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Bo-wen Shen - One of the best experts on this subject based on the ideXlab platform.
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Homoclinic Orbits and Solitary Waves within the Nondissipative Lorenz Model and KdV Equation
International Journal of Bifurcation and Chaos, 2020Co-Authors: Bo-wen ShenAbstract:Recent studies using the classical Lorenz Model and generalized Lorenz Models present abundant features of both chaotic and oscillatory solutions that may change our view on the nature of the weather as well as climate. In this study, the mathematical universality of solutions in different physical systems is presented. Specifically, the main goal is to reveal mathematical similarities for solutions of homoclinic orbits and solitary waves within a three-dimensional nondissipative Lorenz Model (3D-NLM), the Korteweg–de Vries (KdV) equation, and the Nonlinear Schrodinger (NLS) equation. A homoclinic orbit for the [Formula: see text], [Formula: see text], and [Formula: see text] state variables of the 3D-NLM connects the unstable and stable manifolds of a saddle point. The [Formula: see text] and [Formula: see text] solutions for the homoclinic orbit can be expressed in terms of a hyperbolic secant function ([Formula: see text]) and a hyperbolic secant squared function ([Formula: see text]), respectively. Interestingly, these two solutions have the same mathematical form as solitary solutions for the NLS and KdV equations, respectively. After introducing new independent variables, the same second-order ordinary differential equation (ODE) and solutions for the [Formula: see text] component and the KdV equation were obtained. Additionally, the ODE for the [Formula: see text] component has the same form as the NLS for the solitary wave envelope. Finally, how a logistic equation, also known as the Lorenz error growth Model, and an improved error growth Model can be derived by simplifying the 3D-NLM is also discussed. Future work will compare the solutions of the 3D-NLM and KdV equation in order to understand the different physical role of nonlinearity in their solutions and the solutions of the error growth Model and the 3D-NLM, as well as other Lorenz Models, to propose an improved error growth Model for better representing error growth at linear and nonlinear stages for both oscillatory and nonoscillatory solutions.
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On the Predictability of 30-Day Global Mesoscale Simulations of African Easterly Waves during Summer 2006: A View with the Generalized Lorenz Model
Geosciences, 2019Co-Authors: Bo-wen ShenAbstract:Recent advances in computational and global Modeling technology have provided the potential to improve weather predictions at extended-range scales. In earlier studies by the author and his coauthors, realistic 30-day simulations of multiple African easterly waves (AEWs) and an averaged African easterly jet (AEJ) were obtained. The formation of hurricane Helene (2006) was also realistically simulated from Day 22 to Day 30. In this study, such extended predictability was further analyzed based on recent understandings of chaos and instability within Lorenz Models and the generalized Lorenz Model. The analysis suggested that a statement of the theoretical predictability of two weeks is not universal. New insight into chaotic and non-chaotic processes revealed by the generalized Lorenz Model (GLM) indicated the potential for extending prediction lead times. Two major features within the GLM included: (1) three types of attractors (that also appeared in the original Lorenz Model) and (2) two kinds of attractor coexistence. The features suggest a refined view on the nature of weather, as follows: The entirety of weather is a superset that consists of chaotic and non-chaotic processes. Better predictability can be obtained for stable, steady-state solutions and nonlinear periodic solutions that occur at small and large Rayleigh parameters, respectively. By comparison, chaotic solutions appear only at moderate Rayleigh parameters. Errors associated with dissipative small-scale processes do not necessarily contaminate the simulations of large scale processes. Based on the nonlinear periodic solutions (also known as limit cycle solutions), here, we propose a hypothetical mechanism for the recurrence (or periodicity) of successive AEWs. The insensitivity of limit cycles to initial conditions implies that AEW simulations with strong heating and balanced nonlinearity could be more predictable. Based on the hypothetical mechanism, the possibility of extending prediction lead times at extended range scales is discussed. Future work will include refining the Model to better examine the validity of the mechanism to explain the recurrence of multiple AEWs.
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Coexistence of Chaotic and Non-chaotic Orbits in a New Nine-Dimensional Lorenz Model
11th Chaotic Modeling and Simulation International Conference, 2019Co-Authors: Bo-wen Shen, Tiffany Reyes, Sara Faghih-nainiAbstract:In this study, we present a new nine-dimensional Lorenz Model (9DLM) that requires a larger critical value for the Rayleigh parameter (a rc of 679.8) for the onset of chaos, as compared to a rc of 24.74 for the 3DLM, a rc of 42.9 for the 5DLM, and a rc 116.9 for the 7DLM. Major features within the 9DLM include: (1) the coexistence of chaotic and non-chaotic orbits with moderate Rayleigh parameters, and (2) the coexistence of limit cycle/torus orbits and spiral sinks with large Rayleigh parameters. Version 2 of the 9DLM, referred to as the 9DLM-V2, is derived to show that: (i) based on a linear stability analysis, two non-trivial critical points are stable for all Rayleigh parameters greater than one; (ii) under non-dissipative and linear conditions, the extended nonlinear feedback loop produces four incommensurate frequencies; and (iii) for a stable orbit, small deviations away from equilibrium (e.g., the stable critical point) do not have a significant impact on orbital stability. Based on our results, we suggest that the entirety of weather is a superset that consists of both chaotic and non-chaotic processes.
