RFixedPoints
✖
RFixedPoints
gives the fixed points for a system of recurrence equations.
Details and Options

- Fixed points are also known as equilibrium points or stationary points.
- RFixedPoints is typically used to locate all fixed points for nonlinear discrete-time systems, such as frequently occur in ecological, economical or technical modeling. The local behavior at these fixed points can be analyzed using RStabilityConditions.
- For a system of recurrence equations
, a point
is a fixed point iff
. In effect, the initial value
remains stationary; if you initialize at
you stay at
.
- RFixedPoints returns a list of the form {{
,
,…},…}, where {
,
,…} is a fixed point.
- RFixedPoints works for linear and nonlinear ordinary difference equations.
- The following options can be given:
-
Assumptions $Assumptions assumptions on parameters
Examples
open allclose allBasic Examples (6)Summary of the most common use cases
Find the fixed point for the recursion :

https://wolfram.com/xid/0dc15a95hab6-dbs8eq

Find the fixed point for the recursion :

https://wolfram.com/xid/0dc15a95hab6-bq4m0r

Find the fixed points for the recursion :

https://wolfram.com/xid/0dc15a95hab6-h7t3wq

Check the stability of the point:

https://wolfram.com/xid/0dc15a95hab6-hoc84k

Plot several solutions for different values of a:

https://wolfram.com/xid/0dc15a95hab6-g84l4m


https://wolfram.com/xid/0dc15a95hab6-bgk4zg

Find the fixed points of a two-dimensional system:

https://wolfram.com/xid/0dc15a95hab6-h0gsn4

Determine the stability conditions:

https://wolfram.com/xid/0dc15a95hab6-k2qlxy

Plot the parameter region for which the system is stable:

https://wolfram.com/xid/0dc15a95hab6-crwdm8

Find the fixed points of a nonlinear recurrence equation:

https://wolfram.com/xid/0dc15a95hab6-fxnim2


https://wolfram.com/xid/0dc15a95hab6-e52jp

Use a cobweb plot to demonstrate the stability:

https://wolfram.com/xid/0dc15a95hab6-8k502


https://wolfram.com/xid/0dc15a95hab6-bnqjs2

https://wolfram.com/xid/0dc15a95hab6-c4rqcs

Calculate the differences and
to generate the vector field plot of the system:

https://wolfram.com/xid/0dc15a95hab6-rc5c4

Scope (14)Survey of the scope of standard use cases
Linear Equations (4)
Find the fixed point for the recursion :

https://wolfram.com/xid/0dc15a95hab6-eispst

A first-order linear inhomogeneous equation:

https://wolfram.com/xid/0dc15a95hab6-e3k2ad

Determine the stability condition:

https://wolfram.com/xid/0dc15a95hab6-hcf1vg


https://wolfram.com/xid/0dc15a95hab6-dp23pr


https://wolfram.com/xid/0dc15a95hab6-93mqi


https://wolfram.com/xid/0dc15a95hab6-h7d45b

Determine the stability conditions:

https://wolfram.com/xid/0dc15a95hab6-bf1260


https://wolfram.com/xid/0dc15a95hab6-bshskp


https://wolfram.com/xid/0dc15a95hab6-pi7f9

Determine the stability conditions:

https://wolfram.com/xid/0dc15a95hab6-iowjkv


https://wolfram.com/xid/0dc15a95hab6-dffej4

Nonlinear Equations (3)
The fixed points of a logistic equation:

https://wolfram.com/xid/0dc15a95hab6-ivrlkt

Analyze the stability of the points:

https://wolfram.com/xid/0dc15a95hab6-genyvv


https://wolfram.com/xid/0dc15a95hab6-de68w9


https://wolfram.com/xid/0dc15a95hab6-cjyz41

Use a cobweb plot to demonstrate the stability:

https://wolfram.com/xid/0dc15a95hab6-kmuphz

The fixed points of a Riccati equation:

https://wolfram.com/xid/0dc15a95hab6-djomsz

Use a cobweb plot to demonstrate the stability:

https://wolfram.com/xid/0dc15a95hab6-eh6snr


https://wolfram.com/xid/0dc15a95hab6-e393xb


https://wolfram.com/xid/0dc15a95hab6-e5v8oe

Linear Systems (5)
The fixed points of a linear system of uncoupled equations:

https://wolfram.com/xid/0dc15a95hab6-ifq44v

Linear system with constant coefficients:

https://wolfram.com/xid/0dc15a95hab6-cqld7o

Use VectorPlot to visualize the fixed point:

https://wolfram.com/xid/0dc15a95hab6-ox87oj

Solve the system with boundary conditions:

