Articoli di riviste: "Unstable periodic orbits (UPOs)"

1

Saiki, Y., e M. Yamada. "Time averaged properties along unstable periodic orbits and chaotic orbits in two map systems". Nonlinear Processes in Geophysics 15, n. 4 (7 agosto 2008): 675–80. http://dx.doi.org/10.5194/npg-15-675-2008.

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Abstract. Unstable periodic orbit (UPO) recently has become a keyword in analyzing complex phenomena in geophysical fluid dynamics and space physics. In this paper, sets of UPOs in low dimensional maps are theoretically or systematically found, and time averaged properties along UPOs are studied, in relation to those of chaotic orbits.

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COY, BENJAMIN. "DIMENSION REDUCTION FOR ANALYSIS OF UNSTABLE PERIODIC ORBITS USING LOCALLY LINEAR EMBEDDING". International Journal of Bifurcation and Chaos 22, n. 01 (gennaio 2012): 1230001. http://dx.doi.org/10.1142/s0218127412300017.

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An autonomous four-dimensional dynamical system is investigated through a topological analysis. This system generates a chaotic attractor for the range of control parameters studied and we determine the organization of the unstable periodic orbits (UPOs) associated with the chaotic attractor. Surrogate UPOs were found in the four-dimensional phase space and pairs of these orbits were embedded in three-dimensions using Locally Linear Embedding. This is a dimensionality reduction technique recently developed in the machine learning community. Embedding pairs of orbits allows the computation of their linking numbers, a topological invariant. A table of linking numbers was computed for a range of control parameter values which shows that the organization of the UPOs is consistent with that of a Lorenz-type branched manifold with rotation symmetry.

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Morena, Matthew A., e Kevin M. Short. "Cupolets: History, Theory, and Applications". Dynamics 4, n. 2 (13 maggio 2024): 394–424. http://dx.doi.org/10.3390/dynamics4020022.

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In chaos control, one usually seeks to stabilize the unstable periodic orbits (UPOs) that densely inhabit the attractors of many chaotic dynamical systems. These orbits collectively play a significant role in determining the dynamics and properties of chaotic systems and are said to form the skeleton of the associated attractors. While UPOs are insightful tools for analysis, they are naturally unstable and, as such, are difficult to find and computationally expensive to stabilize. An alternative to using UPOs is to approximate them using cupolets. Cupolets, a name derived from chaotic, unstable, periodic, orbit-lets, are a relatively new class of waveforms that represent highly accurate approximations to the UPOs of chaotic systems, but which are generated via a particular control scheme that applies tiny perturbations along Poincaré sections. Originally discovered in an application of secure chaotic communications, cupolets have since gone on to play pivotal roles in a number of theoretical and practical applications. These developments include using cupolets as wavelets for image compression, targeting in dynamical systems, a chaotic analog to quantum entanglement, an abstract reducibility classification, a basis for audio and video compression, and, most recently, their detection in a chaotic neuron model. This review will detail the historical development of cupolets, how they are generated, and their successful integration into theoretical and computational science and will also identify some unanswered questions and future directions for this work.

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Dolan, Kevin, Annette Witt, Jürgen Kurths e Frank Moss. "Spatiotemporal Distributions of Unstable Periodic Orbits in Noisy Coupled Chaotic Systems". International Journal of Bifurcation and Chaos 13, n. 09 (settembre 2003): 2673–80. http://dx.doi.org/10.1142/s021812740300817x.

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Techniques for detecting encounters with unstable periodic orbits (UPOs) have been very successful in the analysis of noisy, experimental time series. We present here a technique for applying the topological recurrence method of UPO detection to spatially extended systems. This approach is tested on a network of diffusively coupled chaotic Rössler systems, with both symmetric and asymmetric coupling schemes. We demonstrate how to extract encounters with UPOs from such data, and present a preliminary method for analyzing the results and extracting dynamical information from the data, based on a linear correlation analysis of the spatiotemporal occurrence of encounters with these low period UPOs. This analysis can provide an insight into the coupling structure of such a spatially extended system.

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TIAN, YU-PING, e XINGHUO YU. "STABILIZING UNSTABLE PERIODIC ORBITS OF CHAOTIC SYSTEMS WITH UNKNOWN PARAMETERS". International Journal of Bifurcation and Chaos 10, n. 03 (marzo 2000): 611–20. http://dx.doi.org/10.1142/s0218127400000426.

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A novel adaptive time-delayed control method is proposed for stabilizing inherent unstable periodic orbits (UPOs) in chaotic systems with unknown parameters. We first explore the inherent properties of chaotic systems and use the system state and time-delayed system state to form an asymptotically stable invariant manifold so that when the system state enters the manifold and stays in it thereafter, the resulting motion enables the stabilization of the desired UPOs. We then use the model following concept to construct an identifier for the estimation of the uncertain system parameters. We shall prove that under the developed scheme, the system parameter estimates will converge to their true values. The effectiveness of the method is confirmed by computer simulations.

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Boukabou, A., A. Chebbah e A. Belmahboul. "Stabilizing Unstable Periodic Orbits of the Multi-Scroll Chua's Attractor". Nonlinear Analysis: Modelling and Control 12, n. 4 (25 ottobre 2007): 469–77. http://dx.doi.org/10.15388/na.2007.12.4.14678.

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This paper addresses the control of the n-scroll Chua’s circuit. It will be shown how chaotic systems with multiple unstable periodic orbits (UPOs) detected in the Poincar´e section can be stabilized as well as taking the system dynamics from one UPO to another.

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Maiocchi, Chiara Cecilia, Valerio Lucarini e Andrey Gritsun. "Decomposing the dynamics of the Lorenz 1963 model using unstable periodic orbits: Averages, transitions, and quasi-invariant sets". Chaos: An Interdisciplinary Journal of Nonlinear Science 32, n. 3 (marzo 2022): 033129. http://dx.doi.org/10.1063/5.0067673.

