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N. V. Kuznetsov - One of the best experts on this subject based on the ideXlab platform.
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Attractor as a convex combination of a set of Attractors
Communications in Nonlinear Science and Numerical Simulation, 2021Co-Authors: Mariusf Danca, N. V. Kuznetsov, Michal Fĕckan, Guanrong ChenAbstract:Abstract This paper presents an effective approach to constructing numerical Attractors of a general class of continuous homogenous dynamical systems: decomposing an Attractor as a convex combination of a set of other existing Attractors. For this purpose, the convergent Parameter Switching (PS) numerical method is used to integrate the underlying dynamical system. The method is built on a convergent fixed step-size numerical method for ODEs. The paper shows that the PS algorithm, incorporating two binary operations, can be used to approximate any numerical Attractor via a convex combination of some existing Attractors. Several examples are presented to show the effectiveness of the proposed method.
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finite time lyapunov dimension and hidden Attractor of the rabinovich system
Nonlinear Dynamics, 2018Co-Authors: N. V. Kuznetsov, Awadhesh Prasad, G A Leonov, T N Mokaev, Manish Dev ShrimaliAbstract:The Rabinovich system, describing the process of interaction between waves in plasma, is considered. It is shown that the Rabinovich system can exhibit a hidden Attractor in the case of multistability as well as a classical self-excited Attractor. The hidden Attractor in this system can be localized by analytical/numerical methods based on the continuation and perpetual points. The concept of finite-time Lyapunov dimension is developed for numerical study of the dimension of Attractors. A conjecture on the Lyapunov dimension of self-excited Attractors and the notion of exact Lyapunov dimension are discussed. A comparative survey on the computation of the finite-time Lyapunov exponents and dimension by different algorithms is presented. An adaptive algorithm for studying the dynamics of the finite-time Lyapunov dimension is suggested. Various estimates of the finite-time Lyapunov dimension for the hidden Attractor and hidden transient chaotic set in the case of multistability are given.
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approximating hidden chaotic Attractors via parameter switching
Chaos, 2018Co-Authors: Mariusf Danca, N. V. Kuznetsov, Guanrong ChenAbstract:In this paper, the problem of approximating hidden chaotic Attractors of a general class of nonlinear systems is investigated. The parameter switching (PS) algorithm is utilized, which switches the control parameter within a given set of values with the initial value problem numerically solved. The PS-generated Attractor approximates the Attractor obtained by averaging the control parameter with the switched values, which represents the hidden chaotic Attractor. The hidden chaotic Attractors of a generalized Lorenz system and the Rabinovich-Fabrikant system are simulated for illustration.
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hidden Attractors in dynamical systems
Physics Reports, 2016Co-Authors: Dawid Dudkowski, N. V. Kuznetsov, Tomasz Kapitaniak, G A Leonov, Sajad Jafari, Awadhesh PrasadAbstract:Abstract Complex 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, i.e., 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.
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control of multistability in hidden Attractors
European Physical Journal-special Topics, 2015Co-Authors: Pooja Rani Sharma, Manish Dev Shrimali, Awadhesh Prasad, N. V. Kuznetsov, Georgievich Aleksandr LeonovAbstract:Hidden Attractors have a basin of attraction which is not connected with unstable equilibrium. Certain systems with hidden Attractor show multistability for a range of parameter. Multistability or coexistence of different Attractors in nonlinear systems often creates inconvenience and therefore, needs to be avoided to obtain a desired specific output from the system. We discuss the control of multistability in the hidden Attractor through the scheme of linear augmentation, that can drive the multistable system to a monostable state. With the proper choice of control parameters a shift from multistability to monostability can be achieved. This transition from multiple Attractors to a single Attractor is confirmed by calculating the basin size as a measure. When a nonlinear system with hidden Attractors is coupled with a linear system, two important transitions are observed with the increase of coupling strength: transition from multistability to monostability and then stabilization of newly created equilibrium point via suppression of oscillations.