https://wolfram.com/xid/0dc15a95hab6-e1ny7c


https://wolfram.com/xid/0dc15a95hab6-b4yamt

A first-order system with periodic coefficients:

https://wolfram.com/xid/0dc15a95hab6-iz06a8

Compare to the general solution:

https://wolfram.com/xid/0dc15a95hab6-im5chv

A first-order system with eventually periodic coefficients:

https://wolfram.com/xid/0dc15a95hab6-bpqt4f

10x10 discrete linear system with random constant coefficients:

https://wolfram.com/xid/0dc15a95hab6-cbvzbz

https://wolfram.com/xid/0dc15a95hab6-dgl79g

https://wolfram.com/xid/0dc15a95hab6-d1ar62

https://wolfram.com/xid/0dc15a95hab6-h5ouwh

Nonlinear Systems (2)
A nonlinear first-order system:

https://wolfram.com/xid/0dc15a95hab6-deu221

Determine the stability of the fixed points:

https://wolfram.com/xid/0dc15a95hab6-ra86y

The fixed points and the stability of a linear fractional system:

https://wolfram.com/xid/0dc15a95hab6-fzjyp2


https://wolfram.com/xid/0dc15a95hab6-kcdszw

Use VectorPlot to visualize the stability at point :

https://wolfram.com/xid/0dc15a95hab6-cbon3n

Applications (8)Sample problems that can be solved with this function
Numerical Analyses (3)
Analyze the stability of the Newton–Raphson difference equation for the function x2-a:

https://wolfram.com/xid/0dc15a95hab6-b7xvb

https://wolfram.com/xid/0dc15a95hab6-hr7a8c


https://wolfram.com/xid/0dc15a95hab6-dla0zj

Visualize the stability of the fixed points for :

https://wolfram.com/xid/0dc15a95hab6-dgpopv

Analyze the stability of the Newton–Raphson difference equation for the function x1/3:

https://wolfram.com/xid/0dc15a95hab6-bvz2rl

https://wolfram.com/xid/0dc15a95hab6-b84f0s

The equation has one unstable fixed point at origin:

https://wolfram.com/xid/0dc15a95hab6-4vyxl


https://wolfram.com/xid/0dc15a95hab6-ioa3pt

The instability of the point means that Newton's method cannot be used in this case:

https://wolfram.com/xid/0dc15a95hab6-bxgvcw


https://wolfram.com/xid/0dc15a95hab6-en827f

Consider a system of linear equations , where:

https://wolfram.com/xid/0dc15a95hab6-bdmenu
Construct the Gauss–Seidel difference equation for the system:

https://wolfram.com/xid/0dc15a95hab6-eoonoh

https://wolfram.com/xid/0dc15a95hab6-bcaxde

https://wolfram.com/xid/0dc15a95hab6-fbllam

Find the fixed point of the Gauss–Seidel equation:

https://wolfram.com/xid/0dc15a95hab6-jysmlk

Solve the system using LinearSolve:

https://wolfram.com/xid/0dc15a95hab6-l6zqd

Physics (1)
Consider an object with temperature in the environment with constant temperature
. Let
be the change in temperature of the object over a time interval
. Newton's law of cooling states that the rate of change of the temperature of an object is proportional to the difference of the temperature of the object and its surroundings:

https://wolfram.com/xid/0dc15a95hab6-m83llk

https://wolfram.com/xid/0dc15a95hab6-hhi2c2

The fixed point is stable if :

https://wolfram.com/xid/0dc15a95hab6-ca395


https://wolfram.com/xid/0dc15a95hab6-mhlps


https://wolfram.com/xid/0dc15a95hab6-c5mhmb


https://wolfram.com/xid/0dc15a95hab6-faie64

Ecology and Biology (2)
Stability analysis for a competing species model:

https://wolfram.com/xid/0dc15a95hab6-ejunxb


https://wolfram.com/xid/0dc15a95hab6-eukrfi

Solve the equation for given initial conditions and plot the solution:

https://wolfram.com/xid/0dc15a95hab6-gc28iy

https://wolfram.com/xid/0dc15a95hab6-f9lvdk

Stability analysis for a predator-prey model:

https://wolfram.com/xid/0dc15a95hab6-bq6c6g


https://wolfram.com/xid/0dc15a95hab6-gvmpky

Vector field plot of the model:

https://wolfram.com/xid/0dc15a95hab6-cy1rmo

Use RecurrenceTable to solve the system numerically:

https://wolfram.com/xid/0dc15a95hab6-e2wz7u

https://wolfram.com/xid/0dc15a95hab6-mck818

Economics (2)
Consider a bank account with initial deposit , annual rate
and monthly withdrawal amount
. The amount in the savings account after
months satisfies a recurrence equation:

https://wolfram.com/xid/0dc15a95hab6-mrkpdj
The equation has unstable fixed point :

https://wolfram.com/xid/0dc15a95hab6-gyjsh9


https://wolfram.com/xid/0dc15a95hab6-cbukb6

The amount in the account will increase each month if :

https://wolfram.com/xid/0dc15a95hab6-g5pp3

https://wolfram.com/xid/0dc15a95hab6-jbr6fq

https://wolfram.com/xid/0dc15a95hab6-ccqd9

Stability analysis for the logistic equation:

https://wolfram.com/xid/0dc15a95hab6-fwss2y

Check the stability of the fixed points:

https://wolfram.com/xid/0dc15a95hab6-e8yrlu

Check the stability for given range of the parameter :

https://wolfram.com/xid/0dc15a95hab6-mq4v3


https://wolfram.com/xid/0dc15a95hab6-hiccia


https://wolfram.com/xid/0dc15a95hab6-cjtbgr

Properties & Relations (8)Properties of the function, and connections to other functions
RFixedPoints returns fixed points for recurrence equations:

https://wolfram.com/xid/0dc15a95hab6-ni5fw

Use RStabilityConditions to determine the stability for all fixed points of a recurrence equation:

https://wolfram.com/xid/0dc15a95hab6-iquooh


https://wolfram.com/xid/0dc15a95hab6-b0qy8d

Analyze the stability at specific fixed points:

https://wolfram.com/xid/0dc15a95hab6-qwl0x9


https://wolfram.com/xid/0dc15a95hab6-lt77g4

Use RFixedPoints to find all fixed points of a nonlinear recurrence equation:

https://wolfram.com/xid/0dc15a95hab6-cwtd8

Use Solve to find the fixed points:

https://wolfram.com/xid/0dc15a95hab6-rf7um

The fixed points for an n-order recurrence equation are n-dimensional vectors:

https://wolfram.com/xid/0dc15a95hab6-9rxw0

The fixed points for a system of n first-order recurrence equations are n-dimensional vectors:

https://wolfram.com/xid/0dc15a95hab6-pqjes6

Find the fixed points for a system of two ODEs:

https://wolfram.com/xid/0dc15a95hab6-dfvuec

Use RSolveValue to solve the system using fixed point as the initial condition:

https://wolfram.com/xid/0dc15a95hab6-bqt3d2

Use RSolveValue to solve the system for given initial conditions:

https://wolfram.com/xid/0dc15a95hab6-ew3s3y


https://wolfram.com/xid/0dc15a95hab6-doi666

Analyze the stability of a nonlinear ODE:

https://wolfram.com/xid/0dc15a95hab6-gaxuzv


https://wolfram.com/xid/0dc15a95hab6-clwfub

Solve the ODE using RecurrenceTable:

https://wolfram.com/xid/0dc15a95hab6-fjaa2


https://wolfram.com/xid/0dc15a95hab6-bleeat

Analyze the stability of a recurrence equation with eventually constant coefficients:

https://wolfram.com/xid/0dc15a95hab6-bfipbq


https://wolfram.com/xid/0dc15a95hab6-b79gga

Find a series solution using AsymptoticRSolveValue:

https://wolfram.com/xid/0dc15a95hab6-bh4spa


https://wolfram.com/xid/0dc15a95hab6-cxb1gl

Wolfram Research (2024), RFixedPoints, Wolfram Language function, https://reference.wolfram.com/language/ref/RFixedPoints.html.
Text
Wolfram Research (2024), RFixedPoints, Wolfram Language function, https://reference.wolfram.com/language/ref/RFixedPoints.html.
Wolfram Research (2024), RFixedPoints, Wolfram Language function, https://reference.wolfram.com/language/ref/RFixedPoints.html.
CMS
Wolfram Language. 2024. "RFixedPoints." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RFixedPoints.html.
Wolfram Language. 2024. "RFixedPoints." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RFixedPoints.html.
APA
Wolfram Language. (2024). RFixedPoints. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RFixedPoints.html
Wolfram Language. (2024). RFixedPoints. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RFixedPoints.html
BibTeX
@misc{reference.wolfram_2025_rfixedpoints, author="Wolfram Research", title="{RFixedPoints}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/RFixedPoints.html}", note=[Accessed: 05-June-2025
]}
BibLaTeX
@online{reference.wolfram_2025_rfixedpoints, organization={Wolfram Research}, title={RFixedPoints}, year={2024}, url={https://reference.wolfram.com/language/ref/RFixedPoints.html}, note=[Accessed: 05-June-2025
]}