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Unstable periodic orbits (UPOs) are a valuable tool for studying chaotic dynamical systems, as they allow one to distill their dynamical structure. We consider here the Lorenz 1963 model with the classic parameters’ value. We investigate how a chaotic trajectory can be approximated using a complete set of UPOs up to symbolic dynamics’ period 14. At each instant, we rank the UPOs according to their proximity to the position of the orbit in the phase space. We study this process from two different perspectives. First, we find that longer period UPOs overwhelmingly provide the best local approximation to the trajectory. Second, we construct a finite-state Markov chain by studying the scattering of the orbit between the neighborhood of the various UPOs. Each UPO and its neighborhood are taken as a possible state of the system. Through the analysis of the subdominant eigenvectors of the corresponding stochastic matrix, we provide a different interpretation of the mixing processes occurring in the system by taking advantage of the concept of quasi-invariant sets.

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Saiki, Y. "Numerical detection of unstable periodic orbits in continuous-time dynamical systems with chaotic behaviors". Nonlinear Processes in Geophysics 14, n. 5 (14 settembre 2007): 615–20. http://dx.doi.org/10.5194/npg-14-615-2007.

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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.

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TIAN, YU-PING. "AN OPTIMIZATION APPROACH TO LOCATING AND STABILIZING UNSTABLE PERIODIC ORBITS OF CHAOTIC SYSTEMS". International Journal of Bifurcation and Chaos 12, n. 05 (maggio 2002): 1163–72. http://dx.doi.org/10.1142/s0218127402005017.

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In this paper, a novel method for locating and stabilizing inherent unstable periodic orbits (UPOs) in chaotic systems is proposed. The main idea of the method is to formulate the UPO locating problem as an optimization issue by using some inherent properties of UPOs of chaotic systems. The global optimal solution of this problem yields the desired UPO. To avoid a local optimal solution, the state of the controlled chaotic system is absorbed into the initial condition of the optimization problem. The ergodicity of chaotic dynamics guarantees that the optimization process does not stay forever at any local optimal solution. When the chaotic orbit approaches the global optimal solution, which is the desired UPO, the controller will stabilize it at the UPO, and the optimization process will cease simultaneously. The method has been developed for both discrete-time and continuous-time systems, and validated for some typical chaotic systems such as the Hénon map and the Duffing oscillator, among others.

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Gritsun, A. "Statistical characteristics, circulation regimes and unstable periodic orbits of a barotropic atmospheric model". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 371, n. 1991 (28 maggio 2013): 20120336. http://dx.doi.org/10.1098/rsta.2012.0336.

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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.

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HINO, TORU, SHIGERU YAMAMOTO e TOSHIMITSU USHIO. "STABILIZATION OF UNSTABLE PERIODIC ORBITS OF CHAOTIC DISCRETE-TIME SYSTEMS USING PREDICTION-BASED FEEDBACK CONTROL". International Journal of Bifurcation and Chaos 12, n. 02 (febbraio 2002): 439–46. http://dx.doi.org/10.1142/s0218127402004450.

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In this paper, we consider feedback control that stabilizes unstable periodic orbits (UPOs) of chaotic discrete-time systems. First, we show that there exists a strong necessary condition for stabilization of the UPOs when we use delayed feedback control (DFC) that is known as one of the useful methods for controlling chaotic systems. The condition is similar to that in the fixed point stabilization problem, in which it is impossible to stabilize the target unstable fixed point if the Jacobian matrix of the linearized system around it has an odd number of real eigenvalues greater than unity. In order to stabilize UPOs which cannot be stabilized by the standard DFC, we adopt prediction-based control. We show a necessary and sufficient condition for the stabilization of the UPOs with arbitrary period.

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Wang, Zhen, Yong Xin Li, Xiao Jian Xi e Xian Feng Wang. "Computional Dynamics for Diffusionless Lorenz Equations with Periodic Parametric Perturbation". Advanced Materials Research 905 (aprile 2014): 651–54. http://dx.doi.org/10.4028/www.scientific.net/amr.905.651.

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The dynamics of diffusionless Lorenz equations (DLE) with periodic parametric perturbation is studied through numerical and experimental investigations in this paper. A method for calculating Lyapunov exponents (LEs), Lyapunov dimension (LD) from time series is presented. Furthermore, bifurcation and some complex dynamic behaviors such as periodic, quasi-periodic motion and chaos which occurred in the system are analyzed. And an algorithm for detecting unstable periodic orbits (UPOs) is presented. Also, give some numerical simulation studies of the system in order to verify the analytic results.

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UETA, TETSUSHI, GUANRONG CHEN e TOHRU KAWABE. "A SIMPLE APPROACH TO CALCULATION AND CONTROL OF UNSTABLE PERIODIC ORBITS IN CHAOTIC PIECEWISE-LINEAR SYSTEMS". International Journal of Bifurcation and Chaos 11, n. 01 (gennaio 2001): 215–24. http://dx.doi.org/10.1142/s0218127401002092.

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This paper describes a simple method for calculating unstable periodic orbits (UPOs) and their control in piecewise-linear autonomous systems. The algorithm can be used to obtain any desired UPO embedded in a chaotic attractor, and the UPO can be stabilized by a simple state feedback control. A brief stability analysis of the controlled system is also given.

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Moroz, Irene M. "Template analysis of a nonlinear dynamo". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, n. 2137 (14 settembre 2011): 288–302. http://dx.doi.org/10.1098/rspa.2011.0216.

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In this paper, we extend our previous template analysis of a self-exciting Faraday disc dynamo with a linear series motor to the case of a nonlinear series motor. This introduces two additional nonlinear symmetry-breaking terms into the governing dynamo equations. We investigate the consequences for the identification of a possible template on which the unstable periodic orbits (UPOs) lie. By computing Gauss linking numbers between pairs of UPOs, we show that their values are not incompatible with those for a template for the Lorenz attractor for its classic parameter values.

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DELEANU, DUMITRU. "STABILIZING THE PERIODIC ORBITS IN A CHAOTIC MAPPING DESCRIBING THE DISCRETE HEALTH SYSTEMS VIA PREDICTION-BASED CONTROL". Journal of marine Technology and Environment 2021, n. 2 (1 ottobre 2021): 21–26. http://dx.doi.org/10.53464/jmte.02.2021.04.

