445:), for the study of stability and oscillations in complex nonlinear multidimensional systems, numerical methods are often used. In the multi-dimensional case the integration of trajectories with random initial data is unlikely to provide a localization of a hidden attractor, since a basin of attraction may be very small, and the attractor dimension itself may be much less than the dimension of the considered system. Therefore, for the numerical localization of hidden attractors in multi-dimensional space, it is necessary to develop special analytical-numerical computational procedures, which allow one to choose initial data in the attraction domain of the hidden oscillation (which does not contain neighborhoods of equilibria), and then to perform trajectory computation. There are corresponding effective methods based on
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384:, for classical parameters, the attractor is self-excited with respect to all existing equilibria, and can be visualized by any trajectory from their vicinities; however, for some other parameter values there are two trivial attractors coexisting with a chaotic attractor, which is a self-excited one with respect to the zero equilibrium only. Classical attractors in
453:: a sequence of similar systems is constructed, such that for the first (starting) system, the initial data for numerical computation of an oscillating solution (starting oscillation) can be obtained analytically, and then the transformation of this starting oscillation in the transition from one system to another is followed numerically.
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The classification of attractors as self-exited or hidden ones was a fundamental premise for the emergence of the theory of hidden oscillations, which represents the modern development of
Andronov’s theory of oscillations. It is key to determining the exact boundaries of the global stability, parts
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In general, the problem with hidden attractors is that there are no general straightforward methods to trace or predict such states for the system’s dynamics (see, e.g.). While for two-dimensional systems, hidden oscillations can be investigated using analytical methods (see, e.g., the results on
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For a self-excited attractor, its basin of attraction is connected with an unstable equilibrium and, therefore, the self-excited attractors can be found numerically by a standard computational procedure in which after a transient process, a trajectory, starting in a neighbourhood of an unstable
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on the number and mutual disposition of limit cycles in two-dimensional polynomial systems where the nested stable limit cycles are hidden periodic attractors. The notion of a hidden attractor has become a catalyst for the discovery of hidden attractors in many applied dynamical models.
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and observe how the system’s state, starting from this initial state, after a transient process, visualizes the attractor. The classification of attractors as being hidden or self-excited reflects the difficulties of revealing basins of attraction and searching for the local
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Hidden attractors have basins of attraction which are not connected with equilibria and are “hidden” somewhere in the phase space. For example, the hidden attractors are attractors in the systems without equilibria: e.g. rotating electromechanical dynamical systems with
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theory, the birth of a hidden oscillation in a time-invariant control system with bounded states means crossing a boundary, in the domain of the parameters, where local stability of the stationary states implies global stability (see, e.g.
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with a unique equilibrium point that is globally attractive, the birth of a hidden attractor corresponds to a qualitative change in behaviour from monostability to bi-stability. In the general case, a dynamical system may turn out to be
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for the first time in 2009 year. Similarly, an arbitrary bounded oscillation, not necessarily having an open neighborhood as the basin of attraction in the phase space, is classified as a self-excited or hidden oscillation.
502:
Leonov G.A.; Kuznetsov N.V. (2013). "Hidden attractors in dynamical systems. From hidden oscillations in
Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits".
307:
An attractor is called a hidden attractor if its basin of attraction does not intersect with a certain open neighbourhood of equilibrium points; otherwise it is called a self-excited attractor.
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of which are classified by N. Kuznetsov as trivial (i.e., determined by local bifurcations) or as hidden (i.e., determined by non-local bifurcations and by the birth of hidden oscillations).
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Afraimovich Award's plenary lecture: N. Kuznetsov The theory of hidden oscillations and stability of dynamical systems. Int. Conference on
Nonlinear Dynamics and Complexity, 2021
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does not exceed the
Lyapunov dimension of one of the unstable equilibria, the unstable manifold of which intersects with the basin of attraction and visualizes the attractor.
580:
Leonov, G.A.; Kuznetsov, N.V.; Mokaev, T.N. (2015). "Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion".
