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Study of Nonlinear Analysis and Chaos in Vibrations and Fluids

18 May 2016-
TL;DR: In this paper, the authors studied the chaotic vibration phenomenon in high-dimensional partial differential equations and the emergence of the Navier-Stokes-alpha model for channel flows using numerical simulations.
Abstract: Chaos and turbulence are two important topics in nonlinear dynamics. In this study, two problems related to chaos and turbulence modelling are presented. They are the chaotic vibration phenomenon in high-dimensional partial differential equations and the emergence of the Navier-Stokes-alpha model for channel flows. The study of the chaotic vibration phenomenon in high-dimensional partial differential equations is explained from both the numerical and theoretical aspects. In the numerical perspective, we have studied the chaotic vibration phenomenon of the 2D wave equation through numerical simulations. Based on the finite-volume method, we have built our own solver “img2Foam” in the Computational Fluid Dynamics software OpenFOAM (Open source Field Operation and Manipulation). We have implemented several numerical simulations containing both chaotic and non-chaotic cases. As for the theoretical perspective, we give a rigorous proof for the chaotic vibration phenomenon of the 2D non-strictly hyperbolic equation. After introducing two linear operators, the initial system of the 2D non-strictly hyperbolic equation is converted into a system of two coupled first order equations. By using the method of characteristics, we have found the explicit solution formulas of the new system. We have also found a regime of the parameters when the chaotic vibration phenomenon occurs by applying the period-doubling bifurcation theorem. Numerical simulations are presented to validate the theoretical results. Inspired by the concept of the regular part of the weak attractor of the 3D NavierStokes equations, we concentrate on a restricted class of fluid flows to explore the transition from the Navier-Stokes equations to the Navier-Stokes-alpha model for channel flows. The Navier-Stokes equations have been widely used to describe the
Citations
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Journal ArticleDOI
TL;DR: 5. M. Green, J. Schwarz, and E. Witten, Superstring theory, and An interpretation of classical Yang-Mills theory, Cambridge Univ.
Abstract: 5. M. Green, J. Schwarz, and E. Witten, Superstring theory, Cambridge Univ. Press, 1987. 6. J. Isenberg, P. Yasskin, and P. Green, Non-self-dual gauge fields, Phys. Lett. 78B (1978), 462-464. 7. B. Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential Geometric Methods in Mathematicas Physics, Lecture Notes in Math., vol. 570, SpringerVerlag, Berlin and New York, 1977. 8. C. LeBrun, Thickenings and gauge fields, Class. Quantum Grav. 3 (1986), 1039-1059. 9. , Thickenings and conformai gravity, preprint, 1989. 10. C. LeBrun and M. Rothstein, Moduli of super Riemann surfaces, Commun. Math. Phys. 117(1988), 159-176. 11. Y. Manin, Critical dimensions of string theories and the dualizing sheaf on the moduli space of (super) curves, Funct. Anal. Appl. 20 (1987), 244-245. 12. R. Penrose and W. Rindler, Spinors and space-time, V.2, spinor and twistor methods in space-time geometry, Cambridge Univ. Press, 1986. 13. R. Ward, On self-dual gauge fields, Phys. Lett. 61A (1977), 81-82. 14. E. Witten, An interpretation of classical Yang-Mills theory, Phys. Lett. 77NB (1978), 394-398. 15. , Twistor-like transform in ten dimensions, Nucl. Phys. B266 (1986), 245-264. 16. , Physics and geometry, Proc. Internat. Congr. Math., Berkeley, 1986, pp. 267302, Amer. Math. Soc, Providence, R.I., 1987.

1,252 citations

Journal ArticleDOI
TL;DR: In this paper, the quadratic family has been used to define hyperbolicity in linear algebra and advanced calculus, including the Julia set and the Mandelbrot set.
Abstract: Part One: One-Dimensional Dynamics Examples of Dynamical Systems Preliminaries from Calculus Elementary Definitions Hyperbolicity An example: the quadratic family An Example: the Quadratic Family Symbolic Dynamics Topological Conjugacy Chaos Structural Stability Sarlovskiis Theorem The Schwarzian Derivative Bifurcation Theory Another View of Period Three Maps of the Circle Morse-Smale Diffeomorphisms Homoclinic Points and Bifurcations The Period-Doubling Route to Chaos The Kneeding Theory Geneaology of Periodic Units Part Two: Higher Dimensional Dynamics Preliminaries from Linear Algebra and Advanced Calculus The Dynamics of Linear Maps: Two and Three Dimensions The Horseshoe Map Hyperbolic Toral Automorphisms Hyperbolicm Toral Automorphisms Attractors The Stable and Unstable Manifold Theorem Global Results and Hyperbolic Sets The Hopf Bifurcation The Hnon Map Part Three: Complex Analytic Dynamics Preliminaries from Complex Analysis Quadratic Maps Revisited Normal Families and Exceptional Points Periodic Points The Julia Set The Geometry of Julia Sets Neutral Periodic Points The Mandelbrot Set An Example: the Exponential Function

104 citations

References
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Book
01 Jan 2002
TL;DR: The CLAWPACK software as discussed by the authors is a popular tool for solving high-resolution hyperbolic problems with conservation laws and conservation laws of nonlinear scalar scalar conservation laws.
Abstract: Preface 1. Introduction 2. Conservation laws and differential equations 3. Characteristics and Riemann problems for linear hyperbolic equations 4. Finite-volume methods 5. Introduction to the CLAWPACK software 6. High resolution methods 7. Boundary conditions and ghost cells 8. Convergence, accuracy, and stability 9. Variable-coefficient linear equations 10. Other approaches to high resolution 11. Nonlinear scalar conservation laws 12. Finite-volume methods for nonlinear scalar conservation laws 13. Nonlinear systems of conservation laws 14. Gas dynamics and the Euler equations 15. Finite-volume methods for nonlinear systems 16. Some nonclassical hyperbolic problems 17. Source terms and balance laws 18. Multidimensional hyperbolic problems 19. Multidimensional numerical methods 20. Multidimensional scalar equations 21. Multidimensional systems 22. Elastic waves 23. Finite-volume methods on quadrilateral grids Bibliography Index.

