Looking More Closely at the Rabinovich–Fabrikant System

Danca, Marius‐F.; Fečkan, Michal; Kuznetsov, Nikolay V.; Chen, Guanrong

doi:10.1142/s0218127416500383

Cited by 35 publications

(36 citation statements)

References 26 publications

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“…The system, initially designed as a physical system, describes the stochasticity arising from the modulation instability in a dissipative medium. However, as revealed numerically in [54] and [55], the system of integer order presents unusual and extremely rich dynamics, including multistability, an important ingredient for potential existence of hidden attractor. The equilibria are X * 0 and X * 1,2 ∓x * 1,2 , ±y * 1,2 , z * 1,2 , X * 3,4 ∓x * 3,4 , ±y * 3,4 , z * 3,4 ,…”

Section: Hidden Chaotic Attractor Of the Rabinovich-fabrikant Systemmentioning

confidence: 99%

Hidden chaotic attractors in fractional-order systems

Danca

2017

Nonlinear Dyn

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In this paper, we present a scheme for uncovering hidden chaotic attractors in nonlinear autonomous systems of fractional order. The stability of equilibria of fractional-order systems is analyzed. The underlying initial value problem is numerically integrated with the predictor-corrector Adams-Bashforth-Moulton algorithm for fractional-order differential equations. Three examples of fractional-order systems are considered: a generalized Lorenz system, the Rabinovich-Fabrikant system and a non-smooth Chua system.

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Section: Hidden Chaotic Attractor Of the Rabinovich-fabrikant Systemmentioning

confidence: 99%

Hidden chaotic attractors in fractional-order systems

Danca

2017

Nonlinear Dyn

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“…where a > 0 and b is the bifurcation parameter. The system, revealed numerically in [1], presents unusual and extremely rich dynamics, including multistability (coexistence of multiple attractors for a given set of parameters), which represents an important ingredient for potential existence of hidden attractor.…”

Section: The Rf Systemmentioning

confidence: 99%

“…Due to the complexity of the ODEs (third-order nonlinearities), a complete mathematical analysis such as stability of equilibria, existence of invariant sets, existence and convergence of heteroclinic or homoclinic orbits, has to be done numerically (mostly investigated in [1]).…”

Section: The Rf Systemmentioning

confidence: 99%

“…Contrarily, fixed-step-size schemes, such as the standard RungeKutta method (RK4), or the multi-step predictorcorrector LIL method [38], generally give more accurate results (see details in [1]). As revealed in [1], the RF system presents extremely rich dynamics. In this paper, we are concerned with chaotic attractors only.…”

Section: Hidden Transient Chaotic Attractorsmentioning

confidence: 99%

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Hidden transient chaotic attractors of Rabinovich–Fabrikant system

Danca

2016

Nonlinear Dyn

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In [1], it is shown that the Rabinovich-Fabrikant (RF) system admits self-excited and hidden chaotic attractors. In this paper, we further show that the RF system also admits a pair of symmetric transient hidden chaotic attractors. We reveal more extremely rich dynamics of this system, such as a new kind of "virtual saddles".Keywords Hidden transient chaotic attractor; Hidden attractor; Self-excited attractor; Rabinovich-Fabrikant system 1 Introduction Nowadays, the notion of self-excited and hidden attractor introduced by Leonov and Kuznetsov [2,3,4], has become a common subject (see e.g. [5,6,7,8,9,10,11,12,13,14,15,16]). The main characteristic of hidden attractors is that their basins of attraction do not intersect with arbitrary small neighborhood of any equilibrium point, while a basin of attraction of a self-excited attractor is associated with some unstable equilibrium. In this context, stationary points are less important for finding hidden attractors than for finding self-excited attractors. Self-excited attractors can be localized (excited) by standard computational procedures, by starting from a point in some neighborhood of an unstable equilibrium, while for localization of hidden attractors it is necessary to develop special numerical procedures. Some well-known classical chaotic and regular attractors (such as Lorenz 1 , Chen, Rösler, van der Pol, some Sprott systems, etc.) are self-excited attractors, 1 The possible existence in the Lorenz of a hidden chaotic attractor in the Lorenz system is an open problem [4].

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“…The paradox was first conceptualized as an abstraction of the phenomenon of flashing Brownian ratchets, [2][3][4][5][6][7][8][9] wherein diffusive particles exhibit unexpected drift when exposed to alternating periodic potentials. It has since been applied across a multitude of neighboring disciplines in the physical sciences and engineering-related fields, [10,11] such as diffusive and granular flow dynamics, [12][13][14] information thermodynamics, [15][16][17][18] chaos theory, [19][20][21][22][23][24][25] switching problems, [26][27][28] and quantum phenomena; [29][30][31][32][33][34][35][36][37][38] but a plethora of exciting applications has also been found in biology, which this paper focuses upon. Parrondo's paradox has nourished a synergistic interdisciplinary effort in which the existing mathematical and gametheoretic work is driving its rapid application in biology, where it has inspired new ideas and technical approaches.…”

Section: Introductionmentioning

confidence: 99%

Paradoxical Survival: Examining the Parrondo Effect across Biology

2019

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Parrondo's paradox, in which losing strategies can be combined to produce winning outcomes, has received much attention in mathematics and the physical sciences; a plethora of exciting applications has also been found in biology at an astounding pace. In this review paper, the authors examine a large range of recent developments of Parrondo's paradox in biology, across ecology and evolution, genetics, social and behavioral systems, cellular processes, and disease. Intriguing connections between numerous works are identified and analyzed, culminating in an emergent pattern of nested recurrent mechanics that appear to span the entire biological gamut, from the smallest of spatial and temporal scales to the largest—from the subcellular to the complete biosphere. In analyzing the macro perspective, the pivotal role that the paradox plays in the shaping of biological life becomes apparent, and its identity as a potential universal principle underlying biological diversity and persistence is uncovered. Directions for future research are also discussed in light of this new perspective.

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Looking More Closely at the Rabinovich–Fabrikant System

Cited by 35 publications

References 26 publications

Hidden chaotic attractors in fractional-order systems

Hidden chaotic attractors in fractional-order systems

Hidden transient chaotic attractors of Rabinovich–Fabrikant system

Paradoxical Survival: Examining the Parrondo Effect across Biology

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