Abstract
Systems of differential equations with discontinuous right-hand sides are considered, specifically investigating periodic solutions which simultaneously intersect two or more surfaces of discontinuity. It is shown that the Poincaré mapping along phase trajectories of the system in the neighbourhood of a fixed point, corresponding to periodic motion, is in general piecewise-differentiable: this neighbourhood divides into several sectors in which the Jacobians are different. For such mappings, theorems of stability in the first approximation [1] are not applicable, and one has to devise new stability criteria. Several necessary conditions for stability are obtained, as well as sufficient conditions. The results are used to investigate symmetric modes of motion of a vibro-impact system with two impact pairs.
Highlights
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It is shown that the Poincare mapping along phase trajectories of the system in the neighbourhood of a fixed point, corresponding to periodic motion, is in general piecewise-differentiable: this neighbourhood divides into several sectors in which the Jacobians are diffeN)nt
After calculating the monodromy matrix, the stabi1.ity problem is solved in exactly the same way as in the smooth case. Another type of discontinuity is characteristic for systems with impact: when the phase trajectory reaches the boundary of the domain of continuous motion ;;. 0), it experiences a jump discontinuity in accordance with the formula
Summary
It is shown that the Poincare mapping along phase trajectories of the system in the neighbourhood of a fixed point, corresponding to periodic motion, is in general piecewise-differentiable: this neighbourhood divides into several sectors in which the Jacobians are diffeN)nt. For such mappings, theorems of stability in the first approximation [1] are not applicable, and one has to devise new stability cn1eria. The method of investigating stability in the first approximation was previously applied to discontinuous systems for solutions that intersect one surface of discontinuity [2]. It turned out that under such conditions the Poincare mapping is differentiable, so that Lyapunov's theorems could be used
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