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Aggregated Negative Feedback in a Generalized Lorenz Model
International Journal of Bifurcation and Chaos, 2019Co-Authors: Bo-wen ShenAbstract:In this study, we first present a generalized Lorenz Model (LM) with M modes, where M is an odd number that is greater than three. The generalized LM (GLM) is derived based on a successive extensio...
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Aggregated Negative Feedback in a Generalized Lorenz Model
International Journal of Bifurcation and Chaos, 2019Co-Authors: Bo-wen ShenAbstract:In this study, we first present a generalized Lorenz Model (LM) with [Formula: see text] modes, where [Formula: see text] is an odd number that is greater than three. The generalized LM (GLM) is derived based on a successive extension of the nonlinear feedback loop (NFL) with additional high wavenumber modes. By performing a linear stability analysis with [Formula: see text] and [Formula: see text], we illustrate that: (1) within the 3D, 5D, and 7D LMs, the appearance of unstable nontrivial critical points requires a larger Rayleigh parameter in a higher-dimensional LM and (2) within the 9DLM, nontrivial critical points are stable. By comparing the GLM with various numbers of modes, we discuss the aggregated negative feedback enabled by the extended NFL and its role in stabilizing solutions in high-dimensional LMs. Our analysis indicates that the 9DLM is the lowest order generalized LM with stable nontrivial critical points for all Rayleigh parameters greater than one. As shown by calculations of the ensemble Lyapunov exponent, the 9DLM still produces chaotic solutions. Within the 9DLM, a larger critical value for the Rayleigh parameter, [Formula: see text], is required for the onset of chaos as compared to a [Formula: see text] for the 3DLM, a [Formula: see text] for the 5DLM, and a [Formula: see text] for the 7DLM. In association with stable nontrivial critical points that may lead to steady-state solutions, the appearance of chaotic orbits indicates the important role of a saddle point at the origin in producing the sensitive dependence of solutions on initial conditions. The 9DLM displays the coexistence of chaotic and steady-state solutions at moderate Rayleigh parameters and the coexistence of limit cycle and steady-state solutions at large Rayleigh parameters. The first kind of coexistence appears within a smaller range of Rayleigh parameters in lower-dimensional LMs (i.e. [Formula: see text] within the 3DLM) but in a wider range of Rayleigh parameters within the 9DLM (i.e. [Formula: see text]). The second kind of coexistence has never been reported in high-dimensional Lorenz systems.
Warwick Tucker - One of the best experts on this subject based on the ideXlab platform.
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a database of rigorous and high precision periodic orbits of the Lorenz Model
Computer Physics Communications, 2015Co-Authors: Roberto Barrio, Ángeles Dena, Warwick TuckerAbstract:Abstract A benchmark database of very high-precision numerical and validated initial conditions of periodic orbits for the Lorenz Model is presented. This database is a “computational challenge” and it provides the initial conditions of all periodic orbits of the Lorenz Model up to multiplicity 10 and guarantees their existence via computer-assisted proofs methods. The orbits are computed using high-precision arithmetic and mixing several techniques resulting in 1000 digits of precision on the initial conditions of the periodic orbits, and intervals of size 10 100 that prove the existence of each orbit. Program summary Program title: Lorenz-Database Catalogue identifier: AEWM_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEWM_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 8515 No. of bytes in distributed program, including test data, etc.: 6964501 Distribution format: tar.gz Programming language: Data. Computer: Any computer. Operating system: Any. RAM: Database, no requirements Classification: 4.3, 4.12. Nature of problem: Database of all periodic orbits of the Lorenz Model up to multiplicity 10 with 1000 precision digits. Solution method: Advanced search methods for locating unstable periodic orbits combined with the Taylor series method for multiple precision integration of ODEs and interval methods for providing Computer-Assisted proofs of the periodic orbits. Unusual features: The database gives 100 digits rigorously proved using Computer-Assisted techniques and 1000 digits using an optimal adaptive Taylor series method. Running time: Not Applicable.
