Computational aspects of commutative algebra by Lorenzo Robbiano

By Lorenzo Robbiano

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K=l Similarly, one argues for a complex-conjugate pair of eigenvalues. D 18 1. Autonomous Linear Differential and Difference Equations This result shows, in particular, that Lyapunov exponents alone do not allow us to characterize stability for linear systems. They are related to exponential or, equivalently, to asymptotic stability, and they do not detect polynomial instabilities. The following example is a damped linear oscillator. 11. The second order differential equation x+2bx+x = o is equivalent to the system [ :~ ] [ - ~ -2~ ] [ :~ ] .

The following example is a damped linear oscillator. 11. The second order differential equation x+2bx+x = o is equivalent to the system [ :~ ] [ - ~ -2~ ] [ :~ ] . /b2 - 1. For b > 0 the real parts of the eigenvalues are negative and hence the stable subspace coincides with R. 2 • Hence b is called a damping parameter. Note also that for every solution x( ·) the function y(t) := ebtx(t), t ER, is a solution of the equation ii+(l-b2 )y = 0. 5. The Discrete-Time Case: Linear Difference Equations This section discusses solution formulas and stability properties, in particular, Lyapunov exponents for autonomous linear difference equations.

Then we define H : [O, 1] x §d-i -+ [O, 1] x §d-i using the path from B to A. Here the tvalues remain preserved and on §d-i we use AtA-i. This yields the identity fort= 1 and BA-i fort= 0. Let us make this program precise. Define maps TA, hA: [0, 1] X §d-i-+ FA by hA(t, x) = TA(t, x)x, 2. A 1 where 1A is the map which is affine int and determined by TA(l,x) = and TA(O, x) = 1/ jjA- 1xjjA. Then hA(l, x) = x/ llxllA E SA, the outer boundary of FA, and hA(O, x) is on the inner boundary of FA, since hA(O,x) = TA(O,x)x = x jjA- 1 xjjA =A (A- 1 x/ jjA- 1 xjjA) E ASA.

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