### Abstract

The problem of almost everywhere stability of a nonlinear autonomous ordinary differential equation is studied using a linear transfer operator framework. The infinitesimal generator of a linear transfer operator (Perron-Frobenius) is used to provide stability conditions of an autonomous ordinary differential equation. It is shown that almost everywhere uniform stability of a nonlinear differential equation, is equivalent to the existence of a non-negative solution for a steady state advection type linear partial differential equation. We refer to this non-negative solution, verifying almost everywhere global stability, as Lyapunov density. A numerical method using finite element techniques is used for the computation of Lyapunov density.

Original language | English (US) |
---|---|

Pages (from-to) | 144-156 |

Number of pages | 13 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 368 |

Issue number | 1 |

DOIs | |

State | Published - Aug 2010 |

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### Keywords

- Advection equation
- Almost everywhere stability
- Density function

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Journal of Mathematical Analysis and Applications*,

*368*(1), 144-156. https://doi.org/10.1016/j.jmaa.2010.02.032

**Stability in the almost everywhere sense : A linear transfer operator approach.** / Rajaram, R.; Vaidya, U.; Fardad, Makan; Ganapathysubramanian, B.

Research output: Contribution to journal › Article

*Journal of Mathematical Analysis and Applications*, vol. 368, no. 1, pp. 144-156. https://doi.org/10.1016/j.jmaa.2010.02.032

}

TY - JOUR

T1 - Stability in the almost everywhere sense

T2 - A linear transfer operator approach

AU - Rajaram, R.

AU - Vaidya, U.

AU - Fardad, Makan

AU - Ganapathysubramanian, B.

PY - 2010/8

Y1 - 2010/8

N2 - The problem of almost everywhere stability of a nonlinear autonomous ordinary differential equation is studied using a linear transfer operator framework. The infinitesimal generator of a linear transfer operator (Perron-Frobenius) is used to provide stability conditions of an autonomous ordinary differential equation. It is shown that almost everywhere uniform stability of a nonlinear differential equation, is equivalent to the existence of a non-negative solution for a steady state advection type linear partial differential equation. We refer to this non-negative solution, verifying almost everywhere global stability, as Lyapunov density. A numerical method using finite element techniques is used for the computation of Lyapunov density.

AB - The problem of almost everywhere stability of a nonlinear autonomous ordinary differential equation is studied using a linear transfer operator framework. The infinitesimal generator of a linear transfer operator (Perron-Frobenius) is used to provide stability conditions of an autonomous ordinary differential equation. It is shown that almost everywhere uniform stability of a nonlinear differential equation, is equivalent to the existence of a non-negative solution for a steady state advection type linear partial differential equation. We refer to this non-negative solution, verifying almost everywhere global stability, as Lyapunov density. A numerical method using finite element techniques is used for the computation of Lyapunov density.

KW - Advection equation

KW - Almost everywhere stability

KW - Density function

UR - http://www.scopus.com/inward/record.url?scp=77951207305&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77951207305&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2010.02.032

DO - 10.1016/j.jmaa.2010.02.032

M3 - Article

VL - 368

SP - 144

EP - 156

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -