Hidden attractors in dynamical systems

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• 作者：Dawid Dudkowski^a ; ^{dawid.dudkowski@p.lodz.pl" class="auth_mail" title="E-mail the corresponding author} ; Sajad Jafari^b ; ^{sajadjafari83@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Tomasz Kapitaniak^a ; ^{tomasz.kapitaniak@p.lodz.pl" class="auth_mail" title="E-mail the corresponding author} ; Nikolay V. Kuznetsov^c ; ^d ; ^{nkuznetsov239@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Gennady A. Leonov^c ; ^{g.leonov@spbu.ru" class="auth_mail" title="E-mail the corresponding author} ; Awadhesh Prasad^e ; ^{awadhesh.prasad@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Nonlinear dynamics ; Attractors ; Multistability ; Basins of attraction
• 刊名：Physics Reports
• 出版年：2016
• 出版时间：3 June 2016
• 年：2016
• 卷：637
• 期：Complete
• 页码：1-50
• 全文大小：15386 K

文摘

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.