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Journal ArticleDOI

Hidden attractors in dynamical systems

03 Jun 2016-Physics Reports (Elsevier BV * North-Holland)-Vol. 637, pp 1-50

AbstractComplex dynamical systems, ranging from the climate, ecosystems to financial markets and engineering applications typically have many coexisting attractors This property of the system is called multistability The final state, ie, the attractor on which the multistable system evolves strongly depends on the initial conditions Additionally, such systems are very sensitive towards noise and system parameters so a sudden shift to a contrasting regime may occur To understand the dynamics of these systems one has to identify all possible attractors and their basins of attraction Recently, it has been shown that multistability is connected with the occurrence of unpredictable attractors which have been called hidden attractors The basins of attraction of the hidden attractors do not touch unstable fixed points (if exists) and are located far away from such points Numerical localization of the hidden attractors is not straightforward since there are no transient processes leading to them from the neighborhoods of unstable fixed points and one has to use the special analytical–numerical procedures From the viewpoint of applications, the identification of hidden attractors is the major issue The knowledge about the emergence and properties of hidden attractors can increase the likelihood that the system will remain on the most desirable attractor and reduce the risk of the sudden jump to undesired behavior We review the most representative examples of hidden attractors, discuss their theoretical properties and experimental observations We also describe numerical methods which allow identification of the hidden attractors

Topics: Attractor (57%), Multistability (56%), Dynamical systems theory (50%)

Summary (5 min read)

1. Introduction

  • The climate [1–5], a number of ecosystems (e.g. the Amazon rainforest) [6–9], the human brain [10, 11], arrays of coupled lasers [12–14], financial markets [15, 16] and many applied engineering systems [17–20] are modeled by complex dynamical systems which are characterized by the existence of many coexisting attractors.
  • Such a shift can lead to the catastrophic events ranging from sudden climate changes, serious diseases to financial crises and disasters of commercial devices [32].
  • Recently, it has been shown that multistability is connected with the occurrence of unpredictable attractors [21–31] which have been called the hidden attractors [37–40].
  • The examples of the experimental realizations of the systems with hidden attractors are given in Sec. 6.

2. Hidden attractors: widespread objects in dynamical systems

  • The localization from vicinity of fixed point F0 is shown (two views (a) and (b)).
  • The basin of attraction for a hidden attractor is not connected with any unstable fixed point.
  • The hidden attractors are observed in the systems with no unstable fixed points or with one stable fixed point (a special case of multistability).
  • The authors start with simple flows and continue to present the real physical systems.

2.2. Flows without fixed points (equilibria)

  • Jafari and Sprott [66] have performed systematic search to find the simplest three–dimensional chaotic systems with quadratic nonlinearities and no equilibria.
  • Seventeen simple systems that show chaos have been found.
  • Other examples of chaotic and hyperchaotic systems with no equilibrium and hidden attractors can be found in [51, 67–73].

2.3. Flows with stable fixed point (equilibrium)

  • The example of such unusual chaotic flow (with only one stable equilibrium) has been designed by Wang and Chen [74].
  • Later Jafari and Sprott have identified 23 simple systems with this property.
  • To do that, they have performed systematic computer search for chaos in three–dimensional autonomous systems with quadratic nonlinearities and a single equilibrium (stable according to the Routh–Hurwitz criterion).
  • At least one point attractor coexists with a hidden strange attractor for these types of chaotic flows.

2.4. Flows with a line of fixed points (equilibria)

  • After proposing a chaotic system with any number of equilibria by Wang and Chen in [73], Jafari and Sprott in [83] have introduced simple chaotic systems with a line of equilibria.
  • They have been inspired by the structure of the conservative Sprott case A system [61] and have considered a general parametric form of it with quadratic nonlinearities.
  • With exhaustive computer search, nine simple cases have been found.
  • The strange attractor of this system is hidden, from computational point of view, since there are uncountably many unstable points on the equilibrium line of which only a tiny portion intersects the basin of the chaotic attractor.

2.5. Electromechanical system without equilibria

  • Hidden attractors appear naturally in systems without equilibria, describing various mechanical and electromechanical models with rotation and electrical circuits with cylindrical phase space.
  • He has studied the oscillations caused by a motor driving an unbalanced weight and discovered the resonance capture (Sommerfeld effect).
  • The x variable is the displacement of the cart from its equilibrium, while θ is the rotational angle of the rotor.
  • In the space (x, ẋ, θ̇) there is a hidden attractor for u = 0.49 shown in Fig. 11(a), where the red curve corresponds to the regular start of the system, i.e. zero initial conditions, and is attracted by an attractor.
  • The blue curve corresponds to the regular start of the system, i.e. zero initial conditions and demonstrates the Sommerfeld effect, while the red curve is attracted to the desired operation of the system.