Guanrong Chen - One of the best experts on this subject based on the ideXlab platform.
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Attractor as a convex combination of a set of Attractors
Communications in Nonlinear Science and Numerical Simulation, 2021Co-Authors: Mariusf Danca, N. V. Kuznetsov, Michal Fĕckan, Guanrong ChenAbstract:Abstract This paper presents an effective approach to constructing numerical Attractors of a general class of continuous homogenous dynamical systems: decomposing an Attractor as a convex combination of a set of other existing Attractors. For this purpose, the convergent Parameter Switching (PS) numerical method is used to integrate the underlying dynamical system. The method is built on a convergent fixed step-size numerical method for ODEs. The paper shows that the PS algorithm, incorporating two binary operations, can be used to approximate any numerical Attractor via a convex combination of some existing Attractors. Several examples are presented to show the effectiveness of the proposed method.
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approximating hidden chaotic Attractors via parameter switching
arXiv: Chaotic Dynamics, 2018Co-Authors: Mariusf Danca, Nikolay Kuznetsovc, Guanrong ChenAbstract:In this paper, the problem of approximating hidden chaotic Attractors of a general class of nonlinear systems is investigated. The Parameter Switching (PS) algorithm is utilized, which switches the control parameter within a given set of values with the initial value problem numerically solved. The PS-generated generated Attractor approximates the Attractor obtained by averaging the control parameter with the switched values, which represents the hidden chaotic Attractor. The hidden chaotic Attractors of a generalized Lorenz system and the Rabinovich-Fabrikant system are simulated for illustration
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approximating hidden chaotic Attractors via parameter switching
Chaos, 2018Co-Authors: Mariusf Danca, N. V. Kuznetsov, Guanrong ChenAbstract:In this paper, the problem of approximating hidden chaotic Attractors of a general class of nonlinear systems is investigated. The parameter switching (PS) algorithm is utilized, which switches the control parameter within a given set of values with the initial value problem numerically solved. The PS-generated Attractor approximates the Attractor obtained by averaging the control parameter with the switched values, which represents the hidden chaotic Attractor. The hidden chaotic Attractors of a generalized Lorenz system and the Rabinovich-Fabrikant system are simulated for illustration.
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a four wing chaotic Attractor generated from a new 3 d quadratic autonomous system
Chaos Solitons & Fractals, 2008Co-Authors: Guanrong Chen, Michael Antonie Van Wyk, Barend Jacobus Van Wyk, Yuhui ZhangAbstract:Abstract This paper introduces a new 3-D quadratic autonomous system, which can generate two coexisting single-wing chaotic Attractors and a pair of diagonal double-wing chaotic Attractors. More importantly, the system can generate a four-wing chaotic Attractor with very complicated topological structures over a large range of parameters. Some basic dynamical behaviors and the compound structure of the new 3-D system are investigated. Detailed bifurcation analysis illustrates the evolution processes of the system among two coexisting sinks, two coexisting periodic orbits, two coexisting single-wing chaotic Attractors, major and minor diagonal double-wing chaotic Attractors, and a four-wing chaotic Attractor. Poincare-map analysis shows that the system has extremely rich dynamics. The physical existence of the four-wing chaotic Attractor is verified by an electronic circuit. Finally, spectral analysis shows that the system has an extremely broad frequency bandwidth, which is very desirable for engineering applications such as secure communications.
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inherent frequency and spatial decomposition of the lorenz chaotic Attractor
arXiv: Chaotic Dynamics, 2004Co-Authors: Gonzalo Alvarez, Guanrong ChenAbstract:This letter suggests a new way to investigate 3-D chaos in spatial and frequency domains simultaneously. After spatially decomposing the Lorenz Attractor into two separate scrolls with peaked spectra and a 1-D discrete-time zero-crossing series with a wide-band spectrum, it is found that the Lorenz chaotic Attractor has an inherent frequency uniquely determined by the three system parameters. This result implies that chaos in the Lorenz Attractor is mainly exhibited when the trajectory crosses from one scroll to another, not within the two scrolls. This is also true for some other double-scroll Lorenz-like chaotic Attractors, such as Chua's Attractor. Some possible applications of the inherent frequency and the spatial decomposition are also discussed.