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A database of rigorous and high-precision periodic orbits of the Lorenz Model ☆
Computer Physics Communications, 2015Co-Authors: Roberto Barrio, Ángeles Dena, Warwick TuckerAbstract:Abstract A benchmark database of very high-precision numerical and validated initial conditions of periodic orbits for the Lorenz Model is presented. This database is a “computational challenge” and it provides the initial conditions of all periodic orbits of the Lorenz Model up to multiplicity 10 and guarantees their existence via computer-assisted proofs methods. The orbits are computed using high-precision arithmetic and mixing several techniques resulting in 1000 digits of precision on the initial conditions of the periodic orbits, and intervals of size 10 100 that prove the existence of each orbit. Program summary Program title: Lorenz-Database Catalogue identifier: AEWM_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEWM_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 8515 No. of bytes in distributed program, including test data, etc.: 6964501 Distribution format: tar.gz Programming language: Data. Computer: Any computer. Operating system: Any. RAM: Database, no requirements Classification: 4.3, 4.12. Nature of problem: Database of all periodic orbits of the Lorenz Model up to multiplicity 10 with 1000 precision digits. Solution method: Advanced search methods for locating unstable periodic orbits combined with the Taylor series method for multiple precision integration of ODEs and interval methods for providing Computer-Assisted proofs of the periodic orbits. Unusual features: The database gives 100 digits rigorously proved using Computer-Assisted techniques and 1000 digits using an optimal adaptive Taylor series method. Running time: Not Applicable.
Roberto Barrio - One of the best experts on this subject based on the ideXlab platform.
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a database of rigorous and high precision periodic orbits of the Lorenz Model
Computer Physics Communications, 2015Co-Authors: Roberto Barrio, Ángeles Dena, Warwick TuckerAbstract:Abstract A benchmark database of very high-precision numerical and validated initial conditions of periodic orbits for the Lorenz Model is presented. This database is a “computational challenge” and it provides the initial conditions of all periodic orbits of the Lorenz Model up to multiplicity 10 and guarantees their existence via computer-assisted proofs methods. The orbits are computed using high-precision arithmetic and mixing several techniques resulting in 1000 digits of precision on the initial conditions of the periodic orbits, and intervals of size 10 100 that prove the existence of each orbit. Program summary Program title: Lorenz-Database Catalogue identifier: AEWM_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEWM_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 8515 No. of bytes in distributed program, including test data, etc.: 6964501 Distribution format: tar.gz Programming language: Data. Computer: Any computer. Operating system: Any. RAM: Database, no requirements Classification: 4.3, 4.12. Nature of problem: Database of all periodic orbits of the Lorenz Model up to multiplicity 10 with 1000 precision digits. Solution method: Advanced search methods for locating unstable periodic orbits combined with the Taylor series method for multiple precision integration of ODEs and interval methods for providing Computer-Assisted proofs of the periodic orbits. Unusual features: The database gives 100 digits rigorously proved using Computer-Assisted techniques and 1000 digits using an optimal adaptive Taylor series method. Running time: Not Applicable.
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A database of rigorous and high-precision periodic orbits of the Lorenz Model ☆
Computer Physics Communications, 2015Co-Authors: Roberto Barrio, Ángeles Dena, Warwick TuckerAbstract:Abstract A benchmark database of very high-precision numerical and validated initial conditions of periodic orbits for the Lorenz Model is presented. This database is a “computational challenge” and it provides the initial conditions of all periodic orbits of the Lorenz Model up to multiplicity 10 and guarantees their existence via computer-assisted proofs methods. The orbits are computed using high-precision arithmetic and mixing several techniques resulting in 1000 digits of precision on the initial conditions of the periodic orbits, and intervals of size 10 100 that prove the existence of each orbit. Program summary Program title: Lorenz-Database Catalogue identifier: AEWM_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEWM_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 8515 No. of bytes in distributed program, including test data, etc.: 6964501 Distribution format: tar.gz Programming language: Data. Computer: Any computer. Operating system: Any. RAM: Database, no requirements Classification: 4.3, 4.12. Nature of problem: Database of all periodic orbits of the Lorenz Model up to multiplicity 10 with 1000 precision digits. Solution method: Advanced search methods for locating unstable periodic orbits combined with the Taylor series method for multiple precision integration of ODEs and interval methods for providing Computer-Assisted proofs of the periodic orbits. Unusual features: The database gives 100 digits rigorously proved using Computer-Assisted techniques and 1000 digits using an optimal adaptive Taylor series method. Running time: Not Applicable.
