scholarly journals Detecting unstable periodic orbits from continuous chaotic dynamical systems by dynamical transformation method

2012 ◽  
Vol 61 (17) ◽  
pp. 170510
Author(s):  
Ma Wen-Cong ◽  
Jin Ning-De ◽  
Gao Zhong-Ke
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Transformation Method ◽  
Unstable Periodic Orbits ◽  
Chaotic Dynamical Systems

scholarly journals Statistical characteristics, circulation regimes and unstable periodic orbits of a barotropic atmospheric model

2013 ◽  
Vol 371 (1991) ◽  
pp. 20120336 ◽  
Author(s):  
A. Gritsun
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Atmospheric Model ◽  
Statistical Characteristics ◽  
Local Analysis ◽  
Dynamical System Theory ◽  
Unstable Periodic Orbits ◽  
Attractor Dimension ◽  
Local Maxima ◽  
Chaotic Dynamical Systems

The theory of chaotic dynamical systems gives many tools that can be used in climate studies. The widely used ones are the Lyapunov exponents, the Kolmogorov entropy and the attractor dimension characterizing global quantities of a system. Another potentially useful tool from dynamical system theory arises from the fact that the local analysis of a system probability distribution function (PDF) can be accomplished by using a procedure that involves an expansion in terms of unstable periodic orbits (UPOs). The system measure (or its statistical characteristics) is approximated as a weighted sum over the orbits. The weights are inversely proportional to the orbit instability characteristics so that the least unstable orbits make larger contributions to the PDF. Consequently, one can expect some relationship between the least unstable orbits and the local maxima of the system PDF. As a result, the most probable system trajectories (or ‘circulation regimes’ in some sense) may be explained in terms of orbits. For the special classes of chaotic dynamical systems, there is a strict theory guaranteeing the accuracy of this approach. However, a typical atmospheric model may not qualify for these theorems. In our study, we will try to apply the idea of UPO expansion to the simple atmospheric system based on the barotropic vorticity equation of the sphere. We will check how well orbits approximate the system attractor, its statistical characteristics and PDF. The connection of the most probable states of the system with the least unstable periodic orbits will also be analysed.

Unstable Periodic Orbits Sampling and Its Applications to Climate Models

2021 ◽  
Author(s):  
Chiara Cecilia Maiocchi ◽  
Valerio Lucarini
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Numerical Study ◽  
Building Blocks ◽  
Statistical Properties ◽  
Mathematical Framework ◽  
Multiple Time ◽  
Chain Process ◽  
Unstable Periodic Orbits ◽  
Chaotic Dynamical Systems

<p>Climate can be interpreted as a complex, high dimensional non-equilibrium stationary system characterised by multiple time and space scales spanning various orders of magnitude. Statistical mechanics and dynamical system theory have been key mathematical frameworks in the study of the climate system. In particular, unstable periodic orbits (UPOs) have been proven to provide relevant insight in the understanding of its statistical properties. In a recent paper Lucarini and Gritsun [1] provided an alternative approach for understanding the properties of the atmosphere.</p><p>In general, UPOs decomposition plays a relevant role in the study of chaotic dynamical systems. In fact, UPOs densely populate the attractor of a chaotic system, and can therefore be thought as building blocks to construct the dynamic of the system itself. Since they are dense in the attractor, it is always possible to find a UPO arbitrarily near to a chaotic trajectory: the trajectory will remain close to the UPO, but it will never follow it indefinitely, because of its instability. Loosely speaking, a chaotic trajectory is repelled between neighbourhoods of different UPOs and can thus be approximated in terms of these periodic orbits. The statistical properties of the system can then be reconstructed from the full set of periodic orbits in this fashion.</p><p>The numerical study of UPOs thus represents a relevant problem and an interesting research topic for Climate Science and chaotic dynamical systems in general. In this presentation we address the problem of sampling UPOs for the paradigmatic Lorenz-63 model. First, we present results regarding the measure of the system, thus its statistical properties, using UPOs theory, namely with the trace formulas. Second, we introduce a more innovative approach, considering UPOs as global states of the system. We approximate the exact dynamics by a suitable Markov chain process, describing how the system hops on different UPOs, and we compare the two different approaches.  </p><p>[1] V. Lucarini and A. Gritsun, “A new mathematical framework for atmospheric blocking events,” Climate Dynamics, vol. 54, pp. 575–598, Jan 2020.</p>

Unstable Periodic Orbits Sampling in Climate Models

2020 ◽  
Author(s):  
Chiara Cecilia Maiocchi ◽  
Valerio Lucarini ◽  
Andrey Gritsun ◽  
Grigorios Pavliotis
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Climate Models ◽  
Climate Model ◽  
Climate Science ◽  
Building Blocks ◽  
Mathematical Framework ◽  
Mathematical Tool ◽  
Unstable Periodic Orbits ◽  
Chaotic Dynamical Systems

