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RFixedPoints
✖
RFixedPoints

New in 14.1

✖
RFixedPoints[eqn,a[n],n]

gives the fixed points for a recurrence equation.

✖
RFixedPoints[{eqn1,eqn2,…},{a1[n],a2[n],…},n]

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$Assumptionsassumptions on parameters

Examples

open allclose all

Basic Examples  (6)Summary of the most common use cases

Find the fixed point for the recursion :

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-dbs8eq
Out[1]=1

Find the fixed point for the recursion :

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-bq4m0r
Out[1]=1

Find the fixed points for the recursion :

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-h7t3wq
Out[1]=1

Check the stability of the point:

In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-hoc84k
Out[2]=2

Plot several solutions for different values of a:

In[3]:=3
✖
https://wolfram.com/xid/0dc15a95hab6-g84l4m
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0dc15a95hab6-bgk4zg
Out[4]=4

Find the fixed points of a two-dimensional system:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-h0gsn4
Out[1]=1

Determine the stability conditions:

In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-k2qlxy
Out[2]=2

Plot the parameter region for which the system is stable:

In[3]:=3
✖
https://wolfram.com/xid/0dc15a95hab6-crwdm8
Out[3]=3

Find the fixed points of a nonlinear recurrence equation:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-fxnim2
Out[1]=1

Determine the stability:

In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-e52jp
Out[2]=2

Use a cobweb plot to demonstrate the stability:

In[3]:=3
✖
https://wolfram.com/xid/0dc15a95hab6-8k502
Out[3]=3

Consider a system , :

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-bnqjs2
In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-c4rqcs
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0dc15a95hab6-rc5c4
Out[3]=3

Scope  (14)Survey of the scope of standard use cases

Linear Equations  (4)

Find the fixed point for the recursion :

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-eispst
Out[1]=1

A first-order linear inhomogeneous equation:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-e3k2ad
Out[1]=1

Determine the stability condition:

In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-hcf1vg
Out[2]=2

Plot the solution for :

In[3]:=3
✖
https://wolfram.com/xid/0dc15a95hab6-dp23pr
Out[3]=3

Plot the solution for :

In[4]:=4
✖
https://wolfram.com/xid/0dc15a95hab6-93mqi
Out[4]=4

Second-order linear equation:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-h7d45b
Out[1]=1

Determine the stability conditions:

In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-bf1260
Out[2]=2

Plot the stability region:

In[3]:=3
✖
https://wolfram.com/xid/0dc15a95hab6-bshskp
Out[3]=3

Third-order linear equation:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-pi7f9
Out[1]=1

Determine the stability conditions:

In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-iowjkv
Out[2]=2

Plot the stability region:

In[3]:=3
✖
https://wolfram.com/xid/0dc15a95hab6-dffej4
Out[3]=3

Nonlinear Equations  (3)

The fixed points of a logistic equation:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-ivrlkt
Out[1]=1

Analyze the stability of the points:

In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-genyvv
Out[2]=2

Plot the solution:

In[3]:=3
✖
https://wolfram.com/xid/0dc15a95hab6-de68w9
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0dc15a95hab6-cjyz41
Out[4]=4

Use a cobweb plot to demonstrate the stability:

In[5]:=5
✖
https://wolfram.com/xid/0dc15a95hab6-kmuphz
Out[5]=5

The fixed points of a Riccati equation:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-djomsz
Out[1]=1

Use a cobweb plot to demonstrate the stability:

In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-eh6snr
Out[2]=2

Higher-order equations:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-e393xb
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-e5v8oe
Out[2]=2

Linear Systems  (5)

The fixed points of a linear system of uncoupled equations:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-ifq44v
Out[1]=1

Linear system with constant coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-cqld7o
Out[1]=1

Use VectorPlot to visualize the fixed point:

In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-ox87oj
Out[2]=2

Solve the system with boundary conditions:

In[3]:=3
✖
https://wolfram.com/xid/0dc15a95hab6-e1ny7c
Out[3]=3

Plot the solution:

In[4]:=4
✖
https://wolfram.com/xid/0dc15a95hab6-b4yamt
Out[4]=4

A first-order system with periodic coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-iz06a8
Out[1]=1

Compare to the general solution:

In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-im5chv
Out[2]=2

A first-order system with eventually periodic coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-bpqt4f
Out[1]=1

10x10 discrete linear system with random constant coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-cbvzbz
In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-dgl79g
In[3]:=3
✖
https://wolfram.com/xid/0dc15a95hab6-d1ar62
In[4]:=4
✖
https://wolfram.com/xid/0dc15a95hab6-h5ouwh
Out[4]=4

Nonlinear Systems  (2)

A nonlinear first-order system:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-deu221
Out[1]=1

Determine the stability of the fixed points:

In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-ra86y
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-fzjyp2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-kcdszw
Out[2]=2

Use VectorPlot to visualize the stability at point :

In[3]:=3
✖
https://wolfram.com/xid/0dc15a95hab6-cbon3n
Out[3]=3

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:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-b7xvb
In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-hr7a8c
Out[2]=2

Find fixed points:

In[3]:=3
✖
https://wolfram.com/xid/0dc15a95hab6-dla0zj
Out[3]=3

Visualize the stability of the fixed points for :

In[4]:=4
✖
https://wolfram.com/xid/0dc15a95hab6-dgpopv
Out[6]=6

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

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-bvz2rl
In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-b84f0s
Out[2]=2

The equation has one unstable fixed point at origin:

In[3]:=3
✖
https://wolfram.com/xid/0dc15a95hab6-4vyxl
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0dc15a95hab6-ioa3pt
Out[4]=4

