By G. Rega, F. Vestroni
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Extra resources for Chaotic Dynamics and Control of Systems and Processes in Mechanics
Vol III Part I, 105-128, Pl. 2, 1830.  J. M. Bloxam, “On the mathematical theory and practical defects of clock escapements, with a description of a new escapement; and some observations for astronomical and scientific purposes”, Memoirs of the Royal Astronomical Society, Vol. XXII, 103-150, 1854.  E. B. a. Lord Grimthorpe), A Rudimentary Treatise on Clocks and Watches and Bells, 1868.  A. A. Andronov, A. A. Vitt and S. E. Khaikin, Theory of Oscillators, Pergamon Press, Oxford. , 1987.
Vector function ϕ and columns of matrix functions and are locally Lipschitz continuous, & is some constant matrix of appropriate dimension. Thus, for any initial condition [ ∈ Ω [ ⊂ # Q and any ∈Ω θ (where Ω [ some known, probably compact, set), solution of (2) [ ( 9 [ ) is well defined at the least locally (further we will omit dependence of [ and if it is clear from the context and will simply write [ ( 9 ) ). For transmitter it is naturally to suppose , that its solution is bounded and defined for all 9 ≥ .
We will need the following lemma that can be easily proved by integration. L e m m a 1 . Let us consider time-varying linear dynamic system S = − D ( W ) S + E( W ) , W ≥ , (4) where S ∈ 5 , S( W )∈ 5 and functions D 5≥ → 5 , E 5≥ → 5 are Lebesgue measurable and essentially bounded. Then: A. Solution of system (4) is defined for all W ≥ W : S( W ) ≤ [ S( W ) + E H ]H , 9 , ( 9 −9 ) . B. If function D is (µ ∆ ) –PA for some µ > , ∆ > , then this solution is bounded and the following upper estimate holds: ⎧[ S( W ) + E H ( + ∆ ) ]H ∆ LI W ≤ W + ∆ ⎪ S ( W ) ≤ ⎨ S( W )H −µ ( − ) + µ − E H −µ (∆ + ) + ⎪ + D − E H − ( − ∆ ) LI W ≥ W + ∆ ⎩ , 9 9 9 , 9 , 9 additionally, if E( W )→ for W → + ∞ , then also S( W )→ asymptotically.
Chaotic Dynamics and Control of Systems and Processes in Mechanics by G. Rega, F. Vestroni