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In the paper the problem of location and stabilization of unstable periodic orbits (UPOs) in discrete systems is investigated via the prediction-based control (PBC). It involves using the state of the free system one period ahead as reference for the control signal. Two types of control gains are tested, the first requiring the knowledge of the UPO to be stabilized and the second depending only on the actual state of the trajectory. The effectiveness of PBC is demonstrated on a chaotic mapping describing the malignant tumor growth. When the results obtained with the two control laws are compared with each other, it is found that the second variant is qualitatively superior, both in terms of convergence and the number of stabilized UPOs, especially for long-period orbits.

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Deleanu, Dumitru. "Detecting and stabilizing periodic orbits of chaotic Henon map through predictive control". Annals Constanta Maritime University 27, n. 2018 (2018): 73–78. http://dx.doi.org/10.38130/cmu.2067.100/42/12.

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The predictive control method is one of the proposed techniques based on the location and stabilization of the unstable periodic orbits (UPOs) embedded in the strange attractor of a nonlinear mapping. It assumes the addition of a small control term to the uncontrolled state of the discrete system. This term depends on the predictive state ps + 1 and p(s + 1) + 1 iterations forward, where s is the length of the UPO, and p is a large enough nonnegative integer. In this paper, extensive numerical simulations on the Henon map are carried out to confirm the ability of the predictive control to detect and stabilize all the UPOs up to a maximum length of the period. The role played by each involved parameter is investigated and additional results to those reported in the literature are presented.

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Zhu, Qunxi, Xin Li e Wei Lin. "Leveraging neural differential equations and adaptive delayed feedback to detect unstable periodic orbits based on irregularly sampled time series". Chaos: An Interdisciplinary Journal of Nonlinear Science 33, n. 3 (marzo 2023): 031101. http://dx.doi.org/10.1063/5.0143839.

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Detecting unstable periodic orbits (UPOs) based solely on time series is an essential data-driven problem, attracting a great deal of attention and arousing numerous efforts, in nonlinear sciences. Previous efforts and their developed algorithms, though falling into a category of model-free methodology, dealt with the time series mostly with a regular sampling rate. Here, we develop a data-driven and model-free framework for detecting UPOs in chaotic systems using the irregularly sampled time series. This framework articulates the neural differential equations (NDEs), a recently developed and powerful machine learning technique, with the adaptive delayed feedback (ADF) technique. Since the NDEs own the exceptional capability of accurate reconstruction of chaotic systems based on the observational time series with irregular sampling rates, UPOs detection in this scenario could be enhanced by an integration of the NDEs and the ADF technique. We demonstrate the effectiveness of the articulated framework on representative examples.

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Lee, Byoung-Cheon, Ki-Hak Lee e Bo-Hyeun Wang. "Control Bifurcation Structure of Return Map Control in Chua's Circuit". International Journal of Bifurcation and Chaos 07, n. 04 (aprile 1997): 903–9. http://dx.doi.org/10.1142/s0218127497000704.

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We demonstrate that return map control and adaptive tracking can be used together to locate, stabilize, and track unstable periodic orbits (UPOs). Through bifurcation studies as a function of some control parameters of return map control, we observe the control bifurcation (CB) phenomenon which exhibits another route to chaos. Nearby an UPO there are a lot of driven periodic orbits (DPOs) along the CB route. DPOs are not embedded in the original chaotic attractor, but they are generated artificially by driving the system slightly in a direction with feedback control. Based on the CB phenomenon, our adaptive tracking algorithm searches for the location and the exact control condition of the UPO by minimizing feedback perturbations. We discuss the universality of the CB phenomenon and the possibility of immediate control which does not require much prior analysis of the system.

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AKATSU, SATOSHI, HIROYUKI TORIKAI e TOSHIMICHI SAITO. "ZERO-CROSS INSTANTANEOUS STATE SETTING FOR CONTROL OF A BIFURCATING H-BRIDGE INVERTER". International Journal of Bifurcation and Chaos 17, n. 10 (ottobre 2007): 3571–75. http://dx.doi.org/10.1142/s021812740701938x.

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This paper studies stabilization of low-period unstable periodic orbits (UPOs) in a simplified model of a current mode H-bridge inverter. The switching of the inverter is controlled by pulse-width modulation signal depending on the sampled inductor current. The inverter can exhibit rich nonlinear phenomena including period doubling bifurcation and chaos. Our control method is realized by instantaneous opening of inductor at a zero-crossing moment of an objective UPO and can stabilize the UPO instantaneously as far as the UPO crosses zero in principle. Typical system operations can be confirmed by numerical experiments.

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Oteski, L., Y. Duguet e L. R. Pastur. "Lagrangian chaos in confined two-dimensional oscillatory convection". Journal of Fluid Mechanics 759 (27 ottobre 2014): 489–519. http://dx.doi.org/10.1017/jfm.2014.583.

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AbstractThe chaotic advection of passive tracers in a two-dimensional confined convection flow is addressed numerically near the onset of the oscillatory regime. We investigate here a differentially heated cavity with aspect ratio 2 and Prandtl number 0.71 for Rayleigh numbers around the first Hopf bifurcation. A scattering approach reveals different zones depending on whether the statistics of return times exhibit exponential or algebraic decay. Melnikov functions are computed and predict the appearance of the main mixing regions via the break-up of the homoclinic and heteroclinic orbits. The non-hyperbolic regions are characterised by a larger number of Kolmogorov–Arnold–Moser (KAM) tori. Based on the numerical extraction of many unstable periodic orbits (UPOs) and their stable/unstable manifolds, we suggest a coarse-graining procedure to estimate numerically the spatial fraction of chaos inside the cavity as a function of the Rayleigh number. Mixing is almost complete before the first transition to quasi-periodicity takes place. The algebraic mixing rate is estimated for tracers released from a localised source near the hot wall.

