RFixedPoints
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:
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Assumptions $Assumptions assumptions on parameters
Examples
open allclose allBasic Examples (6)
Find the fixed point for the recursion :
Find the fixed point for the recursion :
Find the fixed points for the recursion :
Check the stability of the point:
Plot several solutions for different values of a:
Find the fixed points of a two-dimensional system:
Determine the stability conditions:
Plot the parameter region for which the system is stable:
Find the fixed points of a nonlinear recurrence equation:
Use a cobweb plot to demonstrate the stability:
Calculate the differences and to generate the vector field plot of the system:
Scope (14)
Linear Equations (4)
Nonlinear Equations (3)
Linear Systems (5)
The fixed points of a linear system of uncoupled equations:
Linear system with constant coefficients:
Use VectorPlot to visualize the fixed point:
Solve the system with boundary conditions:
A first-order system with periodic coefficients:
Compare to the general solution:
A first-order system with eventually periodic coefficients:
10x10 discrete linear system with random constant coefficients:
Nonlinear Systems (2)
A nonlinear first-order system:
Determine the stability of the fixed points:
The fixed points and the stability of a linear fractional system:
Use VectorPlot to visualize the stability at point :
Applications (8)
Numerical Analyses (3)
Analyze the stability of the Newton–Raphson difference equation for the function x2-a:
Visualize the stability of the fixed points for :
Analyze the stability of the Newton–Raphson difference equation for the function x1/3:
The equation has one unstable fixed point at origin:
The instability of the point means that Newton's method cannot be used in this case:
Consider a system of linear equations , where:
Construct the Gauss–Seidel difference equation for the system:
Find the fixed point of the Gauss–Seidel equation:
Solve the system using LinearSolve:
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:
Ecology and Biology (2)
Stability analysis for a competing species model:
Solve the equation for given initial conditions and plot the solution:
Stability analysis for a predator-prey model:
Vector field plot of the model:
Use RecurrenceTable to solve the system numerically:
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 :
The amount in the account will increase each month if :
Stability analysis for the logistic equation:
Check the stability of the fixed points:
Properties & Relations (8)
RFixedPoints returns fixed points for recurrence equations:
Use RStabilityConditions to determine the stability for all fixed points of a recurrence equation:
Analyze the stability at specific fixed points:
Use RFixedPoints to find all fixed points of a nonlinear recurrence equation:
Use Solve to find the fixed points:
The fixed points for an n-order recurrence equation are n-dimensional vectors:
The fixed points for a system of n first-order recurrence equations are n-dimensional vectors:
Find the fixed points for a system of two ODEs:
Use RSolveValue to solve the system using fixed point as the initial condition:
Use RSolveValue to solve the system for given initial conditions:
Analyze the stability of a nonlinear ODE:
Solve the ODE using RecurrenceTable:
Analyze the stability of a recurrence equation with eventually constant coefficients:
Find a series solution using AsymptoticRSolveValue:
Text
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.
APA
Wolfram Language. (2024). RFixedPoints. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RFixedPoints.html