# Hidden attractors in dynamical systems

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

## Summary (5 min read)

Jump to: [1. Introduction] – [2. Hidden attractors: widespread objects in dynamical systems] – [2.2. Flows without fixed points (equilibria)] – [2.3. Flows with stable fixed point (equilibrium)] – [2.4. Flows with a line of fixed points (equilibria)] – [2.5. Electromechanical system without equilibria] – [2.6. Electromechanical model of the drilling system] – [2.9. Rabinovich–Fabricant model] – [3. Rare attractors and basin stability] – [4.1.1. Synthesis of scenario of hidden attractor birth based on homotopy and continuation] – [4.1.3. Gluhovsky–Dolzhansky system: from self–excited to hidden chaotic attractor] – [4.1.4. Analytical localization of attractor in dissipative dynamical systems] – [4.1.6. Perpetual points and its connection with hidden attractors] – [4.2. Controlling evolution of hidden attractors] – [5. Dynamics of coupled systems with hidden attractors] – [5.1. Identical systems – types of synchronization, clustering, chimera states with hidden heads] – [5.1.1. System of two coupled oscillators] – [5.1.2. Network of oscillators] – [5.2. Influence of parameters mismatch] – [6. Experimental observations of hidden attractors] and [7. Conclusions]

### 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.

### 2.2. 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.

### 4.1.4. 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.

### 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.

### 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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