In the

bifurcation theory Bifurcation theory is the Mathematics, mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential e ...

, a bounded

oscillation Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...

that is born without loss of stability of stationary set is called a hidden oscillation. In

nonlinear control Nonlinear control theory is the area of control theory which deals with systems that are nonlinear system, nonlinear, time-variant system, time-variant, or both. Control theory is an interdisciplinary branch of engineering and mathematics that ...

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 Kalman's conjecture or Kalman problem is a disproved conjecture on absolute stability of nonlinear control system with one scalar nonlinearity, which belongs to the sector of linear stability. Kalman's conjecture is a strengthening of Aizerman ...

). If a hidden oscillation (or a set of such hidden oscillations filling a compact subset of the

phase space The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...

of the

dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...

) attracts all nearby oscillations, then it is called a hidden attractor. For a

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

attractor In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain c ...

s in the phase space. While trivial attractors, i.e. stable equilibrium points, can be easily found analytically or numerically, the search of

periodic Periodicity or periodic may refer to: Mathematics * Bott periodicity theorem, addresses Bott periodicity: a modulo-8 recurrence relation in the homotopy groups of classical groups * Periodic function, a function whose output contains values tha ...

and

chaotic Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program aired on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kids, Cartoon Netwo ...

attractors can turn out to be a challenging problem (see, e.g. the second part of Hilbert's 16th problem).

Classification of attractors as being hidden or self-excited

To identify a local attractor in a physical or numerical experiment, one needs to choose an initial system’s state in attractor’s basin of attraction 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

s in the

Definition A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitio ...

. ''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.'' The classification of attractors as being hidden or self-excited was introduced by G. Leonov and N. Kuznetsov in connection with the discovery of the hidden

Chua attractor Chua may refer to: * Hokkien or Teochew Romanisation of Cai (surname) People named Chua * Leon O. Chua (born 1936), American electrical engineer and computer scientist * Amy Chua (born 1962), American corporate lawyer, legal academic and autho ...

for the first time in 2009. 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. Self-Excited attractor in Chua circuit

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 Self-oscillation is the generation and maintenance of a periodic motion by a source of power that lacks any corresponding periodicity. The oscillator itself controls the phase with which the external power acts on it. Self-oscillators are therefor ...

process). Thus, self-excited attractors, even coexisting in the case of

multistability In a dynamical system, multistability is the property of having multiple Stability theory, stable equilibrium points in the vector space spanned by the states in the system. By mathematical necessity, there must also be unstable equilibrium points ...

, can be easily revealed and visualized numerically. In the

Lorenz system The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the ...

, 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 Van der Pol (also "Van de Pol", "Van de Poll", "Van den Pol" or "Van Pol") is a Dutch, toponymic surname A toponymic surname or habitational surname or byname is a surname or byname derived from a place name,Beluosov–Zhabotinsky,

Rössler Rössler is a surname and may refer to: * Ervin Rössler (1876–1933), Croatian zoologist * Fritz Rössler (1912–1987), German Nazi politician * Günter Rössler (1926–2012), German photographer and photo-journalist * Jaroslav Rössler (1902� ...

, 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.

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 the

Sommerfeld effect In mechanics, Sommerfeld effect is a phenomenon arising from feedback in the energy exchange between vibrating systems: for example, when for the rocking table, under given conditions, energy transmitted to the motor resulted not in higher revoluti ...

(1902), in systems with only one equilibrium, which is stable e.g. the counterexamples to

Aizerman's conjecture In nonlinear control, Aizerman's conjecture or Aizerman problem states that a linear system in feedback with a sector nonlinearity would be stable if the linear system is stable for any linear gain of the sector. This conjecture, proposed by Mark Ar ...

(1949) and

(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. 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 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, 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

homotopy In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. ...

and

numerical continuation Numerical continuation is a method of computing approximate solutions of a system of parameterized nonlinear equations, :F(\mathbf u,\lambda) = 0. The parameter \lambda is usually a real scalar and the ''solution'' \mathbf u is an ''n''-vector. Fo ...

: 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.

Theory of hidden oscillations

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 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).

References

{{Reflist, 2

Books

* Chaotic Systems with Multistability and Hidden Attractors (Eds.: Wang, Kuznetsov, Chen), Springer, 2021
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
doi:10.1007/978-3-319-71243-7

Selected lectures

N.Kuznetsov, Invited lecture ''The theory of hidden oscillations and stability of dynamical systems'', Int. Workshop on Applied Mathematics, Czech Republic, 2021

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
Dynamical systems Oscillation Chaos theory Nonlinear systems Hidden oscillation