacArthur predatorprey model:
The chemostat model represents biological systems in which microorganisms grow on abiotic resources:
Chemistry (1)
Control Systems (2)
Analyze a satellite's attitude dynamics starting from Euler's equations of motion:
Euler’s equations with principal moments of inertia , , :
Analyze the stability of the equation for fixed values of , , :
Choose the fixed point as an operating point:
Construct a statespace model:
The satellite’s attitude is unregulated if disturbed:
Verify the controllability of the model:
Study an inverted pendulum using the Lagrangian:
The kinetic energy of the cart and pendulum:
Properties & Relations (9)
DStabilityConditions returns fixed points and stability conditions for differential equations:
Use DFixedPoints to find all fixed points of a differential equation:
Analyze the stability at specific fixed points:
Use DFixedPoints to find all fixed points of a nonlinear ODE:
Use Solve to find the fixed points:
Linearize the equation near the first fixed point:
Check the stability near the first fixed point:
Linearize the equation near the second fixed point:
Check the stability near the second fixed point:
Determine the stability of a nonlinear equation using DStabilityConditions:
The fixed points for the norder differential equation are ndimensional vectors:
The fixed points for the system of n firstorder differential equations are ndimensional vectors:
Analyze the stability of a system of two ODEs:
Use DSolveValue to solve the system using a fixed point as initial condition:
Use DSolveValue to solve the system for given initial conditions:
Analyze the stability of a nonlinear ODE:
Solve the ODE using NDSolve:
Analyze the stability of a nonlinear ODE:
Calculate an asymptotic solution of the ODE using the first fixed point as initial condition:
Calculate an asymptotic solution of the ODE for another initial condition:
Find the fixed points for the system of two nonlinear ODEs:
Calculate the Jacobian matrix of the system:
Calculate the eigenvalues of the Jacobian matrix for each fixed point:
The system is locally stable near the fixed point if all of the eigenvalues have negative real parts:
Check the stability of the points using DStabilityConditions:
Possible Issues (2)
Sometimes the conditions for stability are not the simplest possible:
Additional simplification can be achieved by further processing:
DStabilityConditions fails because the given point is not a fixed point:
Use DFixedPoints to find all fixed points of the equation first:
Neat Examples (2)
The van der Pol oscillator is a nonconservative, oscillating system with nonlinear damping:
Analyze the stability of the system:
Animate the trajectories of the system for various values of the parameter :
The FitzHugh–Nagumo model is an example of a relaxation oscillator:
If the external stimulus s exceeds a certain threshold value, the system will exhibit a characteristic excursion in phase space, before the variables x and y relax back to their rest values.
Visualize the trajectories of the system for various values of the parameter s:
Text
Wolfram Research (2024), DStabilityConditions, Wolfram Language function, https://reference.wolfram.com/language/ref/DStabilityConditions.html.
CMS
Wolfram Language. 2024. "DStabilityConditions." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DStabilityConditions.html.
APA
Wolfram Language. (2024). DStabilityConditions. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DStabilityConditions.html