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In this paper we are concerned with the simplest normal form computation of the systems ẋ = 2xf(x, y + z), ẏ = z + yf(x, y + z), ż = −y + zf(x, y + z), (0.1) where f is a formal function with real coefficients and without any constant term. These are the classical normal forms of a larger family of systems with Hopf-Zero singularity. Indeed, these are(More)
In this paper, we study some relationships between the detection and estimation theories for a binary composite hypothesis test <i>H</i><sub>0</sub> against <i>H</i><sub>1</sub> and a related estimation problem. We start with a one-dimensional (1D) space for the unknown parameter space and one-sided hypothesis problems and then extend out results into more(More)
a r t i c l e i n f o a b s t r a c t This paper extends recent developments on spectral sequence approach on the simplest normal form theory to parametric cases. In this paper, the method is first applied to two simple examples (parametric single zero and parametric resonant saddle singulari-ties) and then, to parametric generalized Hopf singularity. We(More)
In this paper, we introduce a suitable algebraic structure for efficient computation of the para-metric normal form of Hopf singularity based on a notion of formal decompositions. Our para-metric state and time spaces are respectively graded parametric Lie algebra and graded ring. As a consequence, the parametric state space is also a graded module.(More)
We introduce a formal decomposition method for efficiently computing the parametric normal form of nonlinear dynamical systems with multiple parameters. Recently introduced notions of formal basis style and costyle are applied through formal decomposition method to obtain the simplest parametric normal form for degenerate nonlinear parametric center. The(More)
Let <i>x</i> = (<i>x</i><sub>1</sub>,...,<i>x</i><sub><i>n</i></sub>) &#8712; R <sup><i>n</i></sup> and &#955; &#8712; R. A smooth map <i>f</i>(<i>x</i>,&#955;) is called <i>Z</i><sub>2</sub>-equivariant (<i>Z</i><sub>2</sub>-invariant) if <i>f</i>(&minus;<i>x</i>, &#955;) = &minus;<i>f</i>(<i>x</i>,&#955;) (<i>f</i>(&minus;<i>x</i>, &#955;) =(More)
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