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LEKEBUSCH, A., A. FÖRSTER e F. W. SCHNEIDER. "CHAOS CONTROL BY ELECTRIC CURRENT IN AN ENZYMATIC REACTION". International Journal of Neural Systems 07, n. 04 (settembre 1996): 393–97. http://dx.doi.org/10.1142/s0129065796000361.

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We apply the continuous delayed feedback method of Pyragas to control chaos in the enzymatic Peroxidase-Oxidase (PO) reaction, using the electric current as the control parameter. At each data point in the time series, a time delayed feedback function applies a small amplitude perturbation to inert platinum electrodes, which causes redox processes on the surface of the electrodes. These perturbations are calculated as the difference between the previous (time delayed) signal and the actual signal. Unstable periodic P1, 11, and 12 orbits (UPOs) were stabilized in the CSTR (continuous stirred tank reactor) experiments. The stabilization is demonstrated by at least three conditions: A minimum in the experimental dispersion function, the equality of the delay time with the period of the stabilized attractor and the embedment of the stabilized periodic attractor in the chaotic attractor.

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Kazantsev, E. "Sensitivity of the attractor of the barotropic ocean model to external influences: approach by unstable periodic orbits". Nonlinear Processes in Geophysics 8, n. 4/5 (31 ottobre 2001): 281–300. http://dx.doi.org/10.5194/npg-8-281-2001.

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Abstract. A description of a deterministic chaotic system in terms of unstable periodic orbits (UPO) is used to develop a method of an a priori estimate of the sensitivity of statistical averages of the solution to small external influences. This method allows us to determine the forcing perturbation which maximizes the norm of the perturbation of a statistical moment of the solution on the attractor. The method was applied to the barotropic ocean model in order to determine the perturbation of the wind field which provides the greatest perturbation of the model's climate. The estimates of perturbations of the model's time mean solution and its mean variance were compared with directly calculated values. The comparison shows that some 20 UPOs is sufficient to realize this approach and to obtain a good accuracy.

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Miino, Yuu, Daisuke Ito, Tetsushi Ueta e Hiroshi Kawakami. "Locating and Stabilizing Unstable Periodic Orbits Embedded in the Horseshoe Map". International Journal of Bifurcation and Chaos 31, n. 04 (30 marzo 2021): 2150110. http://dx.doi.org/10.1142/s0218127421501108.

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Based on the theory of symbolic dynamical systems, we propose a novel computation method to locate and stabilize the unstable periodic points (UPPs) in a two-dimensional dynamical system with a Smale horseshoe. This method directly implies a new framework for controlling chaos. By introducing the subset based correspondence between a planar dynamical system and a symbolic dynamical system, we locate regions sectioned by stable and unstable manifolds comprehensively and identify the specified region containing a UPP with the particular period. Then Newton’s method compensates the accurate location of the UPP with the regional information as an initial estimation. On the other hand, the external force control (EFC) is known as an effective method to stabilize the UPPs. By applying the EFC to the located UPPs, robust controlling chaos is realized. In this framework, we never use ad hoc approaches to find target UPPs in the given chaotic set. Moreover, the method can stabilize UPPs with the specified period regardless of the situation where the targeted chaotic set is attractive. As illustrative numerical experiments, we locate and stabilize UPPs and the corresponding unstable periodic orbits in a horseshoe structure of the Duffing equation. In spite of the strong instability of UPPs, the controlled orbit is robust and the control input retains being tiny in magnitude.

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Katsanikas, M., Víctor J. García-Garrido e S. Wiggins. "Detection of Dynamical Matching in a Caldera Hamiltonian System Using Lagrangian Descriptors". International Journal of Bifurcation and Chaos 30, n. 09 (luglio 2020): 2030026. http://dx.doi.org/10.1142/s0218127420300268.

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The goal of this paper is to apply the method of Lagrangian descriptors to reveal the phase space mechanism by which a Caldera-type potential energy surface (PES) exhibits the dynamical matching phenomenon. Using this technique, we can easily establish that the nonexistence of dynamical matching is a consequence of heteroclinic connections between the unstable manifolds of the unstable periodic orbits (UPOs) of the upper index-1 saddles (entrance channels to the Caldera) and the stable manifolds of the family of UPOs of the central minimum of the Caldera, resulting in the temporary trapping of trajectories. Moreover, dynamical matching will occur when there is no heteroclinic connection, which allows trajectories to enter and exit the Caldera without interacting with the shallow region of the central minimum. Knowledge of this phase space mechanism is relevant because it allows us to effectively predict the existence, and nonexistence, of dynamical matching. In this work, we explore a stretched Caldera potential by means of Lagrangian descriptors, allowing us to accurately compute the critical value for the stretching parameter for which dynamical matching behavior occurs in the system. This approach is shown to provide a tremendous advantage for exploring this mechanism in comparison to other methods from nonlinear dynamics that use phase space dividing surfaces.

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Ivan, Cosmin, e Mihai Catalin Arva. "Nonlinear Time Series Analysis in Unstable Periodic Orbits Identification-Control Methods of Nonlinear Systems". Electronics 11, n. 6 (18 marzo 2022): 947. http://dx.doi.org/10.3390/electronics11060947.

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The main purpose of this paper is to present a solution to the well-known problems generated by classical control methods through the analysis of nonlinear time series. Among the problems analyzed, for which an explanation has been sought for a long time, we list the significant reduction in control power and the identification of unstable periodic orbits (UPOs) in chaotic time series. To accurately identify the type of behavior of complex systems, a new solution is presented that involves a method of two-dimensional representation specific to the graphical point of view, and in particular the recurrence plot (RP). An example of the issue studied is presented by applying the recurrence graph to identify the UPO in a chaotic attractor. To identify a certain type of behavior in the numerical data of chaotic systems, nonlinear time series will be used, as a novelty element, to locate unstable periodic orbits. Another area of use for the theories presented above, following the application of these methods, is related to the control of chaotic dynamical systems by using RP in control techniques. Thus, the authors’ contributions are outlined by using the recurrence graph, which is used to identify the UPO from a chaotic attractor, in the control techniques that modify a system variable. These control techniques are part of the closed loop or feedback strategies that describe control as a function of the current state of the UPO stabilization system. To exemplify the advantages of the methods presented above, the use of the recurrence graph in the control of a buck converter through the application of a phase difference signal was analyzed. The study on the command of a direct current motor using a buck converter shows, through a final concrete application, the advantages of using these analysis methods in controlling dynamic systems.