2.6. Electromechanical model of the drilling system

  • In the works [89, 90] a double–mass mathematical model of the drilling system is studied by the group of researchers from the Eindhoven University of Technology.
  • It consists of upper and lower discs connected with each other by a steel string.
  • Here θu and θl are angular displacements of upper and lower discs; Ju and Jl are constant inertia torques; b is rotational friction ; kθ is the torsional spring stiffness; km is the motor constant; v is the constant input voltage; Tfu(θ̇u) and Tfl(θ̇l) are friction torques acting on the upper and the lower discs.

2.9. Rabinovich–Fabricant model

  • Mikhail Rabinovich and Anatoly Fabrikant [104] introduced and analyzed from a physical point of view a model describing the stochasticity arising from the modulation instability in a non–equilibrium dissipative medium.
  • This is a simplification of a complex nonlinear parabolic equation modelling different physical systems, such as: the Tollmien–Schlichting waves in hydrodynamic flows, wind waves on water, concentration waves during chemical reactions in a medium, where diffusion occurs, Langmuir waves in plasma, etc. [104].
  • According to the article [105], the authors consider a = 0.1, b ∈ (bmin, b∗) and take the values of b for which in the phase space there are a chaotic attractor besides the stable equilibria X∗1,2.
  • In order to integrate system (18) the authors use the LIL method of order 4 (MATLAB code taken from [106]).

3. Rare attractors and basin stability

  • The system in non–autonomous and its attractor can be identified as the projection on (x, ẋ) plane.
  • In each subfigure, the basins of attraction of coexisting attractors for fixed ω value are shown.
  • In Fig. 17(a) the basins of attractors AT , A7T , two A9T and A11T are shown in green, yellow, blue, red and pink colors respectively.
  • The second most probable attractor is the periodic one A35T (white basin) with p(A35T ) = 0.1207, while the previously dominant chaotic state ACh is now the rare one with probability at p(ACh) = 0.0785.

4.1.1. Synthesis of scenario of hidden attractor birth based on homotopy and continuation

  • One of the effective methods for the numerical localization of hidden attractors is based on a homotopy and numerical continuation.
  • The authors construct a sequence of similar systems such that the initial data for numerically computing the oscillating solution ( oscillation) can be obtained analytically for the first system.
  • That is, instead of analyzing the scenario of a transition into chaos, the authors can synthesize it.
  • Then the authors numerically investigate the changes to the shape of the attractor obtained for λ1 = a1.
  • The application of this method is demonstrated below.

4.1.3. Gluhovsky–Dolzhansky system: from self–excited to hidden chaotic attractor

  • Let us construct a line segment on the plane (a, r) that intersects a boundary of the stability domain of equilibria S1,2 (see Fig. 21).
  • Here for the selected path and partition, the authors can visualize a hidden attractor of system (35) (see Fig. 22).

4.1.4. Analytical localization of attractor in dissipative dynamical systems

  • In the previous sections, the authors have considered the numerical localization of various self–excited and hidden attractors of system (35).
  • Consider an autonomous differential equation (2) (cf. Sec. 2).
  • The authors can effectively prove dissipativity by constructing Lyapunov function [139, 140].
  • Note that for system (35) the ellipsoidal absorbing set B0 can be improved using special additional transformations and Yudovich’s theorem (see, e.g. [145]), similarly to [146] for the Lorenz system.

4.1.6. Perpetual points and its connection with hidden attractors

  • The fixed points of any dynamical system are the ones where velocity and acceleration of the system simultaneously become zero.
  • The results of their calculations are shown in Fig. 24.
  • As the background the authors have used the basins of attraction of system (43) (these change periodically with the period of the excitation), where each color correspond to the basin of different attractors.
  • This suggest that the connection between the perpetual points and the hidden attractors may be correct in one way, i.e. if the system has perpetual points, then they may lead to its hidden states.
  • As it can be seen, the perpetual points give only some possibilities in the research of dynamical systems.

