Sensitivity analysis for oscillating dynamical systems.

*(English)*Zbl 1201.65122This paper is concerned with the sensitivity analysis of systems governed by differential equations with respect to initial conditions and model parameters. For a given \( n_y\)-dimensional differential system with the state variable \( {\mathbf y} \) whose flow \( {\mathbf f} ( {\mathbf y}, {\mathbf p})\) depends on the \( n_p\)-dimensional parameter \( {\mathbf p}, \) defined by the differential equation \( \partial_t {\mathbf y} = {\mathbf f} ( {\mathbf y}, {\mathbf p})\) the dependence of the solution \( {\mathbf y}(t; {\mathbf p}, {\mathbf y}_0 ( {\mathbf p} ) )\) on the parameter \( {\mathbf p}\) and initial condition \( {\mathbf y}(0)= {\mathbf y}_0 ( {\mathbf p} )\) is studied for two types of oscillatory systems: The so called limit-cycle oscillators (LCOs) in which for each value of \( {\mathbf p} \) there is an isolated periodic orbit that is an attracting orbit of neighboring orbits and secondly the nonlimit-cycle oscillator (NLCO) in which all orbits are periodic with period depending on initial conditions and parameters.

For these kinds of problems the first order sensitivity matrices \( {\mathbf S}_0({\mathbf p}) \equiv \partial {\mathbf y}_0/ \partial {\mathbf p}\) and \( \partial T/ \partial {\mathbf p}\) where \( T = T( {\mathbf p} )\) is the period are given as solutions of some boundary value problems by introducing a suitable phase locking condition. Some comments on the numerical techniques that make easier the solution of these boundary value problems that arise in the calculation of different types of sensitivities in several practical problems are presented.

For these kinds of problems the first order sensitivity matrices \( {\mathbf S}_0({\mathbf p}) \equiv \partial {\mathbf y}_0/ \partial {\mathbf p}\) and \( \partial T/ \partial {\mathbf p}\) where \( T = T( {\mathbf p} )\) is the period are given as solutions of some boundary value problems by introducing a suitable phase locking condition. Some comments on the numerical techniques that make easier the solution of these boundary value problems that arise in the calculation of different types of sensitivities in several practical problems are presented.

Reviewer: Manuel Calvo (Zaragoza)

##### MSC:

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems |

34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |

34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |

34C25 | Periodic solutions to ordinary differential equations |

37C27 | Periodic orbits of vector fields and flows |