DFixedPoints
DFixedPoints[eqn,x[t],t]
gives the fixed points for a differential equation.
DFixedPoints[{eqn1,eqn2,…},{x1[t],x2[t],…},t]
gives the fixed points for a system of differential equations.
Details and Options
![](Files/DFixedPoints.en/details_1.png)
- Fixed points are also known as stationary points or equilibrium points of the differential equation.
- DFixedPoints is typically used to locate all fixed points for nonlinear continuous-time systems that frequently occur in ecological, economical or technical modeling. The local behavior at these fixed points can be analyzed using DStabilityConditions.
- 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
.
- DFixedPoints returns a list of the form {{
,
,…},…}, where {
,
,…} is a fixed point of the system.
- DFixedPoints works for linear and nonlinear ordinary differential equations.
- The following options can be given:
-
Assumptions $Assumptions assumptions on parameters
Examples
open allclose allBasic Examples (6)
Find the fixed point for the equation :
Find the fixed point for the equation :
Find the fixed points and conditions for stability for the equation :
Plot several solutions for different values of a:
Fixed points and stability analysis of a two-dimensional 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:
Scope (19)
Linear Equations (5)
Nonlinear Equations (2)
The stability of a first-order nonlinear equation:
Plot the solution for the initial value :
Use StreamPlot to demonstrate the stability at point :
Consider a second-order nonlinear ODE:
DSolve is unable to solve this equation:
Find the fixed points for the equation using DFixedPoints:
Linear Systems (9)
Nonlinear Systems (3)
A nonlinear first-order system:
Analyze the stability of the points:
Use StreamPlot to visualize the stability:
A nonlinear system with periodic fixed points:
Options (1)
Assumptions (1)
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)
Stability analysis for the spring-mass system with damping:
Use assumptions to simplify the stability conditions:
Solve the spring-mass 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 (3)
Stability analysis for the predator-prey model (Lotka–Volterra equations):
Plot the phase portrait of the system:
Solve the system for the initial conditions ,
:
The Rosenzweig–MacArthur predator-prey 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:
![](Files/DFixedPoints.en/Image_26.gif)
Euler’s equations with principal moments of inertia ,
,
:
Find the fixed points of the equation for fixed values of ,
,
:
Choose the fixed point as an operating point:
Construct a state-space model:
The satellite’s attitude is unregulated if disturbed:
Verify the controllability of the model:
Study an inverted pendulum using the Lagrangian:
![](Files/DFixedPoints.en/Image_33.gif)
The kinetic energy of the cart and pendulum:
Properties & Relations (8)
DFixedPoints returns fixed points for differential equations:
Use DFixedPoints to find all fixed points of a differential equation:
Use DStabilityConditions to 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:
The fixed points for the n-order differential equation are n-dimensional vectors:
The fixed points for the system of n first-order differential equations are n-dimensional vectors:
Find the fixed points of a system of two ODEs:
Use DSolveValue to solve the system using the fixed point as initial condition:
Use DSolveValue to solve the system for given initial conditions:
Analyze the fixed points of a nonlinear ODE:
Solve the ODE using NDSolve:
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 if all of the eigenvalues have negative real parts:
Check the stability of the points using DStabilityConditions:
Text
Wolfram Research (2024), DFixedPoints, Wolfram Language function, https://reference.wolfram.com/language/ref/DFixedPoints.html.
CMS
Wolfram Language. 2024. "DFixedPoints." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DFixedPoints.html.
APA
Wolfram Language. (2024). DFixedPoints. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DFixedPoints.html