1,753 research outputs found

    Control of Hopf Bifurcation in Autonomous System Based on Washout Filter

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    In order to further understand a Lorenz-like system, we study the stability of the equilibrium points and the existence of Hopf bifurcation by center manifold theorem and normal form theory. More precisely, we designed a washout controller such that the equilibrium E0 undergoes a controllable Hopf bifurcation, and by adjusting the controller parameters, we delayed Hopf bifurcation phenomenon of the equilibrium E+. Besides, numerical simulation is given to illustrate the theoretical analysis. Finally, two possible electronic circuits are given to realize the uncontrolled and the controlled systems

    Effective Control and Bifurcation Analysis in a Chaotic System with Distributed Delay Feedback

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    We discuss the dynamic behavior of a new Lorenz-like chaotic system with distributed delayed feedback by the qualitative analysis and numerical simulations. It is verified that the equilibria are locally asymptotically stable when α∈(0,α0) and unstable when α∈(α0,∞); Hopf bifurcation occurs when α crosses a critical value α0 by choosing α as a bifurcation parameter. Meanwhile, the explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by normal form theorem and center manifold argument. Furthermore, regarding α as a bifurcation parameter, we explore variation tendency of the dynamics behavior of a chaotic system with the increase of the parameter value α

    Symmetries in the Lorenz-96 model

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    The Lorenz-96 model is widely used as a test model for various applications, such as data assimilation methods. This symmetric model has the forcing F∈RF\in\mathbb{R} and the dimension n∈Nn\in\mathbb{N} as parameters and is Zn\mathbb{Z}_n equivariant. In this paper, we unravel its dynamics for F<0F<0 using equivariant bifurcation theory. Symmetry gives rise to invariant subspaces, that play an important role in this model. We exploit them in order to generalise results from a low dimension to all multiples of that dimension. We discuss symmetry for periodic orbits as well. Our analysis leads to proofs of the existence of pitchfork bifurcations for F<0F<0 in specific dimensions nn: In all even dimensions, the equilibrium (F,…,F)(F,\ldots,F) exhibits a supercritical pitchfork bifurcation. In dimensions n=4kn=4k, k∈Nk\in\mathbb{N}, a second supercritical pitchfork bifurcation occurs simultaneously for both equilibria originating from the previous one. Furthermore, numerical observations reveal that in dimension n=2qpn=2^qp, where q∈N∪{0}q\in\mathbb{N}\cup\{0\} and pp is odd, there is a finite cascade of exactly qq subsequent pitchfork bifurcations, whose bifurcation values are independent of nn. This structure is discussed and interpreted in light of the symmetries of the model.Comment: 31 pages, 9 figures and 3 table

    Kneadings, Symbolic Dynamics and Painting Lorenz Chaos. A Tutorial

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    A new computational technique based on the symbolic description utilizing kneading invariants is proposed and verified for explorations of dynamical and parametric chaos in a few exemplary systems with the Lorenz attractor. The technique allows for uncovering the stunning complexity and universality of bi-parametric structures and detect their organizing centers - codimension-two T-points and separating saddles in the kneading-based scans of the iconic Lorenz equation from hydrodynamics, a normal model from mathematics, and a laser model from nonlinear optics.Comment: Journal of Bifurcations and Chaos, 201

    Stabilizing unstable periodic orbits in the Lorenz equations using time-delayed feedback control

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    For many years it was believed that an unstable periodic orbit with an odd number of real Floquet multipliers greater than unity cannot be stabilized by the time-delayed feedback control mechanism of Pyragus. A recent paper by Fiedler et al uses the normal form of a subcritical Hopf bifurcation to give a counterexample to this theorem. Using the Lorenz equations as an example, we demonstrate that the stabilization mechanism identified by Fiedler et al for the Hopf normal form can also apply to unstable periodic orbits created by subcritical Hopf bifurcations in higher-dimensional dynamical systems. Our analysis focuses on a particular codimension-two bifurcation that captures the stabilization mechanism in the Hopf normal form example, and we show that the same codimension-two bifurcation is present in the Lorenz equations with appropriately chosen Pyragus-type time-delayed feedback. This example suggests a possible strategy for choosing the feedback gain matrix in Pyragus control of unstable periodic orbits that arise from a subcritical Hopf bifurcation of a stable equilibrium. In particular, our choice of feedback gain matrix is informed by the Fiedler et al example, and it works over a broad range of parameters, despite the fact that a center-manifold reduction of the higher-dimensional problem does not lead to their model problem.Comment: 21 pages, 8 figures, to appear in PR

    Attempts to relate the Navier-Stokes equations to turbulence

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    The present talk is designed as a survey, is slanted to my personal tastes, but I hope it is still representative. My intention is to keep the whole discussion pretty elementary by touching large numbers of topics and avoiding details as well as technical difficulties in any one of them. Subsequent talks will go deeper into some of the subjects we discuss today. The main goal is to link up the statistics, entropy, correlation functions, etc., in the engineering side with a "nice" mathematical model of turbulence
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