By Albert C. J. Luo, Dennis M. O'Connor
This e-book describes process dynamics with discontinuity attributable to process interactions and provides the speculation of circulation singularity and switchability on the boundary in discontinuous dynamical structures. according to this sort of idea, the authors handle dynamics and movement mechanism of engineering discontinuous structures because of interplay. balance and bifurcations of mounted issues in nonlinear discrete dynamical platforms are provided, and mapping dynamics are built for analytical predictions of periodic motions in engineering discontinuous dynamical structures. eventually, the ebook offers another method to talk about the periodic and chaotic behaviors in discontinuous dynamical systems.
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Extra info for System Dynamics with Interaction Discontinuity
For an Á À Ã Â arbitrarily small ε > 0, there are two time intervals tmÀε , tm and tm , tmþε . Suppose xðiÞ ðtmÀ Þ ¼ xm ¼ xð jÞ ðtmþ Þ. The two vector fields F(i)(x, t, pi) and r F( j )(x, t, pj) are Crðti mÀε ;tm and Cðtj m ;tmþε -continuous ( r α ! 1, α ¼ i, j ) for time t, respectively. drα þ1 xðαÞ =dtrα þ1 < 1. The resultant flow of two flows x(i)(t) and x( j )(t) at the point (xm, tm) to the boundary ∂Ωij is semi-passable from the domain Ωi to Ωj if and only if 9 T n∂Ω Á FðiÞ ðtmÀ Þ > 0 and = ij for n∂Ωij !
Kind from g ∂Ω ij to ∂Ω ij (or simply called the sliding fragmentation bifurcation) if T T n∂Ω Á x_ ð jÞ ðtmÇ Þ ¼ 0 and n∂Ω Á x_ ðiÞ ðtmÀ Þ 6¼ 0; ij ij ð2:88Þ Â Ã 9 T n∂Ω Á xð jÞ ðtmÀ Þ À xð jÞ ðtmÀε Þ < 0 > > ij > = Â ð jÞ Ã T ð jÞ n Á x ðt Þ À x ðt Þ > 0 for n∂Ωij ! Ω j ; either ∂Ωij mþε mþ > > Â ðiÞ Ã > ; T ðiÞ n∂Ωij Á x ðtmÀ Þ À x ðtmÀε Þ > 0 ð2:89Þ Â Ã 9 T n∂Ω Á xð jÞ ðtmÀ Þ À xð jÞ ðtmÀε Þ > 0 > > ij > = Â ð jÞ Ã T ð jÞ or n∂Ωij Á x ðtmþε Þ À x ðtmþ Þ < 0 for n∂Ωij ! 21 For a discontinuous dynamical system in Eq.
Only the sliding flow exists on such a boundary. 6 Switching Bifurcations of Non-passable Flows Fig. 10 The source fragmentation bifurcation to ^ in the source boundary ∂Ω ij domain: (a) Ωj and (b) Ωi. Four points xðαÞ ðtmÆε Þ, xðβÞ ðtmþε Þ and xm lie in the corresponding domains and on the boundary ∂Ωi j , respectively. α, β 2 fi; jg but α 6¼ β and n1 þ n2 ¼ n 49 a Ωj x(j)(tm+ε) xm xm1 x(j)(tm−ε) x(i)(tm+ε) xn2 Ωi xm2 xn1 b x( j )(tm+ε) xm1 Ωj xm x( i )(tm−ε) x( i )(tm+ε) xn2 Ωi xm2 xn1 sliding and semi-passable motions.