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

In[5]:=5
✖
https://wolfram.com/xid/0dc15a95hab6-bxgvcw
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0dc15a95hab6-en827f

Consider a system of linear equations , where:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-bdmenu

Construct the Gauss–Seidel difference equation for the system:

In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-eoonoh
In[3]:=3
✖
https://wolfram.com/xid/0dc15a95hab6-bcaxde
In[4]:=4
✖
https://wolfram.com/xid/0dc15a95hab6-fbllam
Out[4]=4

Find the fixed point of the Gauss–Seidel equation:

In[5]:=5
✖
https://wolfram.com/xid/0dc15a95hab6-jysmlk
Out[5]=5

Solve the system using LinearSolve:

In[6]:=6
✖
https://wolfram.com/xid/0dc15a95hab6-l6zqd
Out[6]=6

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:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-m83llk

Find the fixed points:

In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-hhi2c2
Out[2]=2

The fixed point is stable if :

In[3]:=3
✖
https://wolfram.com/xid/0dc15a95hab6-ca395
Out[3]=3

Solution of the equation:

In[4]:=4
✖
https://wolfram.com/xid/0dc15a95hab6-mhlps
Out[4]=4

Simulate the cooling process:

In[5]:=5
✖
https://wolfram.com/xid/0dc15a95hab6-c5mhmb
Out[5]=5

Simulate the heating process:

In[6]:=6
✖
https://wolfram.com/xid/0dc15a95hab6-faie64
Out[6]=6

Ecology and Biology  (2)

Stability analysis for a competing species model:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-ejunxb
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-eukrfi
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0dc15a95hab6-gc28iy
In[4]:=4
✖
https://wolfram.com/xid/0dc15a95hab6-f9lvdk
Out[4]=4

Stability analysis for a predator-prey model:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-bq6c6g
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-gvmpky
Out[2]=2

Vector field plot of the model:

In[3]:=3
✖
https://wolfram.com/xid/0dc15a95hab6-cy1rmo
Out[3]=3

Use RecurrenceTable to solve the system numerically:

In[4]:=4
✖
https://wolfram.com/xid/0dc15a95hab6-e2wz7u
In[5]:=5
✖
https://wolfram.com/xid/0dc15a95hab6-mck818
Out[5]=5

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:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-mrkpdj

The equation has unstable fixed point :

In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-gyjsh9
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0dc15a95hab6-cbukb6
Out[3]=3

The amount in the account will increase each month if :

In[4]:=4
✖
https://wolfram.com/xid/0dc15a95hab6-g5pp3
In[5]:=5
✖
https://wolfram.com/xid/0dc15a95hab6-jbr6fq
In[6]:=6
✖
https://wolfram.com/xid/0dc15a95hab6-ccqd9
Out[6]=6

Stability analysis for the logistic equation:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-fwss2y
Out[1]=1

Check the stability of the fixed points:

In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-e8yrlu
Out[2]=2

Check the stability for given range of the parameter :

In[3]:=3
✖
https://wolfram.com/xid/0dc15a95hab6-mq4v3
Out[3]=3

Visualize the stability for :

In[4]:=4
✖
https://wolfram.com/xid/0dc15a95hab6-hiccia
Out[4]=4

Visualize the stability for :

In[5]:=5
✖
https://wolfram.com/xid/0dc15a95hab6-cjtbgr
Out[5]=5

Properties & Relations  (8)Properties of the function, and connections to other functions

RFixedPoints returns fixed points for recurrence equations:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-ni5fw
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-iquooh
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-b0qy8d
Out[2]=2

Analyze the stability at specific fixed points:

In[3]:=3
✖
https://wolfram.com/xid/0dc15a95hab6-qwl0x9
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0dc15a95hab6-lt77g4
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-cwtd8
Out[1]=1

Use Solve to find the fixed points:

In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-rf7um
Out[2]=2

The fixed points for an n^(th)-order recurrence equation are n-dimensional vectors:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-9rxw0
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-pqjes6
Out[1]=1

Find the fixed points for a system of two ODEs:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-dfvuec
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-bqt3d2
Out[2]=2

Use RSolveValue to solve the system for given initial conditions:

In[3]:=3
✖
https://wolfram.com/xid/0dc15a95hab6-ew3s3y
Out[3]=3

Plot the solution:

In[4]:=4
✖
https://wolfram.com/xid/0dc15a95hab6-doi666
Out[4]=4

Analyze the stability of a nonlinear ODE:

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-gaxuzv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-clwfub
Out[2]=2

Solve the ODE using RecurrenceTable:

In[3]:=3
✖
https://wolfram.com/xid/0dc15a95hab6-fjaa2
Out[3]=3

Plot the solution:

In[4]:=4
✖
https://wolfram.com/xid/0dc15a95hab6-bleeat
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0dc15a95hab6-bfipbq
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dc15a95hab6-b79gga
Out[2]=2

Find a series solution using AsymptoticRSolveValue:

In[3]:=3
✖
https://wolfram.com/xid/0dc15a95hab6-bh4spa
Out[3]=3

Plot the asymptotic solution:

In[4]:=4
✖
https://wolfram.com/xid/0dc15a95hab6-cxb1gl
Out[4]=4
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.

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: 20-June-2025 ]}

✖
@misc{reference.wolfram_2025_rfixedpoints, author="Wolfram Research", title="{RFixedPoints}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/RFixedPoints.html}", note=[Accessed: 20-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: 20-June-2025 ]}

✖
@online{reference.wolfram_2025_rfixedpoints, organization={Wolfram Research}, title={RFixedPoints}, year={2024}, url={https://reference.wolfram.com/language/ref/RFixedPoints.html}, note=[Accessed: 20-June-2025 ]}