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Encyclopedia of Complexity and Systems Science pp 1–24Cite as

Perturbation of Systems with Nilpotent Real Part

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Definition of the Subject

The main goal of this entry is to dwell upon the influence of the presence (explicit and/or hidden) of nontrivial real nilpotent perturbations appearing in problems in dynamical systems, partial differential equations, and mathematical physics. Under the term nilpotent perturbation, we will mean, broadly speaking, a classical linear algebra typesetting: we start with an object (vector field or map near a fixed point, first-order singular partial differential equations, system of evolution partial differential equations) whose “linear part” A is semisimple (diagonalizable) and we add a (small) nilpotent part N. The problems of interest might be summarized as follows: are the “relevant properties” (in suitable functional framework) of the initial “object” stable under the perturbation N? If instabilities occur, to classify, if possible, the novel features of the perturbed systems.

Broadly speaking, the cases when the instabilities occur are rare; they form some...

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Abbreviations

Perturbation:

Typically, one starts with an “initial” system S 0, which is usually simple and/or well understood. We perturb the system by adding a (small) perturbation R so that the new object becomes S 0 + R. In our context the typical examples for S 0 will be systems of linear ordinary differential equations with constant coefficients in \( {\mathbb{K}}^n \) or the associated linear vector fields.

Nilpotent Linear Transformation:

Let \( A:{\mathbb{K}}^n\to {\mathbb{K}}^n \) be a linear map, where \( \mathbb{K}=\mathrm{\mathbb{R}} \) or \( \mathbb{K}=\mathrm{\mathbb{C}} \). We call A nilpotent if there exists a positive integer r such that the rth iteration A r becomes the zero map, in short A r = 0.

Gevrey Spaces:

Let Ω be an open domain in ℝn and let σ ≥ 1. The Gevrey space G σ(Ω) stands for the set of all functions f∈C ∞(Ω) such that for every compact subset K ⊂ ⊂ Ω, one can find C = C K,f > 0 such that

$$ \underset{x\in K}{ \sup}\left|{\partial}_x^{\alpha }f(x)\right|\le {C}^{\left|\alpha \right|+1}\alpha {!}^{\sigma } $$
(1)

for all α = (α 1, …, α n ) ∈ ℤ n+ , α! = α1!…α n !, | α | : = α 1 + ⋯ + α n . If σ = 1 we recapture the space of real analytic functions in Ω while the scale G σ(Ω), σ > 1, serves as an intermediate space between the real analytic functions and the set of all C ∞ functions in Ω. By the Stirling formula, one may replace α!σ by | α | !σ, | α | σ | α | or Γ(σ | α |), where Γ(z) stands for the Euler gamma function; cf. the book of Rodino (1993) for more details on the Gevrey spaces.

One associates also Gevrey index to formal power series, namely, given a (formal) power series

$$ f(x)={\displaystyle \sum_{\alpha }{f}_{\alpha }}{x}^{\alpha } $$

this is in the formal Gevrey space \( {G}_f^{\sigma}\left({\mathbb{K}}^n\right) \) if there exist C > 0 and R > 0 such that

$$ \left|{f}_{\alpha}\right|\le {C}^{\left|\alpha \right|+1}\left|\alpha \right|{!}^{\sigma -1} $$
(2)

for all α ∈ ℤ n+ .

In fact, one can find in the literature another definition of the formal Gevrey spaces G f τ of index τ, namely, replacing σ−1 by τ (see, e.g., Ramis 1984).

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  1. Dipartimento di Matematica e Informatica, Università di Cagliari, Cagliari, Italy

    Todor Gramchev

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Correspondence to Todor Gramchev .

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  1. Ramtech Ltd., Larkspur, California, USA

    Robert A. Meyers

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Gramchev, T. (2013). Perturbation of Systems with Nilpotent Real Part. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-3-642-27737-5_395-4

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  • DOI: https://doi.org/10.1007/978-3-642-27737-5_395-4

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