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Lucarini, Valerio, e Andrey Gritsun. "A new mathematical framework for atmospheric blocking events". Climate Dynamics 54, n. 1-2 (1 novembre 2019): 575–98. http://dx.doi.org/10.1007/s00382-019-05018-2.

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Abstract We use a simple yet Earth-like hemispheric atmospheric model to propose a new framework for the mathematical properties of blocking events. Using finite-time Lyapunov exponents, we show that the occurrence of blockings is associated with conditions featuring anomalously high instability. Longer-lived blockings are very rare and have typically higher instability. In the case of Atlantic blockings, predictability is especially reduced at the onset and decay of the blocking event, while a relative increase of predictability is found in the mature phase. The opposite holds for Pacific blockings, for which predictability is lowest in the mature phase. Blockings are realised when the trajectory of the system is in the neighbourhood of a specific class of unstable periodic orbits (UPOs), natural modes of variability that cover the attractor the system. UPOs corresponding to blockings have, indeed, a higher degree of instability compared to UPOs associated with zonal flow. Our results provide a rigorous justification for the classical Markov chains-based analysis of transitions between weather regimes. The analysis of UPOs elucidates that the model features a very severe violation of hyperbolicity, due to the presence of a substantial variability in the number of unstable dimensions, which explains why atmospheric states can differ a lot in term of their predictability. Additionally, such a variability explains the need for performing data assimilation in a state space that includes not only the unstable and neutral subspaces, but also some stable modes. The lack of robustness associated with the violation of hyperbolicity might be a basic cause contributing to the difficulty in representing blockings in numerical models and in predicting how their statistics will change as a result of climate change. This corresponds to fundamental issues limiting our ability to construct very accurate numerical models of the atmosphere, in term of predictability of the both the first and of the second kind in the sense of Lorenz.

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Keeler, Jack S., Alice B. Thompson, Grégoire Lemoult, Anne Juel e Andrew L. Hazel. "The influence of invariant solutions on the transient behaviour of an air bubble in a Hele-Shaw channel". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, n. 2232 (dicembre 2019): 20190434. http://dx.doi.org/10.1098/rspa.2019.0434.

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We hypothesize that dynamical systems concepts used to study the transition to turbulence in shear flows are applicable to other transition phenomena in fluid mechanics. In this paper, we consider a finite air bubble that propagates within a Hele-Shaw channel containing a depth-perturbation. Recent experiments revealed that the bubble shape becomes more complex, quantified by an increasing number of transient bubble tips, with increasing flow rate. Eventually, the bubble changes topology, breaking into multiple distinct entities with non-trivial dynamics. We demonstrate that qualitatively similar behaviour to the experiments is exhibited by a previously established, depth-averaged mathematical model and arises from the model’s intricate solution structure. For the bubble volumes studied, a stable asymmetric bubble exists for all flow rates of interest, while a second stable solution branch develops above a critical flow rate and transitions between symmetric and asymmetric shapes. The region of bistability is bounded by two Hopf bifurcations on the second branch. By developing a method for a numerical weakly nonlinear stability analysis we show that unstable periodic orbits (UPOs) emanate from the first Hopf bifurcation. Moreover, as has been found in shear flows, the UPOs are edge states that influence the transient behaviour of the system.

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28

EL AROUDI, A., M. DEBBAT, R. GIRAL, G. OLIVAR, L. BENADERO e E. TORIBIO. "BIFURCATIONS IN DC–DC SWITCHING CONVERTERS: REVIEW OF METHODS AND APPLICATIONS". International Journal of Bifurcation and Chaos 15, n. 05 (maggio 2005): 1549–78. http://dx.doi.org/10.1142/s0218127405012946.

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This paper presents, in a tutorial manner, nonlinear phenomena such as bifurcations and chaotic behavior in DC–DC switching converters. Our purpose is to present the different modeling approaches, the main results found in the last years and some possible practical applications. A comparison of the different models is given and their accuracy in predicting nonlinear behavior is discussed. A general Poincaré map is considered to model any multiple configuration of DC–DC switching converters and its Jacobian matrix is derived for stability analysis. More emphasis is done in the discrete-time approach as it gives more accurate prediction of bifurcations. The results are reproduced for different examples of DC–DC switching converters studied in the literature. Some methods of controlling bifurcations are applied to stabilize Unstable Periodic Orbits (UPOs) embedded in the dynamics of the system. Statistical analysis of these systems working in the chaotic regime is discussed. An extensive list of references is included.

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29

Eccles, F. J. R., P. L. Read e T. W. N. Haine. "Synchronization and chaos control in a periodically forced quasi-geostrophic two-layer model of baroclinic instability". Nonlinear Processes in Geophysics 13, n. 1 (22 febbraio 2006): 23–39. http://dx.doi.org/10.5194/npg-13-23-2006.

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Abstract. Cyclic forcing on many timescales is believed to have a significant effect on various quasi-periodic, geophysical phenomena such as El Niño, the Quasi-Biennial Oscillation, and glacial cycles. This variability has been investigated by numerous previous workers, in models ranging from simple energy balance constructions to full general circulation models. We present a numerical study in which periodic forcing is applied to a highly idealised, two-layer, quasi-geostrophic model on a β-plane. The bifurcation structure and (unforced) behaviour of this particular model has been extensively examined by Lovegrove et al. (2001) and Lovegrove et al. (2002). We identify from their work three distinct regimes on which we perform our investigations: a steady, travelling wave regime, a quasi-periodic, modulated wave regime and a chaotic regime. In the travelling wave regime a nonlinear resonance is found. In the periodic regime, Arnol'd tongues, frequency locking and a Devil's staircase is seen for small amplitudes of forcing. As the forcing is increased the Arnol'd tongues undergo a period doubling route to chaos, and for larger forcings still, the parameter space we explored is dominated by either period 1 behaviour or chaotic behaviour. In the chaotic regime we extract unstable periodic orbits (UPOs) and add the periodic forcing at periods corresponding to integer multiples of the UPO periods. We find regions of synchronization, similar to Arnol'd tongue behaviour but more skewed and centred approximately on these periods. The regions where chaos suppression took place are smaller than the synchronization regions, and are contained within them.