Manish Dev Shrimali - One of the best experts on this subject based on the ideXlab platform.
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finite time lyapunov dimension and hidden Attractor of the rabinovich system
Nonlinear Dynamics, 2018Co-Authors: N. V. Kuznetsov, Awadhesh Prasad, G A Leonov, T N Mokaev, Manish Dev ShrimaliAbstract:The Rabinovich system, describing the process of interaction between waves in plasma, is considered. It is shown that the Rabinovich system can exhibit a hidden Attractor in the case of multistability as well as a classical self-excited Attractor. The hidden Attractor in this system can be localized by analytical/numerical methods based on the continuation and perpetual points. The concept of finite-time Lyapunov dimension is developed for numerical study of the dimension of Attractors. A conjecture on the Lyapunov dimension of self-excited Attractors and the notion of exact Lyapunov dimension are discussed. A comparative survey on the computation of the finite-time Lyapunov exponents and dimension by different algorithms is presented. An adaptive algorithm for studying the dynamics of the finite-time Lyapunov dimension is suggested. Various estimates of the finite-time Lyapunov dimension for the hidden Attractor and hidden transient chaotic set in the case of multistability are given.
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control of multistability in hidden Attractors
European Physical Journal-special Topics, 2015Co-Authors: Pooja Rani Sharma, Manish Dev Shrimali, Awadhesh Prasad, N. V. Kuznetsov, Georgievich Aleksandr LeonovAbstract:Hidden Attractors have a basin of attraction which is not connected with unstable equilibrium. Certain systems with hidden Attractor show multistability for a range of parameter. Multistability or coexistence of different Attractors in nonlinear systems often creates inconvenience and therefore, needs to be avoided to obtain a desired specific output from the system. We discuss the control of multistability in the hidden Attractor through the scheme of linear augmentation, that can drive the multistable system to a monostable state. With the proper choice of control parameters a shift from multistability to monostability can be achieved. This transition from multiple Attractors to a single Attractor is confirmed by calculating the basin size as a measure. When a nonlinear system with hidden Attractors is coupled with a linear system, two important transitions are observed with the increase of coupling strength: transition from multistability to monostability and then stabilization of newly created equilibrium point via suppression of oscillations.
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finite time lyapunov dimension and hidden Attractor of the rabinovich system
arXiv: Chaotic Dynamics, 2015Co-Authors: N. V. Kuznetsov, Awadhesh Prasad, G A Leonov, T N Mokaev, Manish Dev ShrimaliAbstract:The Rabinovich system, describing the process of interaction between waves in plasma, is considered. It is shown that the Rabinovich system can exhibit a {hidden Attractor} in the case of multistability as well as a classical {self-excited Attractor}. The hidden Attractor in this system can be localized by analytical-numerical methods based on the {continuation} and {perpetual points}. For numerical study of the Attractors' dimension the concept of {finite-time Lyapunov dimension} is developed. A conjecture on the Lyapunov dimension of self-excited Attractors and the notion of {exact Lyapunov dimension} are discussed. A comparative survey on the computation of the finite-time Lyapunov exponents by different algorithms is presented and an approach for a reliable numerical estimation of the finite-time Lyapunov dimension is suggested. Various estimates of the finite-time Lyapunov dimension for the hidden Attractor and hidden transient chaotic set in the case of multistability are given.
Awadhesh Prasad - One of the best experts on this subject based on the ideXlab platform.