351:. Trajectories with initial data in neighborhoods of two saddle points (blue) tend (red arrow) to infinity or tend (black arrow) to stable zero equilibrium point (orange).
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Kuznetsov, N.V.; Leonov, G.A.; Mokaev, T.N.; Prasad, A.; Shrimali, M.D. (2018). "Finite-time
Lyapunov dimension and hidden attractor of the Rabinovich system".
84:
951:
Chen, G.; Kuznetsov, N.V.; Leonov, G.A.; Mokaev, T.N. (2015). "Hidden attractors on one path: Glukhovsky-Dolzhansky, Lorenz, and
Rabinovich systems".
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Kuznetsov N. V.; Leonov G. A. (2014). "Hidden attractors in dynamical systems: systems with no equilibria, multistability and coexisting attractors".
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339:. Trajectories with initial data in neighborhoods of two saddle points (blue) and zero equilibrium point (orange) tend (green) to attractor.
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Two hidden chaotic attractors and one hidden periodic attractor coexist with two trivial attractors in Chua circuit (from the IJBC cover)
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Stankevich N. V.; Kuznetsov N. V.; Leonov G. A.; Chua L. (2017). "Scenario of the birth of hidden attractors in the Chua circuit".
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Nonlinear
Dynamical Systems with Self-Excited and Hidden Attractors (Eds.: Pham, Vaidyanathan, Volos et al.), Springer, 2018 (
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540:"Algorithms for Finding Hidden Oscillations in Nonlinear Systems. The Aizerman and Kalman Conjectures and Chua's Circuits"
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To identify a local attractor in a physical or numerical experiment, one needs to choose an initial system’s state in
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914:"Hidden attractors in dynamical models of phase-locked loop circuits: limitations of simulation in MATLAB and SPICE"
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Help add sources such as review articles, monographs, or textbooks. Please also establish the relevance for any
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432:(1957) on the monostability of nonlinear control systems. One of the first related theoretical problems is
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Chaotic
Systems with Multistability and Hidden Attractors (Eds.: Wang, Kuznetsov, Chen), Springer, 2021 (
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Dudkowski D.; Jafari S.; Kapitaniak T.; Kuznetsov N. V.; Leonov G. A.; Prasad A. (2016).
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equilibrium, is attracted to the state of oscillation and then traces it (see, e.g.
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The classification of attractors as being hidden or self-excited was introduced by
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International
Journal of Bifurcation and Chaos in Applied Sciences and Engineering
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Journal of Bifurcation and Chaos in Applied Sciences and Engineering
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International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
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Kuznetsov, N.V.; Mokaev, T.N.; Kuznetsova, O.A.; Kudryashova, E.V. (2020).
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cited. Unsourced or poorly sourced material may be challenged and removed.
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Kuznetsov, N.V.; Leonov, G.A.; Yuldashev, M.V.; Yuldashev, R.V. (2017).
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process). Thus, self-excited attractors, even coexisting in the case of
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that is born without loss of stability of stationary set is called a
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The theory of hidden oscillations and stability of dynamical systems
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The theory of hidden oscillations and stability of dynamical systems
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Bragin V.O.; Vagaitsev V.I.; Kuznetsov N.V.; Leonov G.A. (2011).
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attractors can turn out to be a challenging problem (see, e.g.
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Communications in Nonlinear Science and Numerical Simulation
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Classification of attractors as being hidden or self-excited
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IFAC Proceedings Volumes (IFAC World Congress Proceedings)
243:) attracts all nearby oscillations, then it is called a
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Journal of Computer and Systems Sciences International
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Journal of Computer and Systems Sciences International
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Leonov G.A.; Vagaitsev V.I.; Kuznetsov N.V. (2012).
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Leonov G.A.; Vagaitsev V.I.; Kuznetsov N.V. (2011).
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Kuznetsov N.V.; Leonov G.A.; Vagaitsev V.I. (2010).
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702:"Hidden attractor in smooth Chua systems"
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457:Theory of hidden oscillations
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