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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

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

Find the fixed point for the recursion :

Out[1]=1

Find the fixed point for the recursion :

Out[1]=1

Find the fixed points for the recursion :

Out[1]=1

Check the stability of the point:

Out[2]=2

Plot several solutions for different values of a:

Out[3]=3
Out[4]=4

Find the fixed points of a two-dimensional system:

Out[1]=1

Determine the stability conditions:

Out[2]=2

Plot the parameter region for which the system is stable:

Out[3]=3

Find the fixed points of a nonlinear recurrence equation:

Out[1]=1

Determine the stability:

Out[2]=2

Use a cobweb plot to demonstrate the stability:

Out[3]=3

Consider a system , :

Out[2]=2

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

Out[3]=3

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

Linear Equations  (4)

Find the fixed point for the recursion :

Out[1]=1

A first-order linear inhomogeneous equation:

Out[1]=1

Determine the stability condition:

Out[2]=2

Plot the solution for :

Out[3]=3

Plot the solution for :

Out[4]=4

Second-order linear equation:

Out[1]=1

Determine the stability conditions:

Out[2]=2

Plot the stability region:

Out[3]=3

Third-order linear equation:

Out[1]=1

Determine the stability conditions:

Out[2]=2

Plot the stability region:

Out[3]=3

Nonlinear Equations  (3)

The fixed points of a logistic equation:

Out[1]=1

Analyze the stability of the points:

Out[2]=2

Plot the solution:

Out[3]=3
Out[4]=4

Use a cobweb plot to demonstrate the stability:

Out[5]=5

The fixed points of a Riccati equation:

Out[1]=1

Use a cobweb plot to demonstrate the stability:

Out[2]=2

Higher-order equations:

Out[1]=1
Out[2]=2

Linear Systems  (5)

The fixed points of a linear system of uncoupled equations:

Out[1]=1

Linear system with constant coefficients:

Out[1]=1

Use VectorPlot to visualize the fixed point:

Out[2]=2

Solve the system with boundary conditions:

Out[3]=3

Plot the solution:

Out[4]=4

A first-order system with periodic coefficients:

Out[1]=1

Compare to the general solution:

Out[2]=2

A first-order system with eventually periodic coefficients:

Out[1]=1

10x10 discrete linear system with random constant coefficients:

Out[4]=4

Nonlinear Systems  (2)

A nonlinear first-order system:

Out[1]=1

Determine the stability of the fixed points:

Out[2]=2

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

Out[1]=1
Out[2]=2

Use VectorPlot to visualize the stability at point :

Out[3]=3

Applications  (8)Sample problems that can be solved with this function

Numerical Analyses  (3)

Analyze the stability of the NewtonRaphson difference equation for the function x2-a:

Out[2]=2

Find fixed points:

Out[3]=3

Visualize the stability of the fixed points for :

Out[6]=6

Analyze the stability of the NewtonRaphson difference equation for the function x1/3:

Out[2]=2

The equation has one unstable fixed point at origin:

Out[3]=3
Out[4]=4

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

Out[5]=5

Consider a system of linear equations , where:

Construct the GaussSeidel difference equation for the system:

Out[4]=4

Find the fixed point of the GaussSeidel equation:

Out[5]=5

Solve the system using LinearSolve:

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:

Find the fixed points:

Out[2]=2

The fixed point is stable if :

Out[3]=3

Solution of the equation:

Out[4]=4

Simulate the cooling process:

Out[5]=5

Simulate the heating process:

Out[6]=6

Ecology and Biology  (2)

Stability analysis for a competing species model:

Out[1]=1
Out[2]=2

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

Out[4]=4

Stability analysis for a predator-prey model:

Out[1]=1
Out[2]=2

Vector field plot of the model:

Out[3]=3

Use RecurrenceTable to solve the system numerically:

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:

The equation has unstable fixed point :

Out[2]=2
Out[3]=3

The amount in the account will increase each month if :

Out[6]=6

Stability analysis for the logistic equation:

Out[1]=1

Check the stability of the fixed points:

Out[2]=2

Check the stability for given range of the parameter :

Out[3]=3

Visualize the stability for :

Out[4]=4

Visualize the stability for :

Out[5]=5

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

RFixedPoints returns fixed points for recurrence equations:

Out[1]=1

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

Out[1]=1
Out[2]=2

Analyze the stability at specific fixed points:

Out[3]=3
Out[4]=4

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

Out[1]=1

Use Solve to find the fixed points:

Out[2]=2

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

Out[1]=1

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

Out[1]=1

Find the fixed points for a system of two ODEs:

Out[1]=1

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

Out[2]=2

Use RSolveValue to solve the system for given initial conditions:

Out[3]=3

Plot the solution:

Out[4]=4

Analyze the stability of a nonlinear ODE:

Out[1]=1
Out[2]=2

Solve the ODE using RecurrenceTable:

Out[3]=3

Plot the solution:

Out[4]=4

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

Out[1]=1
Out[2]=2

Find a series solution using AsymptoticRSolveValue:

Out[3]=3

Plot the asymptotic solution:

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: 05-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: 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 ]}

@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 ]}