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Dong, Chengwei, e Lian Jia. "Periodic orbits analysis for the Zhou system via variational approach". Modern Physics Letters B 33, n. 19 (8 luglio 2019): 1950212. http://dx.doi.org/10.1142/s0217984919502129.

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We proposed a general method for the systematic calculation of unstable cycles in the Zhou system. The variational approach is employed for the cycle search, and we establish interesting symbolic dynamics successfully based on the orbits circuiting property with respect to different fixed points. Upon the defined symbolic rule, cycles with topological length up to five are sought and ordered. Further, upon parameter changes, the homotopy evolution of certain selected cycles are investigated. The topological classification methodology could be widely utilized in other low-dimensional dissipative systems.

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31

Savi, Marcelo A., Francisco Heitor I. Pereira-Pinto e Armando M. Ferreira. "Chaos Control in Mechanical Systems". Shock and Vibration 13, n. 4-5 (2006): 301–14. http://dx.doi.org/10.1155/2006/545842.

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Chaos has an intrinsically richness related to its structure and, because of that, there are benefits for a natural system of adopting chaotic regimes with their wide range of potential behaviors. Under this condition, the system may quickly react to some new situation, changing conditions and their response. Therefore, chaos and many regulatory mechanisms control the dynamics of living systems, conferring a great flexibility to the system. Inspired by nature, the idea that chaotic behavior may be controlled by small perturbations of some physical parameter is making this kind of behavior to be desirable in different applications. Mechanical systems constitute a class of system where it is possible to exploit these ideas. Chaos control usually involves two steps. In the first, unstable periodic orbits (UPOs) that are embedded in the chaotic set are identified. After that, a control technique is employed in order to stabilize a desirable orbit. This contribution employs the close-return method to identify UPOs and a semi-continuous control method, which is built up on the OGY method, to stabilize some desirable UPO. As an application to a mechanical system, a nonlinear pendulum is considered and, based on parameters obtained from an experimental setup, analyses are carried out. Signals are generated by numerical integration of the mathematical model and two different situations are treated. Firstly, it is assumed that all state variables are available. After that, the analysis is done from scalar time series and therefore, it is important to evaluate the effect of state space reconstruction. Delay coordinates method and extended state observers are employed with this aim. Results show situations where these techniques may be used to control chaos in mechanical systems.

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32

Guha, Anirban, e Firdaus E. Udwadia. "Nonlinear dynamics induced by linear wave interactions in multilayered flows". Journal of Fluid Mechanics 816 (6 marzo 2017): 412–27. http://dx.doi.org/10.1017/jfm.2017.84.

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Using simple kinematics, we propose a general theory of linear wave interactions between the interfacial waves of a two-dimensional (2D), inviscid, multilayered fluid system. The strength of our formalism is that one does not have to specify the physics of the waves in advance. Wave interactions may lead to instabilities, which may or may not be of the familiar ‘normal-mode’ type. Contrary to intuition, the underlying dynamical system describing linear wave interactions is found to be nonlinear. Specifically, a saw-tooth jet profile with three interfaces possessing kinematic and geometric symmetry is explored. Fixed points of the system for different ranges of a Froude number like control parameter $\unicode[STIX]{x1D6FE}$ are derived, and their stability evaluated. Depending upon the initial condition and $\unicode[STIX]{x1D6FE}$, the dynamical system may reveal transient growth, weakly positive Lyapunov exponents, as well as different nonlinear phenomena such as the formation of periodic and pseudo-periodic orbits. All these occur for those ranges of $\unicode[STIX]{x1D6FE}$ where normal-mode theory predicts neutral stability. Such rich nonlinear phenomena are not observed in a 2D dynamical system resulting from the two-wave problem, which reveals only stable and unstable nodes.

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33

So, Paul. "Unstable periodic orbits". Scholarpedia 2, n. 2 (2007): 1353. http://dx.doi.org/10.4249/scholarpedia.1353.

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34

Chizhevsky, V. N., e P. Glorieux. "Targeting unstable periodic orbits". Physical Review E 51, n. 4 (1 aprile 1995): R2701—R2704. http://dx.doi.org/10.1103/physreve.51.r2701.

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35

Carpintero, D. D., e J. C. Muzzio. "The Lyapunov exponents and the neighbourhood of periodic orbits". Monthly Notices of the Royal Astronomical Society 495, n. 2 (16 maggio 2020): 1608–12. http://dx.doi.org/10.1093/mnras/staa1227.

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ABSTRACT We show that the Lyapunov exponents of a periodic orbit can be easily obtained from the eigenvalues of the monodromy matrix. It turns out that the Lyapunov exponents of simply stable periodic orbits are all zero, simply unstable periodic orbits have only one positive Lyapunov exponent, doubly unstable periodic orbits have two different positive Lyapunov exponents, and the two positive Lyapunov exponents of complex unstable periodic orbits are equal. We present a numerical example for periodic orbits in a realistic galactic potential. Moreover, the centre manifold theorem allowed us to show that stable, simply unstable, and doubly unstable periodic orbits are the mothers of families of, respectively, regular, partially, and fully chaotic orbits in their neighbourhood.

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36

Pawelzik, K., e H. G. Schuster. "Unstable periodic orbits and prediction". Physical Review A 43, n. 4 (1 febbraio 1991): 1808–12. http://dx.doi.org/10.1103/physreva.43.1808.