<p>Unstable periodic orbits (UPOs) have been proved to be a relevant mathematical tool in the study of Climate Science. In a recent paper Lucarini and Gritsun [1] provided an alternative approach for understanding the properties of the atmosphere. Climate can be interpreted as a non-equilibrium steady state system and, as such, statistical mechanics can provide us with tools for its study.</p><p>UPOs decomposition plays a relevant role in the study of chaotic dynamical systems. In fact, UPOs densely populate the attractor of a chaotic system, and can therefore be thought as building blocks to construct the dynamic of the system itself. Since they are dense in the attractor, it is always possible to find a UPO arbitrarily near to a chaotic trajectory: the trajectory will remain close to the UPO, but it will never follow it indefinitely, because of its instability. Loosely speaking, a chaotic trajectory is repelled between neighbourhoods of different UPOs and can thus be approximated in terms of these periodic orbits. The characteristics of the system can then be reconstructed from the full set of periodic orbits in this fashion.</p><p>The sampling of UPOs is therefore a relevant problem for describing chaotic dynamical systems and can represent an interesting topic for the study of Climate Science. In this work we address this problem and present an algorithm to numerically extract UPOs from the attractor of a simple Climate Model such as Lorenz-63.</p><p>[1] V. Lucarini and A. Gritsun, “A new mathematical framework for atmospheric blocking events,” Climate Dynamics, vol. 54, pp. 575–598, Jan 2020.</p>

scholarly journals Numerical detection of unstable periodic orbits in continuous-time dynamical systems with chaotic behaviors

2007 ◽  
Vol 14 (5) ◽  
pp. 615-620 ◽  
Author(s):  
Y. Saiki
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Infinite Number ◽  
Continuous Time ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Complex Phenomenon ◽  
Unstable Periodic Orbits ◽  
Newton Raphson ◽  
The Lorenz System

Abstract. An infinite number of unstable periodic orbits (UPOs) are embedded in a chaotic system which models some complex phenomenon. Several algorithms which extract UPOs numerically from continuous-time chaotic systems have been proposed. In this article the damped Newton-Raphson-Mees algorithm is reviewed, and some important techniques and remarks concerning the practical numerical computations are exemplified by employing the Lorenz system.

scholarly journals Feedback control modulation for controlling chaotic maps

2021 ◽  
Vol 26 (3) ◽  
pp. 419-439
Author(s):  
Roberta Hansen ◽  
Graciela A. González
Keyword(s):  
Dynamical Systems ◽  
Feedback Control ◽  
Periodic Orbits ◽  
Waiting Time ◽  
Control Parameter ◽  
Chaotic Maps ◽  
Noise Sensitivity ◽  
Control Methods ◽  
Unstable Periodic Orbits ◽  
Discrete Maps

Based on existing feedback control methods such as OGY and Pyragas, alternative new schemes are proposed for stabilization of unstable periodic orbits of chaotic and hyperchaotic dynamical systems by suitable modulation of a control parameter. Their performances are improved with respect to: (i) robustness, (ii) rate of convergences, (iii) reduction of waiting time, (iv) reduction of noise sensitivity. These features are analytically investigated, the achievements are rigorously proved and supported by numerical simulations. The proposed methods result successful for stabilizing unstable periodic orbits in some classical discrete maps like 1-D logistic and standard 2-D Hénon, but also in the hyperchaotic generalized n-D Hénon-like maps.

scholarly journals Intermittent chaos driven by nonlinear Alfvén waves

2004 ◽  
Vol 11 (5/6) ◽  
pp. 691-700 ◽  
Author(s):  
E. L. Rempel ◽  
A. C.-L. Chian ◽  
A. J. Preto ◽  
S. Stephany
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Systems Approach ◽  
Alfven Waves ◽  
Space Plasmas ◽  
Alfvén Waves ◽  
Unstable Periodic Orbits ◽  
Phase Dynamics ◽  
Intermittent Chaos

Abstract. We investigate the relevance of chaotic saddles and unstable periodic orbits at the onset of intermittent chaos in the phase dynamics of nonlinear Alfvén waves by using the Kuramoto-Sivashinsky (KS) equation as a model for phase dynamics. We focus on the role of nonattracting chaotic solutions of the KS equation, known as chaotic saddles, in the transition from weak chaos to strong chaos via an interior crisis and show how two of these unstable chaotic saddles can interact to produce the plasma intermittency observed in the strongly chaotic regimes. The dynamical systems approach discussed in this work can lead to a better understanding of the mechanisms responsible for the phenomena of intermittency in space plasmas.

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