By Univ.-Prof. Dr. Hans Troger, Univ.-Ass. Dr. Alois Steindl (auth.)

Every scholar in engineering or in different fields of the technologies who has undergone his curriculum understands that the remedy of nonlin ear difficulties has been both refrained from thoroughly or is constrained to big classes the place quite a few varied ad-hoc equipment are awarded. The customary think that no common answer strategies for nonlinear difficulties can be found prevails even at the present time in engineering cir cles. although in a few classes it truly is indicated that during precept nonlinear difficulties are solveable via numerical tools the remedy of nonlinear difficulties, kind of, is taken into account to be an paintings or an highbrow online game. an outstanding instance for this assertion was once the quest for Ljapunov features for nonlinear balance difficulties within the seventies. although issues have replaced. initially of the seventies, commence ing with the paintings of V.1. Arnold, R. Thom etc, new principles which, notwithstanding, have their starting place within the paintings of H. Poincare and A. A. Andronov, within the remedy of nonlinear difficulties seemed. those rules gave start to the time period Bifurcation idea. Bifurcation conception permits to unravel an excellent classification of nonlinear difficulties less than edition of parameters in an easy manner.

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**Sample text**

In a precise way the order symbols O( e) and o{ e) are defined as follows. 1 (Order symbols) there exists a constant K > 0 such that fee) ~ Ke k for all small e . lim fee) = 0 . e_O e k Neglecting terms O( e 2 ) and o( e) may not be the same. In the first case it means that terms starting with e 2 and of higher order are neglected, whereas in the second case, for example, already terms e3 / 2 and higher order terms may be neglected. 15). 16), we find the two fixed points rt = 0 and rt = 2. It is not difficult to check the stability of these fixed points by a linearized stability analysis.

P5tj>ds = O. 29) it is necessary that the boundary term vanishes for all admissible functions <5t/J( s). This can be achieved either if <5t/J = 0, which 36 2 Representation of systems is called a kinematic boundary condition and corresponds to clamped ends, or if 6,p =J 0, which is the case for the rod in Fig. 30) is satisfied. 30) is that the bending moment vanishes at the ends of the rod. 29) reduces to 1L [EJ~s~ + P sin ,p] 6,pds = o. 31) to vanish. Otherwise we always could find a continuous function 6,p( s), such that the integral would not be equal to zero.

To clarify this point, let us consider two examples. In truck dynamics various mechanical models are in use. Some lead to systems of ordinary and some to hybrid systems of ordinary and partial differential equations. If, for example, the stability of motion of a heavy truck is to be investigated, engineering experience shows that a mechanical model of coupled rigid bodies, and consequently, a mathematical model of ordinary differential equations will yield accurate results. If, however, for the motion of the same truck, for design reasons the deformation of the chassis of the truck is of interest, a continuous mechanical model of the truck must be used.