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37

KATSANIKAS, M., P. A. PATSIS e G. CONTOPOULOS. "INSTABILITIES AND STICKINESS IN A 3D ROTATING GALACTIC POTENTIAL". International Journal of Bifurcation and Chaos 23, n. 02 (febbraio 2013): 1330005. http://dx.doi.org/10.1142/s021812741330005x.

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We study the dynamics in the neighborhood of simple and double unstable periodic orbits in a rotating 3D autonomous Hamiltonian system of galactic type. In order to visualize the four-dimensional spaces of section, we use the method of color and rotation. We investigate the structure of the invariant manifolds that we found in the neighborhood of simple and double unstable periodic orbits in 4D spaces of section. We consider orbits in the neighborhood of the families x1v2, belonging to the x1 tree, and the z-axis (the rotational axis of our system). Close to the transition points from stability to simple instability, in the neighborhood of the bifurcated simple unstable x1v2 periodic orbits, we encounter the phenomenon of stickiness as the asymptotic curves of the unstable manifold surround regions of the phase space occupied by rotational tori existing in the region. For larger energies, away from the bifurcating point, the consequents of the chaotic orbits form clouds of points with mixing of color in their 4D representations. In the case of double instability, close to x1v2 orbits, we find clouds of points in the four-dimensional spaces of section. However, in some cases of double unstable periodic orbits belonging to the z-axis family we can visualize the associated unstable eigensurface. Chaotic orbits close to the periodic orbit remain sticky to this surface for long times (of the order of a Hubble time or more). Among the orbits we studied, we found those close to the double unstable orbits of the x1v2 family having the largest diffusion speed. The sticky chaotic orbits close to the bifurcation point of the simple unstable x1v2 orbit and close to the double unstable z-axis orbit that we have examined, have comparable diffusion speeds. These speeds are much slower than of the orbits in the neighborhood of x1v2 simple unstable periodic orbits away from the bifurcating point, or of the double unstable orbits of the same family having very different eigenvalues along the corresponding unstable eigendirections.

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Katsanikas, Matthaios, e Stephen Wiggins. "Phase Space Structure and Transport in a Caldera Potential Energy Surface". International Journal of Bifurcation and Chaos 28, n. 13 (12 dicembre 2018): 1830042. http://dx.doi.org/10.1142/s0218127418300422.

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We study phase space transport in a 2D caldera potential energy surface (PES) using techniques from nonlinear dynamics. The caldera PES is characterized by a flat region or shallow minimum at its center surrounded by potential walls and multiple symmetry related index one saddle points that allow entrance and exit from this intermediate region. We have discovered four qualitatively distinct cases of the structure of the phase space that govern phase space transport. These cases are categorized according to the total energy and the stability of the periodic orbits associated with the family of the central minimum, the bifurcations of the same family, and the energetic accessibility of the index one saddles. In each case, we have computed the invariant manifolds of the unstable periodic orbits of the central region of the potential, and the invariant manifolds of the unstable periodic orbits of the families of periodic orbits associated with the index one saddles. The periodic orbits of the central region are, for the first case, the unstable periodic orbits with period 10 that are outside the stable region of the stable periodic orbits of the family of the central minimum. In addition, the periodic orbits of the central region are, for the second and third cases, the unstable periodic orbits of the family of the central minimum and for the fourth case the unstable periodic orbits with period 2 of a period-doubling bifurcation of the family of the central minimum. We have found that there are three distinct mechanisms determined by the invariant manifold structure of the unstable periodic orbits that govern the phase space transport. The first mechanism explains the nature of the entrance of the trajectories from the region of the low energy saddles into the caldera and how they may become trapped in the central region of the potential. The second mechanism describes the trapping of the trajectories that begin from the central region of the caldera, their transport to the regions of the saddles, and the nature of their exit from the caldera. The third mechanism describes the phase space geometry responsible for the dynamical matching of trajectories originally proposed by Carpenter and described in [Collins et al., 2014] for the two-dimensional caldera PES that we consider.

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Zhang, Yongxiang, e Guanwei Luo. "Detecting unstable periodic orbits and unstable quasiperiodic orbits in vibro-impact systems". International Journal of Non-Linear Mechanics 96 (novembre 2017): 12–21. http://dx.doi.org/10.1016/j.ijnonlinmec.2017.07.011.

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40

Bradley, Elizabeth, e Ricardo Mantilla. "Recurrence plots and unstable periodic orbits". Chaos: An Interdisciplinary Journal of Nonlinear Science 12, n. 3 (settembre 2002): 596–600. http://dx.doi.org/10.1063/1.1488255.

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41

Voros, A. "Unstable periodic orbits and semiclassical quantisation". Journal of Physics A: Mathematical and General 21, n. 3 (7 febbraio 1988): 685–92. http://dx.doi.org/10.1088/0305-4470/21/3/023.

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42

Pisarchik, A. N. "Dynamical tracking of unstable periodic orbits". Physics Letters A 242, n. 3 (maggio 1998): 152–62. http://dx.doi.org/10.1016/s0375-9601(98)00210-2.

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43

Dana, Itzhack. "Hamiltonian transport on unstable periodic orbits". Physica D: Nonlinear Phenomena 39, n. 2-3 (ottobre 1989): 205–30. http://dx.doi.org/10.1016/0167-2789(89)90005-5.

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44

Fazendeiro, L., B. M. Boghosian, P. V. Coveney e J. Lätt. "Unstable periodic orbits in weak turbulence". Journal of Computational Science 1, n. 1 (maggio 2010): 13–23. http://dx.doi.org/10.1016/j.jocs.2010.03.004.

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45

KATSANIKAS, M., P. A. PATSIS e G. CONTOPOULOS. "THE STRUCTURE AND EVOLUTION OF CONFINED TORI NEAR A HAMILTONIAN HOPF BIFURCATION". International Journal of Bifurcation and Chaos 21, n. 08 (agosto 2011): 2321–30. http://dx.doi.org/10.1142/s0218127411029811.

