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**Exponential trichotomy of differential systems.**
*(English)*
Zbl 0651.34052

The notion of exponential trichotomy, which is due to Elaydi and Hajek in an earlier paper, is a natural generalisation of that of exponential dichotomy which is familiar in stability questions for dynamical systems. It is stronger than the notion of trichotomy introduced by Sacker and Sell in 1976. It is shown that an upper triangular system possesses an exponential trichotomy if and only if its diagonal process does. If a linear system has an exponential trichotomy then certain stability results for small nonlinear perturbations of the linear system are established.

Reviewer: J.F.Toland

### MSC:

34D10 | Perturbations of ordinary differential equations |

34D05 | Asymptotic properties of solutions to ordinary differential equations |

34A30 | Linear ordinary differential equations and systems |

34A34 | Nonlinear ordinary differential equations and systems |

37-XX | Dynamical systems and ergodic theory |

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\textit{S. Elaydi} and \textit{O. Hajek}, J. Math. Anal. Appl. 129, No. 2, 362--374 (1988; Zbl 0651.34052)

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### References:

[1] | Bylov, B. F., Siberian Math. J., 7, 600-625 (1966) · Zbl 0161.05902 |

[2] | Coppel, W. A., Dichotomies in Stability Theory, (Lecture Notes in Mathematics, Vol. 629 (1978), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York) · Zbl 0376.34001 |

[3] | Elaydi, S.; Hajek, O., Exponential dichotomy of nonlinear systems of ordinary differential equations, (Lakshmikantham, V., Trends in Theory and Practice of Nonlinear Analysis (1985), North-Holland: North-Holland Amsterdam), 145-153 · Zbl 0582.34059 |

[5] | Kulik, A. N.; Kulik, V. L., Lyapunov functions and dichotomy of linear differential equations, Differentsial’nye Uravneniya, 20, No. 2, 233-241 (1984), [Russian] · Zbl 0537.34057 |

[6] | Palmer, K. J., Exponential dichotomy, integral separation and diagonalizability of linear systems of ordinary differential equations, J. Differential Equations, 43, 184-203 (1982) · Zbl 0443.34007 |

[7] | Sacker, R. J.; Sell, G. R., Existence of dichotomies and invariant splittings for linear differential systems, III, J. Differential Equations, 22, 497-522 (1976) · Zbl 0338.58016 |

[8] | Sacker, R. J.; Sell, G. R., Existence of dichotomies and invariant splittings for linear differential systems, I, J. Differential Equations, 15, 429-458 (1974) · Zbl 0294.58008 |

[9] | Sacker, R. J.; Sell, G. R., The spectrum of an invariant submanifold, J. Differential Equations, 38, 320-358 (1980) · Zbl 0415.58015 |

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