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 $Assumptions assumptions 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 :

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 x^1/3:

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 :

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-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:

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RStabilityConditions

Details and Options

Examples

Basic Examples (8)