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Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion
In this tutorial, we discuss self-excited and hidden attractors for systems
of differential equations. We considered the example of a Lorenz-like system
derived from the well-known Glukhovsky--Dolghansky and Rabinovich systems, to
demonstrate the analysis of self-excited and hidden attractors and their
characteristics. We applied the fishing principle to demonstrate the existence
of a homoclinic orbit, proved the dissipativity and completeness of the system,
and found absorbing and positively invariant sets. We have shown that this
system has a self-excited attractor and a hidden attractor for certain
parameters. The upper estimates of the Lyapunov dimension of self-excited and
hidden attractors were obtained analytically.Comment: submitted to EP
Post-buckling behavior of a beam-column on a nonlinear elastic foundation with a gap
The structural behavior of an elastic beam-column placed with a gap between two nonlinearity elastic layers each resting on a rigid foundation was examined. The beam-column was laterally supported at both ends and subjected to a uniform transverse load and axial compression. Its slenderness was such that the axial compressive force exceeds the amount that would be necessary to buckle it as a simple supported column. The elastic layers were represented by an elastic foundation with a strongly nonlinear specific reaction taken as a rapidly increasing function of the layer compression. The analytical model developed simulated the entire pattern of the deflection and stress state including layer and end support reactions, under gradually increasing axial force
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