Math 2.0

Ordinary differential equations Qualitative theory Example

Liénard equation

Builds on Even and odd functionsAutonomous systemHarmonic oscillator

Example

Let f ⁣:ℝ→ℝf \colon \R \to \R be a continuous even function and let g ⁣:ℝ→ℝg \colon \R \to \R be a continuous odd function. The Liénard equation is the second-order ordinary differential equation x′′+f(x) x′+g(x)=0.x'' + f(x)\, x' + g(x) = 0. Let F(x)=∫0xf(s) dsF(x) = \int_0^x f(s)\, ds. With y=x′+F(x)y = x' + F(x), the equation becomes the planar autonomous system x′=y−F(x),y′=−g(x).x' = y - F(x), \qquad y' = -g(x). Check: from x′=y−F(x)x' = y - F(x) follows x′′=y′−f(x)x′=−g(x)−f(x)x′x'' = y' - f(x) x' = -g(x) - f(x) x', which is the Liénard equation.

Remarks

The term f(x)x′f(x) x' is a damping that depends on the position. Where f(x)<0f(x) < 0 it pumps energy into the motion, and where f(x)>0f(x) > 0 it removes energy. The term g(x)g(x) is a restoring force. For f=0f = 0 and g(x)=xg(x) = x, the equation is the harmonic oscillator. For f(x)=−μ(1−x2)f(x) = -\mu (1 - x^2) with μ>0\mu > 0 and g(x)=xg(x) = x, it is the van der Pol equation x′′−μ(1−x2)x′+x=0x'' - \mu (1 - x^2) x' + x = 0. Liénard's theorem gives conditions on ff and gg under which the system has exactly one limit cycle.

Reading path

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Autonomous system, concept, no evidence levelAutonomous systemconceptComplex numbers, object, no evidence levelComplex numbersobjectDifferential equation, concept, no evidence levelDifferential equationconceptEven and odd functions, concept, no evidence levelEven and odd functionsconceptFunction, concept, no evidence levelFunctionconceptInterval, concept, no evidence levelIntervalconceptEuler's formula, statement, no evidence levelEuler's formulastatementOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionOrder of a differential equation, definition, no evidence levelOrder of a differenti…definitionLinear differential equation of order n, definition, no evidence levelLinear differential e…definitionHomogeneous linear equation, definition, no evidence levelHomogeneous linear eq…definitionLinear equation with constant coefficients, definition, no evidence levelLinear equation with …definitionSuperposition principle, statement, no evidence levelSuperposition princip…statementCharacteristic equation, definition, no evidence levelCharacteristic equati…definitionSolution space of a homogeneous linear equation has dimension n, statement, no evidence levelSolution space of a h…statementFundamental system of solutions, definition, no evidence levelFundamental system of…definitionSolutions for distinct real roots, statement, no evidence levelSolutions for distinc…statementSolutions for repeated roots, statement, no evidence levelSolutions for repeate…statementSolutions for complex conjugate roots, statement, no evidence levelSolutions for complex…statementHarmonic oscillator, example, no evidence levelHarmonic oscillatorexampleLiénard equation, example, no evidence levelLiénard equationexample
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Autonomous system concept
  2. Complex numbers object
  3. Differential equation concept
  4. Even and odd functions concept
  5. Function concept
  6. Interval concept
  7. Euler's formula statement
  8. Ordinary differential equation definition
  9. Order of a differential equation definition
  10. Linear differential equation of order n definition
  11. Homogeneous linear equation definition
  12. Linear equation with constant coefficients definition
  13. Superposition principle statement
  14. Characteristic equation definition
  15. Solution space of a homogeneous linear equation has dimension n statement
  16. Fundamental system of solutions definition
  17. Solutions for distinct real roots statement
  18. Solutions for repeated roots statement
  19. Solutions for complex conjugate roots statement
  20. Harmonic oscillator example
  21. Liénard equation example

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Li%C3%A9nard_equation
About this page
TagD2KC8NEAPS
URIm2://D2KC8NEAPS
Kindexample
WritingW0 stub
MSC34C15, 34C07
Instance ofOrdinary differential equation
UsesEven and odd functions, Autonomous system, Harmonic oscillator
Changed2 November 2026

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