55 research outputs found
Hidden attractors in fundamental problems and engineering models
Recently a concept of self-excited and hidden attractors was suggested: an
attractor is called a self-excited attractor if its basin of attraction
overlaps with neighborhood of an equilibrium, otherwise it is called a hidden
attractor. For example, hidden attractors are attractors in systems with no
equilibria or with only one stable equilibrium (a special case of
multistability and coexistence of attractors). While coexisting self-excited
attractors can be found using the standard computational procedure, there is no
standard way of predicting the existence or coexistence of hidden attractors in
a system. In this plenary survey lecture the concept of self-excited and hidden
attractors is discussed, and various corresponding examples of self-excited and
hidden attractors are considered
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
A New Class of Two-dimensional Chaotic Maps with Closed Curve Fixed Points
This is the author accepted manuscript. The final version is available from World Scientific Publishing via the DOI in this recordThis paper constructs a new class of two-dimensional maps with closed curve fixed points.
Firstly, the mathematical model of these maps is formulated by introducing a nonlinear function. Different types of fixed points which form a closed curve are shown by choosing proper
parameters of the nonlinear function. The stabilities of these fixed points are studied to show
that these fixed points are all non-hyperbolic. Then a computer search program is employed to
explore the chaotic attractors in these maps, and several simple maps whose fixed points form
different shapes of closed curves are presented. Complex dynamical behaviours of these maps are
investigated by using the phase-basin portrait, Lyapunov exponents, and bifurcation diagrams.National Natural Science Foundation of ChinaNatural Science Foundation of Jiangsu Province of China5th 333 High-level Personnel Training Project of Jiangsu Province of ChinaExcellent Scientific and Technological Innovation Team of Jiangsu UniversityJiangsu Key Laboratory for Big Data of Psychology and Cognitive Scienc
Theory of differential inclusions and its application in mechanics
The following chapter deals with systems of differential equations with
discontinuous right-hand sides. The key question is how to define the solutions
of such systems. The most adequate approach is to treat discontinuous systems
as systems with multivalued right-hand sides (differential inclusions). In this
work three well-known definitions of solution of discontinuous system are
considered. We will demonstrate the difference between these definitions and
their application to different mechanical problems. Mathematical models of
drilling systems with discontinuous friction torque characteristics are
considered. Here, opposite to classical Coulomb symmetric friction law, the
friction torque characteristic is asymmetrical. Problem of sudden load change
is studied. Analytical methods of investigation of systems with such
asymmetrical friction based on the use of Lyapunov functions are demonstrated.
The Watt governor and Chua system are considered to show different aspects of
computer modeling of discontinuous systems
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