RStabilityConditions [eqn,a[n],n]
gives the fixed points and stability conditions for a recurrence equation.
RStabilityConditions [{eqn1,eqn2,…},{a1[n],a2[n],…},n]
gives the fixed points and stability conditions for a system of recurrence equations.
RStabilityConditions [{eqn1,eqn2,…},{a1[n],a2[n],…},n,{pnt1,pnt2,…}]
gives stability conditions only for the given fixed points.
RStabilityConditions
RStabilityConditions [eqn,a[n],n]
gives the fixed points and stability conditions for a recurrence equation.
RStabilityConditions [{eqn1,eqn2,…},{a1[n],a2[n],…},n]
gives the fixed points and stability conditions for a system of recurrence equations.
RStabilityConditions [{eqn1,eqn2,…},{a1[n],a2[n],…},n,{pnt1,pnt2,…}]
gives stability conditions only for the given fixed points.
Details and Options
- Stability is also known as asymptotic stability and fixed points are also known as equilibrium points or stationary points.
- RStabilityConditions is typically used to qualitatively analyze long-term behavior near fixed points. If the system is stable, then solutions converge to the fixed point if you are close enough.
- 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 .
- A fixed point is asymptotically stable iff for and you have TemplateBox[{{a, (, n, )}, n, infty}, Limit2Arg]=a^* for sufficiently small.
- RStabilityConditions returns a list of the form {{{,,…},cond},…}, where {,,…} is a fixed point.
- RStabilityConditions gives sufficient conditions for local stability of fixed points. For linear systems, these conditions are also conditions for global stability.
- RStabilityConditions works for linear and nonlinear ordinary difference equations.
- The following options can be given:
-
Examples
open all close allBasic Examples (5)
Find the fixed point and determine its stability for the recursion :
Find the fixed point and determine the stability for the recursion :
Find the fixed points and conditions for stability for the recursion :
Plot several solutions for different values of a:
Stability analysis of a two-dimensional system:
Plot the parameter region for which the system is stable:
Find the fixed points for a nonlinear system of three ODEs:
Study the stability of the first point:
Study the stability of the second point:
Scope (15)
Linear Equations (4)
Find the fixed point and determine its stability for the recursion :
A first-order linear inhomogeneous equation:
Plot the solution for :
Plot the solution for :
Second-order linear equation:
Plot the stability region:
Third-order linear equation:
Plot the stability region:
Nonlinear Equations (4)
Stability analysis of a nonlinear recurrence equation:
Use a cobweb plot to demonstrate the stability:
The stability of a logistic equation:
Plot the solution:
Use a cobweb plot to demonstrate the stability:
The stability of a Riccati equation:
Use a cobweb plot to demonstrate the stability:
Higher-order equations:
Linear Systems (5)
The stability of a linear system of uncoupled equations:
The stability of a linear system with constant coefficients:
Use VectorPlot to visualize the stability:
Solve the system with boundary conditions:
Plot the solution:
A first-order system with periodic coefficients:
Compare with the general solution:
A first-order system with eventually periodic coefficients:
Analyze the stability of a 10x10 discrete linear system with random constant coefficients:
Nonlinear Systems (2)
A nonlinear first-order system:
Check only the stability of the first fixed point:
Check only the stability of the second fixed point:
The stability of a linear fractional system:
Use VectorPlot to visualize the stability at point :
Options (1)
Assumptions (1)
Without Assumptions , there are conditions on parameters for stability:
Using Assumptions can often result in simplified conditions:
Applications (8)
Numerical Analyses (3)
Analyze the stability of the Newton–Raphson difference equation for the function x2-a:
The recurrence equation for the method is:
Study the stability of the system:
Visualize the stability of fixed points for :
Analyze the stability of the Newton–Raphson difference equation for the function x1/3:
The recurrence equation for the method is:
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:
Analyze the stability 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:
The equation is stable if :
Solution of the equation:
Simulate the cooling process:
Simulate the heating process:
Ecology and Biology (2)
Stability analysis for the competing species model:
Solve the equation for given initial conditions and plot the solution:
Stability analysis for the predator-prey model:
Vector field plot of the model:
Solve the system and plot solutions:
Economics (2)
Consider a bank account with initial deposit , annual rate and monthly withdrawal amount . The amount in the saving account after months satisfies a recurrence equation:
The equation has an 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:
Check the stability for given range of the parameter :
Visualize the stability for :
Visualize the stability for :
Properties & Relations (9)
RStabilityConditions returns fixed points and stability conditions for recurrence equations:
Use RFixedPoints to find 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:
Linearize the equation near the first fixed point:
Check the stability near the first fixed point:
Linearize the equation near the second fixed point:
Check the stability near the second fixed point:
Determine the stability of the nonlinear equation using RStabilityConditions :
The fixed points for an n^(th)-order recurrence equation are n-dimensional vectors:
The fixed points for a system of n first-order recurrence equations are n-dimensional vectors:
Consider a system , :
Calculate the differences and to generate the vector field plot of the system:
Analyze the stability of a system of two ODEs:
Use RSolveValue to solve the system using a fixed point as initial condition:
Use RSolveValue to solve the system for given initial conditions:
Plot the solution:
Analyze the stability of a nonlinear ODE:
Solve the ODE using RecurrenceTable :
Plot the solution:
Analyze the stability of a recurrence equation with eventually constant coefficients:
Find a series solution using AsymptoticRSolveValue :
Plot the asymptotic solution:
Possible Issues (2)
Sometimes the conditions for stability are not the simplest possible:
Additional simplification can be achieved by further processing:
RStabilityConditions fails because the given point is not a fixed point:
Use RFixedPoints to find all fixed points of the equation first:
Related Guides
History
Text
Wolfram Research (2024), RStabilityConditions, Wolfram Language function, https://reference.wolfram.com/language/ref/RStabilityConditions.html.
CMS
Wolfram Language. 2024. "RStabilityConditions." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RStabilityConditions.html.
APA
Wolfram Language. (2024). RStabilityConditions. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RStabilityConditions.html
BibTeX
@misc{reference.wolfram_2025_rstabilityconditions, author="Wolfram Research", title="{RStabilityConditions}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/RStabilityConditions.html}", note=[Accessed: 05-December-2025]}
BibLaTeX
@online{reference.wolfram_2025_rstabilityconditions, organization={Wolfram Research}, title={RStabilityConditions}, year={2024}, url={https://reference.wolfram.com/language/ref/RStabilityConditions.html}, note=[Accessed: 05-December-2025]}