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We study the orbital behavior at the neighborhood of complex unstable periodic orbits in a 3D autonomous Hamiltonian system of galactic type. At a transition of a family of periodic orbits from stability to complex instability (also known as Hamiltonian Hopf Bifurcation) the four eigenvalues of the stable periodic orbits move out of the unit circle. Then the periodic orbits become complex unstable. In this paper, we first integrate initial conditions close to the ones of a complex unstable periodic orbit, which is close to the transition point. Then, we plot the consequents of the corresponding orbit in a 4D surface of section. To visualize this surface of section we use the method of color and rotation [Patsis & Zachilas, 1994]. We find that the consequents are contained in 2D "confined tori". Then, we investigate the structure of the phase space in the neighborhood of complex unstable periodic orbits, which are further away from the transition point. In these cases we observe clouds of points in the 4D surfaces of section. The transition between the two types of orbital behavior is abrupt.

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46

KAMIYAMA, KYOHEI, MOTOMASA KOMURO e TETSURO ENDO. "BIFURCATION OF QUASI-PERIODIC OSCILLATIONS IN MUTUALLY COUPLED HARD-TYPE OSCILLATORS: DEMONSTRATION OF UNSTABLE QUASI-PERIODIC ORBITS". International Journal of Bifurcation and Chaos 22, n. 06 (giugno 2012): 1230022. http://dx.doi.org/10.1142/s0218127412300224.

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In this paper, we obtain bifurcations of quasi-periodic orbits occuring in mutually coupled hard-type oscillators by using our recently developed computer algorithm to directly determine the unstable quasi-periodic orbits. So far, there is no computer algorithm to draw unstable invariant closed curves on a Poincare map representing quasi-periodic orbits. Recently, we developed a new algorithm to draw unstable invariant closed curves by using the bisection method. The results of this new algorithm are compared with the previously obtained averaging method results. Several new results are found, which could not be clarified by the averaging method.

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Parker, Jeremy P., e Tobias M. Schneider. "Invariant tori in dissipative hyperchaos". Chaos: An Interdisciplinary Journal of Nonlinear Science 32, n. 11 (novembre 2022): 113102. http://dx.doi.org/10.1063/5.0119642.

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One approach to understand the chaotic dynamics of nonlinear dissipative systems is the study of non-chaotic yet dynamically unstable invariant solutions embedded in the system’s chaotic attractor. The significance of zero-dimensional unstable fixed points and one-dimensional unstable periodic orbits capturing time-periodic dynamics is widely accepted for high-dimensional chaotic systems, including fluid turbulence, while higher-dimensional invariant tori representing quasiperiodic dynamics have rarely been considered. We demonstrate that unstable 2-tori are generically embedded in the hyperchaotic attractor of a dissipative system of ordinary differential equations; tori can be numerically identified via bifurcations of unstable periodic orbits and their parameteric continuation and characterization of stability properties are feasible. As higher-dimensional tori are expected to be structurally unstable, 2-tori together with periodic orbits and equilibria form a complete set of relevant invariant solutions on which to base a dynamical description of chaos.

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48

DHAMALA, MUKESHWAR, e YING-CHENG LAI. "THE NATURAL MEASURE OF NONATTRACTING CHAOTIC SETS AND ITS REPRESENTATION BY UNSTABLE PERIODIC ORBITS". International Journal of Bifurcation and Chaos 12, n. 12 (dicembre 2002): 2991–3005. http://dx.doi.org/10.1142/s0218127402006308.

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The natural measure of a chaotic set in a phase-space region can be related to the dynamical properties of all unstable periodic orbits embedded in the chaotic set contained in that region. This result has been shown to be valid for hyperbolic chaotic invariant sets. The aim of this paper is to examine whether this result applies to nonhyperbolic, nonattracting chaotic saddles which lead to transient chaos in physical systems. In particular, we examine, quantitatively, the closeness of the natural measure obtained from a long trajectory on the chaotic saddle to that evaluated from unstable periodic orbits embedded in the set. We also analyze the difference between the long-time average values of physical quantities evaluated with respect to a dense trajectory and those computed from unstable periodic orbits. Results with both the Hénon map and the Ikeda–Hammel–Jones–Moloney map for which periodic orbits can be enumerated lend credence to the conjecture that the unstable periodic-orbit theory of the natural measure is applicable to nonhyperbolic chaotic saddles.

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Kazantsev, E. "Unstable periodic orbits and attractor of the barotropic ocean model". Nonlinear Processes in Geophysics 5, n. 4 (31 dicembre 1998): 193–208. http://dx.doi.org/10.5194/npg-5-193-1998.

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Abstract. A numerical method for detection of unstable periodic orbits on attractors of nonlinear models is proposed. The method requires similar techniques to data assimilation. This fact facilitates its implementation for geophysical models. This method was used to find numerically several low-period orbits for the barotropic ocean model in a square. Some numerical particularities of application of this method are discussed. Knowledge of periodic orbits of the model helps to explain some of these features like bimodality of probability density functions (PDF) of principal parameters. These PDFs have been reconstructed as weighted averages of periodic orbits with weights proportional to the period of the orbit and inversely proportional to the sum of positive Lyapunov exponents. The fraction of time spent in the vicinity of each periodic orbit has been compared with its instability characteristics. The relationship between these values shows the 93% correlation. The attractor dimension of the model has also been approximated as a weighted average of local attractor dimensions in vicinities of periodic orbits.

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OGORZAŁEK, MACIEJ J., e ZBIGNIEW GALIAS. "CHARACTERISATION OF CHAOS IN CHUA'S OSCILLATOR IN TERMS OF UNSTABLE PERIODIC ORBITS". Journal of Circuits, Systems and Computers 03, n. 02 (giugno 1993): 411–29. http://dx.doi.org/10.1142/s0218126693000253.

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We present a picture book of unstable periodic orbits embedded in typical chaotic attractors found in the canonical Chua's circuit. These include spiral Chua's, double-scroll Chua's and double hook attractors. The "skeleton" of unstable periodic orbits is specific for the considered attractor and provides an invariant characterisation of its geometry.

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