hidden attractor
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In the bifurcation theory, a bounded oscillation that is born without loss of stability of stationary set is called a hidden oscillation. In nonlinear control 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. Kalman's conjecture). If a hidden oscillation (or a set of such hidden oscillations filling a compact subset of the phase space of the dynamical system) attracts all nearby oscillations, then it is called a hidden attractor. For a dynamical system 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 multistable and have coexisting local attractors in the phase space. While trivial attractors, i.e. stable equilibrium points, can be easily found analytically or numerically, the search of periodic and chaotic attractors can turn out to be a challenging problem (see, e.g. the second part of Hilbert's 16th problem).
= Self-excited attractors =
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 equilibrium, is attracted to the state of oscillation and then traces it (see, e.g. self-oscillation process). Thus, self-excited attractors, even coexisting in the case of multistability, can be easily revealed and visualized numerically. In the Lorenz system, 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 Van der Pol, Beluosov–Zhabotinsky, Rössler, Chua, Hénon dynamical systems are self-excited.
A conjecture is that the Lyapunov dimension of a self-excited attractor 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.{{Cite journal |
first1=N.V. |last1=Kuznetsov|first2=G.A. |last2=Leonov|first3=T.N. |last3=Mokaev|first4=A. |last4=Prasad|first5=M.D. |last5=Shrimali |
title=Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system|
journal=Nonlinear Dynamics|
volume=92 | issue=2 |year=2018 |pages=267–285 |
doi=10.1007/s11071-018-4054-z|arxiv=1504.04723 |bibcode=2018NonDy..92..267K |s2cid=54706479}}
= Hidden attractors =
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 Sommerfeld effect (1902), in the systems with only one equilibrium, which is stable: e.g. counterexamples to the Aizerman's conjecture (1949) and Kalman's conjecture (1957) on the monostability of nonlinear control systems. One of the first related theoretical problems is the second part of Hilbert's 16th problem 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.{{cite journal |
author1=Kuznetsov N. V. | author2=Leonov G. A. |
year = 2014 |
title = Hidden attractors in dynamical systems: systems with no equilibria, multistability and coexisting attractors|
journal = IFAC Proceedings Volumes (IFAC World Congress Proceedings) |
volume = 47 | issue =3 | pages = 5445–5454|
doi =10.3182/20140824-6-ZA-1003.02501}}
author1=Dudkowski D. |author2= Jafari S. |author3= Kapitaniak T.|author4=Kuznetsov N. V. | author5=Leonov G. A. |author6=Prasad A.|
year = 2016 |
title = Hidden attractors in dynamical systems |
journal = Physics Reports |
volume = 637 | pages = 1–50|
doi =10.1016/j.physrep.2016.05.002|bibcode= 2016PhR...637....1D |url= http://urn.fi/URN:NBN:fi:jyu-201607013450 }}
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.{{cite journal
| last1=Kuznetsov | first1=N.V.
| last2=Leonov | first2=G.A.
| last3=Yuldashev | first3=M.V.
| last4=Yuldashev | first4=R.V.
| title=Hidden attractors in dynamical models of phase-locked loop circuits: limitations of simulation in MATLAB and SPICE
| journal=Communications in Nonlinear Science and Numerical Simulation
| volume=51 | pages=39–49
| doi=10.1016/j.cnsns.2017.03.010
| year=2017
| bibcode=2017CNSNS..51...39K
| url=http://urn.fi/URN:NBN:fi:jyu-201704031870
}}). While for two-dimensional systems, hidden oscillations can be investigated using analytical methods (see, e.g., the results on the second part of Hilbert's 16th problem), 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,{{Cite journal |
first1=G. |last1=Chen |first2=N.V. |last2=Kuznetsov| first3=G.A. |last3=Leonov |first4=T.N. |last4=Mokaev|
title=Hidden attractors on one path: Glukhovsky-Dolzhansky, Lorenz, and Rabinovich systems|
journal=International Journal of Bifurcation and Chaos in Applied Sciences and Engineering|
volume=27 |issue=8 |year=2015 |pages=art. num. 1750115|
doi=10.1142/S0218127417501152|arxiv=1705.06183|s2cid=21425647 }}
There are corresponding effective methods based on
homotopy and numerical continuation: 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.
References
{{Reflist|2}}
Books
- Chaotic Systems with Multistability and Hidden Attractors (Eds.: Wang, Kuznetsov, Chen), Springer, 2021 ([https://www.springer.com/gp/book/9783030758202 doi:10.1007/978-3-030-75821-9])
- Nonlinear Dynamical Systems with Self-Excited and Hidden Attractors (Eds.: Pham, Vaidyanathan, Volos et al.), Springer, 2018 ([https://dx.doi.org/10.1007/978-3-319-71243-7 doi:10.1007/978-3-319-71243-7])
Selected lectures
- [https://www.youtube.com/watch?v=l1hI2U-Bl28 N.Kuznetsov, Invited lecture The theory of hidden oscillations and stability of dynamical systems, Int. Workshop on Applied Mathematics, Czech Republic, 2021]
- [https://www.youtube.com/watch?v=WbfXz4iBN0I 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]