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finite time lyapunov dimension and hidden Attractor of the rabinovich system
Nonlinear Dynamics, 2018Co-Authors: N. V. Kuznetsov, Awadhesh Prasad, G A Leonov, T N Mokaev, Manish Dev ShrimaliAbstract:The Rabinovich system, describing the process of interaction between waves in plasma, is considered. It is shown that the Rabinovich system can exhibit a hidden Attractor in the case of multistability as well as a classical self-excited Attractor. The hidden Attractor in this system can be localized by analytical/numerical methods based on the continuation and perpetual points. The concept of finite-time Lyapunov dimension is developed for numerical study of the dimension of Attractors. A conjecture on the Lyapunov dimension of self-excited Attractors and the notion of exact Lyapunov dimension are discussed. A comparative survey on the computation of the finite-time Lyapunov exponents and dimension by different algorithms is presented. An adaptive algorithm for studying the dynamics of the finite-time Lyapunov dimension is suggested. Various estimates of the finite-time Lyapunov dimension for the hidden Attractor and hidden transient chaotic set in the case of multistability are given.
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hidden Attractors in dynamical systems
Physics Reports, 2016Co-Authors: Dawid Dudkowski, N. V. Kuznetsov, Tomasz Kapitaniak, G A Leonov, Sajad Jafari, Awadhesh PrasadAbstract:Abstract Complex 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, i.e., 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.
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control of multistability in hidden Attractors
European Physical Journal-special Topics, 2015Co-Authors: Pooja Rani Sharma, Manish Dev Shrimali, Awadhesh Prasad, N. V. Kuznetsov, Georgievich Aleksandr LeonovAbstract:Hidden Attractors have a basin of attraction which is not connected with unstable equilibrium. Certain systems with hidden Attractor show multistability for a range of parameter. Multistability or coexistence of different Attractors in nonlinear systems often creates inconvenience and therefore, needs to be avoided to obtain a desired specific output from the system. We discuss the control of multistability in the hidden Attractor through the scheme of linear augmentation, that can drive the multistable system to a monostable state. With the proper choice of control parameters a shift from multistability to monostability can be achieved. This transition from multiple Attractors to a single Attractor is confirmed by calculating the basin size as a measure. When a nonlinear system with hidden Attractors is coupled with a linear system, two important transitions are observed with the increase of coupling strength: transition from multistability to monostability and then stabilization of newly created equilibrium point via suppression of oscillations.
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finite time lyapunov dimension and hidden Attractor of the rabinovich system
arXiv: Chaotic Dynamics, 2015Co-Authors: N. V. Kuznetsov, Awadhesh Prasad, G A Leonov, T N Mokaev, Manish Dev ShrimaliAbstract:The Rabinovich system, describing the process of interaction between waves in plasma, is considered. It is shown that the Rabinovich system can exhibit a {hidden Attractor} in the case of multistability as well as a classical {self-excited Attractor}. The hidden Attractor in this system can be localized by analytical-numerical methods based on the {continuation} and {perpetual points}. For numerical study of the Attractors' dimension the concept of {finite-time Lyapunov dimension} is developed. A conjecture on the Lyapunov dimension of self-excited Attractors and the notion of {exact Lyapunov dimension} are discussed. A comparative survey on the computation of the finite-time Lyapunov exponents by different algorithms is presented and an approach for a reliable numerical estimation of the finite-time Lyapunov dimension is suggested. Various estimates of the finite-time Lyapunov dimension for the hidden Attractor and hidden transient chaotic set in the case of multistability are given.
G A Leonov - One of the best experts on this subject based on the ideXlab platform.
