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:
  • Assumptions $Assumptionsassumptions on parameters

Examples

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Basic Examples  (8)

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:

Stability analysis of a nonlinear recurrence equation:

Use a cobweb plot to demonstrate the stability:

Consider a system , :

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

The stability of a linear system with constant coefficients:

Use VectorPlot to visualize the stability:

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  (14)

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  (3)

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  (8)

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:

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:

Wolfram Research (2024), RStabilityConditions, Wolfram Language function, https://reference.wolfram.com/language/ref/RStabilityConditions.html.

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_2024_rstabilityconditions, author="Wolfram Research", title="{RStabilityConditions}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/RStabilityConditions.html}", note=[Accessed: 04-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_rstabilityconditions, organization={Wolfram Research}, title={RStabilityConditions}, year={2024}, url={https://reference.wolfram.com/language/ref/RStabilityConditions.html}, note=[Accessed: 04-November-2024 ]}