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BEHAVIOR PATTERNS IN MULTIPARAMETRIC DYNAMICAL SYSTEMS: Lorenz Model
International Journal of Bifurcation and Chaos, 2012Co-Authors: Roberto Barrio, Fernando Blesa, Sergio SerranoAbstract:In experimental and theoretical studies of Dynamical Systems, there are usually several parameters that govern the Models. Thus, a detailed study of the global parametric phase space is not easy and normally unachievable. In this paper, we show that a careful selection of one straight line (or other 1D manifold) permits us to obtain a global idea of the evolution of the system in some circumstances. We illustrate this fact with the paradigmatic example of the Lorenz Model, based on a global study, changing all three parameters. Besides, searching in other regions, for all the detected behavior patterns in one straight line, we have been able to see that missing topological structures of the chaotic attractors may be found on the chaotic-saddles.
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Bounds for the chaotic region in the Lorenz Model
Physica D: Nonlinear Phenomena, 2009Co-Authors: Roberto Barrio, Sergio SerranoAbstract:Abstract In a previous paper, the authors made an extensive numerical study of the Lorenz Model, changing all three parameters of the system. We conjectured that the region of parameters where the Lorenz Model is chaotic is bounded for fixed r . In this paper, we give a theoretical proof of the conjecture by obtaining theoretical bounds for the chaotic region and by using Fenichel theory. The theoretical bounds are complemented with numerical studies performed using the Maximum Lyapunov Exponent and OFLI2 techniques, and a comparison of both sets of results is shown. Finally, we provide a complete three-dimensional Model of the chaotic regime depending on the three parameters.
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A three-parametric study of the Lorenz Model
Physica D: Nonlinear Phenomena, 2007Co-Authors: Roberto Barrio, Sergio SerranoAbstract:In this paper we analyse the three-parametric phase diagram of the classical Lorenz Model. This has been done by the combined use of different techniques. The most important one for our objective is the combined use of the recently developed OFLI2 Chaos Indicator and the classical Maximum Lyapunov Exponent, which have permitted a fast and complete location of the chaotic parameter region. As a result, we provide a three-parametric Model of the chaotic regime for the Lorenz system that gives new light on this classical problem.
Lev Ryashko - One of the best experts on this subject based on the ideXlab platform.
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NOISE-INDUCED CHAOS AND BACKWARD STOCHASTIC BIFURCATIONS IN THE Lorenz Model
International Journal of Bifurcation and Chaos, 2013Co-Authors: Irina Bashkirtseva, Lev Ryashko, Pavel StikhinAbstract:We study the phenomena of stochastic D- and P-bifurcations of randomly forced limit cycles for the Lorenz Model. As noise intensity increases, regular multiple limit cycles of this Model in a period-doubling bifurcations zone are deformed to be stochastic attractors that look chaotic (D-bifurcation) and their multiplicity is reduced (P-bifurcation). In this paper for the comparative investigation of these bifurcations, the analysis of Lyapunov exponents and stochastic sensitivity function technique are used. A probabilistic mechanism of backward stochastic bifurcations for cycles of high multiplicity is analyzed in detail. We show that for a limit cycle with multiplicity two and higher, a threshold value of the noise intensity which marks the onset of chaos agrees with the first backward stochastic bifurcation.
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Constructive analysis of noise-induced transitions for coexisting periodic attractors of the Lorenz Model.
Physical review. E Statistical nonlinear and soft matter physics, 2009Co-Authors: Irina Bashkirtseva, Lev RyashkoAbstract:We study the stochastically forced Lorenz Model in the parameter zone admitting two coexisting limit cycles under the transition to chaos via period-doubling bifurcations. Noise-induced transitions between both different parts of the single attractor and two coexisting separate attractors are demonstrated. The effects of structural stabilization and noise symmetrization are discussed. We suggest a stochastic sensitivity function technique for the analysis of noise-induced transitions between two coexisting limit cycles. This approach allows us to construct the dispersion ellipses of random trajectories for any Poincare sections. Possibilities of our descriptive-geometric method for a detailed analysis of noise-induced transitions between two periodic attractors of Lorenz Model are demonstrated.
Nigel G. Stocks - One of the best experts on this subject based on the ideXlab platform.
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Influence of noise on periodic attractors in the Lorenz Model: Zero-frequency spectral peaks and chaos
Physical Review A, 1992Co-Authors: I. I. Fedchenia, N. D. Stein, Riccardo Mannella, Peter V. E. Mcclintock, Nigel G. StocksAbstract:The influence of weak external noise on the Lorenz Model has been studied in parameter ranges where it is characterized by pairs of nonsymmetrical periodic attractors. The investigations, based on electronic analog and digital techniques, have shown that the noise can give rise to pronounced zero-frequency spectral peaks of complicated shape and to chaos.