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finite time lyapunov dimension and hidden Attractor of the rabinovich system
Nonlinear Dynamics, 2018Co-Authors: N. V. Kuznetsov, Awadhesh Prasad, G A Leonov, T N Mokaev, Manish Dev ShrimaliAbstract:The Rabinovich system, describing the process of interaction between waves in plasma, is considered. It is shown that the Rabinovich system can exhibit a hidden Attractor in the case of multistability as well as a classical self-excited Attractor. The hidden Attractor in this system can be localized by analytical/numerical methods based on the continuation and perpetual points. The concept of finite-time Lyapunov dimension is developed for numerical study of the dimension of Attractors. A conjecture on the Lyapunov dimension of self-excited Attractors and the notion of exact Lyapunov dimension are discussed. A comparative survey on the computation of the finite-time Lyapunov exponents and dimension by different algorithms is presented. An adaptive algorithm for studying the dynamics of the finite-time Lyapunov dimension is suggested. Various estimates of the finite-time Lyapunov dimension for the hidden Attractor and hidden transient chaotic set in the case of multistability are given.
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hidden Attractors in dynamical systems
Physics Reports, 2016Co-Authors: Dawid Dudkowski, N. V. Kuznetsov, Tomasz Kapitaniak, G A Leonov, Sajad Jafari, Awadhesh PrasadAbstract:Abstract Complex 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, i.e., 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.
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homoclinic orbits and self excited and hidden Attractors in a lorenz like system describing convective fluid motion
European Physical Journal-special Topics, 2015Co-Authors: G A Leonov, N. V. Kuznetsov, T N MokaevAbstract:In this paper, we discuss self-excited and hidden Attractors for systems of differential equations. We considered the example of a Lorenz-like system derived from the well-known Glukhovsky-Dolghansky and Rabinovich systems, to demonstrate the analysis of self-excited and hidden Attractors and their characteristics. We applied the fishing principle to demonstrate the existence of a homoclinic orbit, proved the dissipativity and completeness of the system, and found absorbing and positively invariant sets. We have shown that this system has a self-excited Attractor and a hidden Attractor for certain parameters. The upper estimates of the Lyapunov dimension of self-excited and hidden Attractors were obtained analytically.
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finite time lyapunov dimension and hidden Attractor of the rabinovich system
arXiv: Chaotic Dynamics, 2015Co-Authors: N. V. Kuznetsov, Awadhesh Prasad, G A Leonov, T N Mokaev, Manish Dev ShrimaliAbstract:The Rabinovich system, describing the process of interaction between waves in plasma, is considered. It is shown that the Rabinovich system can exhibit a {hidden Attractor} in the case of multistability as well as a classical {self-excited Attractor}. The hidden Attractor in this system can be localized by analytical-numerical methods based on the {continuation} and {perpetual points}. For numerical study of the Attractors' dimension the concept of {finite-time Lyapunov dimension} is developed. A conjecture on the Lyapunov dimension of self-excited Attractors and the notion of {exact Lyapunov dimension} are discussed. A comparative survey on the computation of the finite-time Lyapunov exponents by different algorithms is presented and an approach for a reliable numerical estimation of the finite-time Lyapunov dimension is suggested. Various estimates of the finite-time Lyapunov dimension for the hidden Attractor and hidden transient chaotic set in the case of multistability are given.
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hidden Attractors in dynamical systems from hidden oscillations in hilbert kolmogorov aizerman and kalman problems to hidden chaotic Attractor in chua circuits
International Journal of Bifurcation and Chaos, 2013Co-Authors: G A Leonov, N. V. KuznetsovAbstract:From a computational point of view, in nonlinear dynamical systems, Attractors can be regarded as self-excited and hidden Attractors. Self-excited Attractors can be localized numerically by a standard computational procedure, in which after a transient process a trajectory, starting from a point of unstable manifold in a neighborhood of equilibrium, reaches a state of oscillation, therefore one can easily identify it. In contrast, for a hidden Attractor, a basin of attraction does not intersect with small neighborhoods of equilibria. While classical Attractors are self-excited, Attractors can therefore be obtained numerically by the standard computational procedure. For localization of hidden Attractors it is necessary to develop special procedures, since there are no similar transient processes leading to such Attractors. At first, the problem of investigating hidden oscillations arose in the second part of Hilbert's 16th problem (1900). The first nontrivial results were obtained in Bautin's works, which...
