Abstract
We consider a linear dynamical system under the action of potential and circulatory forces. The matrix of potential forces is positive definite, and the main question is when the circulatory forces induce instability to the system. Different approaches to studying the problem are discussed and illustrated by examples. The case of multiple eigenvalues also is considered, and sufficient conditions of instability are obtained. Some issues of the dynamics of a nonlinear system with an unstable linear approximation are discussed. The behavior of trajectories in the case of unstable equilibrium is investigated, and an example of the chaotic behavior versus the case of bounded solutions is presented and discussed.
Highlights
IntroductionIn the theory of dynamical systems, interest in the properties of symmetry is steadily growing
In the theory of dynamical systems, interest in the properties of symmetry is steadily growing.In recent decades, a systematic study of systems possessing the so-called reversible symmetry has become widespread [1,2,3,4,5]
It is related to many real-life applications and is mainly matched with the response of mechanical systems to parametric excitation and flutter phenomenon exhibited by aerospace systems undergoing interaction between a structure and fluid flows, as well as playing a crucial role in the control of walking robots and many others [6,7,8,9,10,11]
Summary
In the theory of dynamical systems, interest in the properties of symmetry is steadily growing. A general stability problem of equilibrium of different mechanical non-conservative (both dissipative and circulatory) systems was studied in numerous papers [12,13,24,25,26]. In our study, we shall focus on the undamped non-gyroscopic systems of the MKN-type (which are reversible systems) The eigenvalues of such a system are symmetrically located on the complex plane with respect to the imaginary axis; stability is possible only if all of them are situated on this axis. Such systems are found in applications, and often, the influence of an arbitrarily small circulation force leads to the instability of the motion. The motion is unstable in the Lyapunov sense, it can be bounded, which can prove to be acceptable enough in engineering applications
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