4.2. Controlling evolution of hidden attractors

  • Over the last few decades the control of oscillating motions in dynamical systems has been the topic of intense research from both theoretical and experimental points of view [149–154].
  • Various approaches have been established to control such motion.
  • One of the important schemes for the stabilization of the fixed point can be done by using the phenomenon of amplitude or oscillation death [168, 169] due to interactions between the coupled oscillators.
  • The hidden chaotic attractor (Fig. 26 (a)) does continue to exist with small modification due to the presence of the linear term (Eq. (47)).
  • Considering the importance of understanding the multistability and its control several attempts have been made in the last decades.

5. Dynamics of coupled systems with hidden attractors

  • The studies of the coupled systems with hidden attractors are the natural consequence of the development of research in the topic of hidden states.
  • The knowledge in this area is still insufficient and the results are in very early stage.
  • Some considerations on the issue have been shown in [68].
  • Originally chimera states have been discovered by Kuramoto [182] in his studies about non–locally coupled phase oscillators.
  • All the phenomena that the authors have observed in the coupled systems with hidden attractors (and also the ones that still need to be identified and described) arise from the complex nature of the systems themselves.

5.1. Identical systems – types of synchronization, clustering, chimera states with hidden heads

  • In their research the authors have focused on the coupled systems which consist of identical units with hidden attractors.
  • The authors present the results obtained for the smallest possible group, i.e., two coupled oscillators, as well as the effects of their investigations in large networks.
  • In all their considerations the authors have used van der Pol–Duffing oscillator as the basic unit of the studied models.

5.1.1. System of two coupled oscillators

  • Let us consider the system of two coupled van der Pol–Duffing oscillators, which are externally excited by the sine functions.
  • The method applied to equations (48) is similar to the one presented in the quoted Section.
  • In the right panel in Fig. 27 the corresponding projections of perpetual points on position–velocity plane are marked by blue dots.
  • Another new states are shown in Fig. 28(c–d), where one attractor of period 4 (in–phase synchronization) and two solutions of period 1 (anti–phase synchronization) are shown respectively.
  • It should be emphasized, that no state shown in Figs. 28–29 has been observed for a single system using perpetual points.

5.1.2. Network of oscillators

  • To investigate more complex behaviour of the coupled dynamical systems with hidden attractors the authors have considered the network of coupled van der Pol–Duffing oscillators introduced above.
  • The examples of the observed states for fixed coupling radius r = 0.3 and changing coupling strength are shown in Fig. 31.
  • What should be noted, the blue attractor is the original state that has been found for the single oscillator (shown in Fig. 30 as yellow dots), while the second periodic attractor and the chaotic one are new states that do not appear in the original system for given parameter values.
  • In each subfigure (a)–(e) the snapshots of position (upper left panel) and mean velocity (lower left panel), and the space–time mean velocity plot (right panel) are shown.
  • It should be emphasized, that for each color group described above, the oscillators that belong to one chosen structure represent the same local dynamics in the qualitative sense.

5.2. Influence of parameters mismatch

  • In Sec. 5.1 the examples of dynamics of coupled systems with hidden attractors have been shown, both for the smallest system of two and for the network of hundred units.
  • The results for fixed coupling radius r = 0.3 and coupling strength d considered as the bifurcation parameter are shown in Fig. 32.
  • In the right panel the bifurcation diagrams for chosen oscillators i (increasing from the top to the bottom, i = 7, 15, 25, 45, 99 respectively) have been presented.
  • Also, the red dots divide into two subgroups and no longer oscillate on the same level.

6. Experimental observations of hidden attractors

  • Hidden attractors can be observed in various nonlinear control systems, like this schematically shown in Fig. 34.
  • As the first example let us consider the circuit shown in Fig.36 Hidden chaotic attractors of the circuit (52) with a line equilibrium in different planes: (a) νc1νc2 , (b) νc1νc3 , (c) νc2νc3 .
  • Appropriate values of the electronic components are used.

7. Conclusions

  • The authors give evidence that the hidden attractors can be expected in a great number of dynamical systems ranging from low–dimensional (2–dimensional) ones to the high–dimensional networks of coupled oscillators.
  • In applied systems this switch is equivalent to the catastrophic bifurcation from the desired to the undesired regime.
  • Contrary to the self–excited attractors the hidden attractors do not touch unstable fixed points in their basins of attractions.
  • The dynamics of the network consisting of such units with hidden attractors can become very complex.
  • The authors would like to thank Dr. Viet Thanh Pham, Professor J. C. Sprott, Professor Seyed Mohammad Reza Hashemi Golpayegani, Dr. Sifeu Takougang Kingni, Mrs. Fahimeh Nazarimehr, for help and comments which enhanced the quality of this paper.

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