By Jan Awrejcewicz, Mariusz M. Holicke
This publication specializes in the improvement of Melnikov-type equipment utilized to excessive dimensional dynamical platforms ruled via usual differential equations. even though the classical Melnikov's method has chanced on a number of purposes in predicting homoclinic intersections, it really is dedicated merely to the research of third-dimensional structures (in the case of mechanics, they symbolize one-degree-of-freedom nonautonomous systems). This ebook extends the classical Melnikov's method of the learn of excessive dimensional dynamical platforms, and makes use of easy versions of dry friction to analytically are expecting the prevalence of either stick-slip and slip-slip chaotic orbits, study that's very infrequently suggested within the latest literature even on one-degree-of-freedom nonautonomous dynamics.
This pioneering try to expect the incidence of deterministic chaos of nonlinear dynamical platforms will allure many researchers together with utilized mathematicians, physicists, in addition to working towards engineers. Analytical formulation are explicitly formulated step by step, even attracting capability readers with out a rigorous mathematical heritage.
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Additional info for Smooth and Nonsmooth High Dimensional Chaos and the Melnidov-Type Methods
4) one gets in the vicinity of equilibrium ϕ¨ + ω02 cos ϕ0 − ω 2 cos 2ϕ0 (ϕ − ϕ0 ) = 0. For ϕ01 = 0 we get ϕ¨ + ω02 − ω 2 (ϕ − ϕ01 ) = 0. Observe that for |ω| < ω0 the equilibrium is a center, whereas for |ω| > ω0 it is a saddle. For ϕ02 = ±π one gets ϕ¨ − ω 2 (ϕ − ϕ0 ) = 0, which defines the saddle. For remaining critical points ϕ03,4 one obtains ω 2 − ω02 ω 2 + ω02 (ϕ − ϕ0 ) = 0. ω2 The mentioned critical points are centers and they exist only for |ω| > ω0 . 4), which is multiplied by ϕ˙ to give dϕ˙ ϕ˙ − 12 ω 2 ϕ˙ sin 2ϕ + ω02 ϕ˙ sin ϕ = 0, dt and after integration one gets ϕ¨ + ϕ˙ 2 1 C + ω 2 cos 2ϕ − ω02 cos ϕ = , 2 4 2 where C is the integration constant.
Stable and unstable differential manifolds associated with the saddle ϕ02 = ±π are defined by the following equation ϕ˙ s2 = ± 2ω02 (1 + cos ϕ) + 12 ω 2 (1 − cos 2ϕ). After some rearrangements one gets dt = ±dϕ 2ω02 (1 + cos ϕ) + ω 2 (1 − cos2 ϕ) . Integration of the last equation and computation of the inverse function yields 2 2 4 ω + ω0 ϕs2 (t) = ± arccos − 1 , 2 2 2ω + ω0 1 + cosh 2 ω 2 + ω02 t where sign “+” is taken for t ≥ 0, whereas sign “-” holds for t < 0. Differentiation with respect to time gives ϕ˙ s2 (t) = ± 4ω02 ω 2 + ω02 cosh ω 2 + ω02 t 2ω 2 + ω02 1 + cosh 2 ω 2 + ω02 t , July 19, 2007 10:48 World Scientific Book - 9in x 6in SmoothNonsmooth Homoclinic Chaos Criterion in a Rotated Froude Pendulum with Dry Friction 31 and hence ϕs2 (t) = arccos 4 ω 2 + ω02 2ω 2 + ω02 1 + cosh 2 ω2 + ω02 t − 1 sgn t and consequently ϕ˙ s2 (t) = ± 4ω02 ω 2 + ω02 cosh ω 2 + ω02 t .
For instance, the phase point movement along the homoclinic orbit within the given parameterization (t0 = const) is associated not only with a change of the system on the homoclinic trajectory, but also the position on the coordinate η (t) is changed. On the other hand, fixation of time instant (t = const), and hence also of variable η (t) = η0 = const and substitution of different values of constant t0 yields (in plane η (t) = η0 ) a graph of the homoclinic orbit. For t0 equal to the time instant corresponding to the intersection of trajectory with plane η (t) = η0 the phase point lies in plane η (t) = η0 at point qH (0).