7,008 research outputs found

    Competition of spatial and temporal instabilities under time delay near codimension-two Turing-Hopf bifurcations

    Full text link
    Competition of spatial and temporal instabilities under time delay near the codimension-two Turing-Hopf bifurcations is studied in a reaction-diffusion equation. The time delay changes remarkably the oscillation frequency, the intrinsic wave vector, and the intensities of both Turing and Hopf modes. The application of appropriate time delay can control the competition between the Turing and Hopf modes. Analysis shows that individual or both feedbacks can realize the control of the transformation between the Turing and Hopf patterns. Two dimensional numerical simulations validate the analytical results.Comment: 13 pages, 6 figure

    Post-Double Hopf Bifurcation Dynamics and Adaptive Synchronization of a Hyperchaotic System

    Full text link
    In this paper a four-dimensional hyperchaotic system with only one equilibrium is considered and its double Hopf bifurcations are investigated. The general post-bifurcation and stability analysis are carried out using the normal form of the system obtained via the method of multiple scales. The dynamics of the orbits predicted through the normal form comprises possible regimes of periodic solutions, two-period tori, and three-period tori in parameter space. Moreover, we show how the hyperchaotic synchronization of this system can be realized via an adaptive control scheme. Numerical simulations are included to show the effectiveness of the designed control

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

    Full text link
    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

    Suppression of Limit Cycle Oscillations using the Nonlinear Tuned Vibration Absorber

    Full text link
    The objective of the present study is to mitigate, or even completely eliminate, the limit cycle oscillations in mechanical systems using a passive nonlinear absorber, termed the nonlinear tuned vibration absorber (NLTVA). An unconventional aspect of the NLTVA is that the mathematical form of its restoring force is not imposed a priori, as it is the case for most existing nonlinear absorbers. The NLTVA parameters are determined analytically using stability and bifurcation analyses, and the resulting design is validated using numerical continuation. The proposed developments are illustrated using a Van der Pol-Duffing primary system

    Time-delayed feedback control of unstable periodic orbits near a subcritical Hopf bifurcation

    Full text link
    We show that Pyragas delayed feedback control can stabilize an unstable periodic orbit (UPO) that arises from a generic subcritical Hopf bifurcation of a stable equilibrium in an n-dimensional dynamical system. This extends results of Fiedler et al. [PRL 98, 114101 (2007)], who demonstrated that such feedback control can stabilize the UPO associated with a two-dimensional subcritical Hopf normal form. Pyragas feedback requires an appropriate choice of a feedback gain matrix for stabilization, as well as knowledge of the period of the targeted UPO. We apply feedback in the directions tangent to the two-dimensional center manifold. We parameterize the feedback gain by a modulus and a phase angle, and give explicit formulae for choosing these two parameters given the period of the UPO in a neighborhood of the bifurcation point. We show, first heuristically, and then rigorously by a center manifold reduction for delay differential equations, that the stabilization mechanism involves a highly degenerate Hopf bifurcation problem that is induced by the time-delayed feedback. When the feedback gain modulus reaches a threshold for stabilization, both of the genericity assumptions associated with a two-dimensional Hopf bifurcation are violated: the eigenvalues of the linearized problem do not cross the imaginary axis as the bifurcation parameter is varied, and the real part of the cubic coefficient of the normal form vanishes. Our analysis of this degenerate bifurcation problem reveals two qualitatively distinct cases when unfolded in a two-parameter plane. In each case, Pyragas-type feedback successfully stabilizes the branch of small-amplitude UPOs in a neighborhood of the original bifurcation point, provided that the phase angle satisfies a certain restriction.Comment: 35 pages, 19 figure
    • …
    corecore