Book•
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
01 Aug 1983-
TL;DR: In this article, the authors introduce differential equations and dynamical systems, including hyperbolic sets, Sympolic Dynamics, and Strange Attractors, and global bifurcations.
Abstract: Contents: Introduction: Differential Equations and Dynamical Systems.- An Introduction to Chaos: Four Examples.- Local Bifurcations.- Averaging and Perturbation from a Geometric Viewpoint.- Hyperbolic Sets, Sympolic Dynamics, and Strange Attractors.- Global Bifurcations.- Local Codimension Two Bifurcations of Flows.- Appendix: Suggestions for Further Reading. Postscript Added at Second Printing. Glossary. References. Index.
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TL;DR: In this paper, the authors introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous feedback, which subsumes the physical properties of a linearizing output and provides another nonlinear extension of Kalman's controllability.
Abstract: We introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous. Their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of Kalman's controllability. The distance to flatness is measured by a non-negative integer, the defect. We utilize differential algebra where flatness- and defect are best defined without distinguishing between input, state, output and other variables. Many realistic classes of examples are flat. We treat two popular ones: the crane and the car with n trailers, the motion planning of which is obtained via elementary properties of plane curves. The three non-flat examples, the simple, double and variable length pendulums, are borrowed from non-linear physics. A high frequency control strategy is proposed such that the averaged systems become flat.
3,025 citations
Cites background from "Nonlinear Oscillations, Dynamical S..."
...Since the averaged system admits a hyperbolic asymptotically stable equilibrium, the perturbed system admits an hyperbolic asymptotically stable limit cycle around (α, p, z) = (αsp, 0, zsp) (Guckenheimer and Holmes 1983, theorem 4.1.1, page 168): such control maintains (z, α) near (zsp, αsp)....
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Book•
01 Jan 1996TL;DR: In this article, the authors present a review of rigor properties of low-dimensional models and their applications in the field of fluid mechanics. But they do not consider the effects of random perturbation on models.
Abstract: Preface Part I. Turbulence: 1. Introduction 2. Coherent structures 3. Proper orthogonal decomposition 4. Galerkin projection Part II. Dynamical Systems: 5. Qualitative theory 6. Symmetry 7. One-dimensional 'turbulence' 8. Randomly perturbed systems Part III. 9. Low-dimensional Models: 10. Behaviour of the models Part IV. Other Applications and Related Work: 11. Some other fluid problems 12. Review: prospects for rigor Bibliography.
2,920 citations
TL;DR: This work develops a novel framework to discover governing equations underlying a dynamical system simply from data measurements, leveraging advances in sparsity techniques and machine learning and using sparse regression to determine the fewest terms in the dynamic governing equations required to accurately represent the data.
Abstract: Extracting governing equations from data is a central challenge in many diverse areas of science and engineering. Data are abundant whereas models often remain elusive, as in climate science, neuroscience, ecology, finance, and epidemiology, to name only a few examples. In this work, we combine sparsity-promoting techniques and machine learning with nonlinear dynamical systems to discover governing equations from noisy measurement data. The only assumption about the structure of the model is that there are only a few important terms that govern the dynamics, so that the equations are sparse in the space of possible functions; this assumption holds for many physical systems in an appropriate basis. In particular, we use sparse regression to determine the fewest terms in the dynamic governing equations required to accurately represent the data. This results in parsimonious models that balance accuracy with model complexity to avoid overfitting. We demonstrate the algorithm on a wide range of problems, from simple canonical systems, including linear and nonlinear oscillators and the chaotic Lorenz system, to the fluid vortex shedding behind an obstacle. The fluid example illustrates the ability of this method to discover the underlying dynamics of a system that took experts in the community nearly 30 years to resolve. We also show that this method generalizes to parameterized systems and systems that are time-varying or have external forcing.
2,784 citations
Cites background or methods from "Nonlinear Oscillations, Dynamical S..."
...Here, we consider dynamical systems (31) of the form...
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...The SINDy algorithm is readily extended to encompass these important parameterized systems, allowing for the discovery of normal forms (31, 50) associated with a bifurcation parameter μ....
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Book•
19 Aug 1998
TL;DR: This chapter establishes the framework of random dynamical systems and introduces the concept of random attractors to analyze models with stochasticity or randomness.
Abstract: I. Random Dynamical Systems and Their Generators.- 1. Basic Definitions. Invariant Measures.- 2. Generation.- II. Multiplicative Ergodic Theory.- 3. The Multiplicative Ergodic Theorem in Euclidean Space.- 4. The Multiplicative Ergodic Theorem on Bundles and Manifolds.- 5. The MET for Related Linear and Affine RDS.- 6. RDS on Homogeneous Spaces of the General Linear Group.- III. Smooth Random Dynamical Systems.- 7. Invariant Manifolds.- 8. Normal Forms.- 9. Bifurcation Theory.- IV. Appendices.- Appendix A. Measurable Dynamical Systems.- A.1 Ergodic Theory.- A.2 Stochastic Processes and Dynamical Systems.- A.3 Stationary Processes.- A.4 Markov Processes.- Appendix B. Smooth Dynamical Systems.- B.1 Two-Parameter Flows on a Manifold.- B.4 Autonomous Case: Dynamical Systems.- B.5 Vector Fields and Flows on Manifolds.- References.
2,663 citations
01 Jan 2000
TL;DR: In this paper, the authors introduce a specific class of linear matrix inequalities (LMI) whose optimal solution can be characterized exactly, i.e., the optimal value equals the spectral radius of the operator.
Abstract: In the first part of this thesis, we introduce a specific class of Linear Matrix Inequalities (LMI) whose optimal solution can be characterized exactly. This family corresponds to the case where the associated linear operator maps the cone of positive semidefinite matrices onto itself. In this case, the optimal value equals the spectral radius of the operator. It is shown that some rank minimization problems, as well as generalizations of the structured singular value ($mu$) LMIs, have exactly this property.
In the same spirit of exploiting structure to achieve computational efficiency, an algorithm for the numerical solution of a special class of frequency-dependent LMIs is presented. These optimization problems arise from robustness analysis questions, via the Kalman-Yakubovich-Popov lemma. The procedure is an outer approximation method based on the algorithms used in the computation of hinf norms for linear, time invariant systems. The result is especially useful for systems with large state dimension.
The other main contribution in this thesis is the formulation of a convex optimization framework for semialgebraic problems, i.e., those that can be expressed by polynomial equalities and inequalities. The key element is the interaction of concepts in real algebraic geometry (Positivstellensatz) and semidefinite programming.
To this end, an LMI formulation for the sums of squares decomposition for multivariable polynomials is presented. Based on this, it is shown how to construct sufficient Positivstellensatz-based convex tests to prove that certain sets are empty. Among other applications, this leads to a nonlinear extension of many LMI based results in uncertain linear system analysis.
Within the same framework, we develop stronger criteria for matrix copositivity, and generalizations of the well-known standard semidefinite relaxations for quadratic programming.
Some applications to new and previously studied problems are presented. A few examples are Lyapunov function computation, robust bifurcation analysis, structured singular values, etc. It is shown that the proposed methods allow for improved solutions for very diverse questions in continuous and combinatorial optimization.
2,269 citations