5,791 citations


"Study of Nonlinear Analysis and Cha..." refers background or methods in this paper

  • ...FVM is a method that represents and evaluates partial differential equations in the form of algebraic equations [31]....

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  • ...“Finite volume” refers to the small volume surrounding each node point [31]....

    [...]

  • ...By applying the divergence theorem, the volume integrals that contain a divergence term in the equation are converted into surface integrals [31]....

    [...]

Book
01 Jan 1986
TL;DR: In this article, the quadratic family has been used to define hyperbolicity in linear algebra and advanced calculus, including the Julia set and the Mandelbrot set.
Abstract: Part One: One-Dimensional Dynamics Examples of Dynamical Systems Preliminaries from Calculus Elementary Definitions Hyperbolicity An example: the quadratic family An Example: the Quadratic Family Symbolic Dynamics Topological Conjugacy Chaos Structural Stability Sarlovskiis Theorem The Schwarzian Derivative Bifurcation Theory Another View of Period Three Maps of the Circle Morse-Smale Diffeomorphisms Homoclinic Points and Bifurcations The Period-Doubling Route to Chaos The Kneeding Theory Geneaology of Periodic Units Part Two: Higher Dimensional Dynamics Preliminaries from Linear Algebra and Advanced Calculus The Dynamics of Linear Maps: Two and Three Dimensions The Horseshoe Map Hyperbolic Toral Automorphisms Hyperbolicm Toral Automorphisms Attractors The Stable and Unstable Manifold Theorem Global Results and Hyperbolic Sets The Hopf Bifurcation The Hnon Map Part Three: Complex Analytic Dynamics Preliminaries from Complex Analysis Quadratic Maps Revisited Normal Families and Exceptional Points Periodic Points The Julia Set The Geometry of Julia Sets Neutral Periodic Points The Mandelbrot Set An Example: the Exponential Function

3,589 citations

Book
01 Jan 1962

2,230 citations


"Study of Nonlinear Analysis and Cha..." refers background in this paper

  • ...According to Poisson’s formula (see [30]; see also [27]), we have, for any a > 0, a ∈ R,...

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Journal ArticleDOI
TL;DR: 5. M. Green, J. Schwarz, and E. Witten, Superstring theory, and An interpretation of classical Yang-Mills theory, Cambridge Univ.
Abstract: 5. M. Green, J. Schwarz, and E. Witten, Superstring theory, Cambridge Univ. Press, 1987. 6. J. Isenberg, P. Yasskin, and P. Green, Non-self-dual gauge fields, Phys. Lett. 78B (1978), 462-464. 7. B. Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential Geometric Methods in Mathematicas Physics, Lecture Notes in Math., vol. 570, SpringerVerlag, Berlin and New York, 1977. 8. C. LeBrun, Thickenings and gauge fields, Class. Quantum Grav. 3 (1986), 1039-1059. 9. , Thickenings and conformai gravity, preprint, 1989. 10. C. LeBrun and M. Rothstein, Moduli of super Riemann surfaces, Commun. Math. Phys. 117(1988), 159-176. 11. Y. Manin, Critical dimensions of string theories and the dualizing sheaf on the moduli space of (super) curves, Funct. Anal. Appl. 20 (1987), 244-245. 12. R. Penrose and W. Rindler, Spinors and space-time, V.2, spinor and twistor methods in space-time geometry, Cambridge Univ. Press, 1986. 13. R. Ward, On self-dual gauge fields, Phys. Lett. 61A (1977), 81-82. 14. E. Witten, An interpretation of classical Yang-Mills theory, Phys. Lett. 77NB (1978), 394-398. 15. , Twistor-like transform in ten dimensions, Nucl. Phys. B266 (1986), 245-264. 16. , Physics and geometry, Proc. Internat. Congr. Math., Berkeley, 1986, pp. 267302, Amer. Math. Soc, Providence, R.I., 1987.

1,252 citations


"Study of Nonlinear Analysis and Cha..." refers background in this paper

  • ...3) comes from the definition of weak global attractor of the NSE as given in [23], however, we remark that even though there are several equivalent ways to define the global attractors for many dissipative systems (see [41], [20]), in some particular systems without having full dissipations, the appropriate notion for attractors should be defined using the boundedness (see [4])....

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Book
01 Jan 1950
TL;DR: In this article, the authors studied the stability of nonlinear oscillations with nonlinear restoring forces free oscillations and nonlinear self-sustained oscillations using the Hill's equation.
Abstract: Linear vibrations free vibrations of undamped systems with nonlinear restoring forces free oscillations with damping and the geometry of integral curves forced oscillations of systems with nonlinear restoring force self-sustained oscillations Hill's equation and its application to the study of the stability of nonlinear oscillations.

704 citations


"Study of Nonlinear Analysis and Cha..." refers background in this paper

  • ...2 Numerical Simulations of a 2D Wave Equation The study of nonlinear vibrations in mechanical and electronic systems has always been an important area of research by scientists and engineers [39]....

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