5,814 research outputs found
Control bifurcations
A parametrized nonlinear differential equation can have multiple equilibria as the parameter is varied. A local bifurcation of a parametrized differential equation occurs at an equilibrium where there is a change in the topological character of the nearby solution curves. This typically happens because some eigenvalues of the parametrized linear approximating differential equation cross the imaginary axis and there is a change in stability of the equilibrium. The topological nature of the solutions is unchanged by smooth changes of state coordinates so these may be used to bring the differential equation into Poincare/spl acute/ normal form. From this normal form, the type of the bifurcation can be determined. For differential equations depending on a single parameter, the typical ways that the system can bifurcate are fully understood, e.g., the fold (or saddle node), the transcritical and the Hopf bifurcation. A nonlinear control system has multiple equilibria typically parametrized by the set value of the control. A control bifurcation of a nonlinear system typically occurs when its linear approximation loses stabilizability. The ways in which this can happen are understood through the appropriate normal forms. We present the quadratic and cubic normal forms of a scalar input nonlinear control system around an equilibrium point. These are the normal forms under quadratic and cubic change of state coordinates and invertible state feedback. The system need not be linearly controllable. We study some important control bifurcations, the analogues of the classical fold, transcritical and Hopf bifurcations
Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations
In this paper we perform the parameter-dependent center manifold reduction
near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf
bifurcations in delay differential equations (DDEs). This allows us to
initialize the continuation of codimension one equilibria and cycle
bifurcations emanating from these codimension two bifurcation points. The
normal form coefficients are derived in the functional analytic perturbation
framework for dual semigroups (sun-star calculus) using a normalization
technique based on the Fredholm alternative. The obtained expressions give
explicit formulas which have been implemented in the freely available numerical
software package DDE-BifTool. While our theoretical results are proven to apply
more generally, the software implementation and examples focus on DDEs with
finitely many discrete delays. Together with the continuation capabilities of
DDE-BifTool, this provides a powerful tool to study the dynamics near
equilibria of such DDEs. The effectiveness is demonstrated on various models
Bifurcations of piecewise smooth flows:perspectives, methodologies and open problems
In this paper, the theory of bifurcations in piecewise smooth flows is critically surveyed. The focus is on results that hold in arbitrarily (but finitely) many dimensions, highlighting significant areas where a detailed understanding is presently lacking. The clearest results to date concern equilibria undergoing bifurcations at switching boundaries, and limit cycles undergoing grazing and sliding bifurcations. After discussing fundamental concepts, such as topological equivalence of two piecewise smooth systems, discontinuity-induced bifurcations are defined for equilibria and limit cycles. Conditions for equilibria to exist in n-dimensions are given, followed by the conditions under which they generically undergo codimension-one bifurcations. The extent of knowledge of their unfoldings is also summarized. Codimension-one bifurcations of limit cycles and boundary-intersection crossing are described together with techniques for their classification. Codimension-two bifurcations are discussed with suggestions for further study
Aspects of Bifurcation Theory for Piecewise-Smooth, Continuous Systems
Systems that are not smooth can undergo bifurcations that are forbidden in
smooth systems. We review some of the phenomena that can occur for
piecewise-smooth, continuous maps and flows when a fixed point or an
equilibrium collides with a surface on which the system is not smooth. Much of
our understanding of these cases relies on a reduction to piecewise linearity
near the border-collision. We also review a number of codimension-two
bifurcations in which nonlinearity is important.Comment: pdfLaTeX, 9 figure
Adaptive Detection of Instabilities: An Experimental Feasibility Study
We present an example of the practical implementation of a protocol for
experimental bifurcation detection based on on-line identification and feedback
control ideas. The idea is to couple the experiment with an on-line
computer-assisted identification/feedback protocol so that the closed-loop
system will converge to the open-loop bifurcation points. We demonstrate the
applicability of this instability detection method by real-time,
computer-assisted detection of period doubling bifurcations of an electronic
circuit; the circuit implements an analog realization of the Roessler system.
The method succeeds in locating the bifurcation points even in the presence of
modest experimental uncertainties, noise and limited resolution. The results
presented here include bifurcation detection experiments that rely on
measurements of a single state variable and delay-based phase space
reconstruction, as well as an example of tracing entire segments of a
codimension-1 bifurcation boundary in two parameter space.Comment: 29 pages, Latex 2.09, 10 figures in encapsulated postscript format
(eps), need psfig macro to include them. Submitted to Physica
An energy-momentum map for the time-reversal symmetric 1:1 resonance with Z_2 X Z_2 symmetry
We present a general analysis of the bifurcation sequences of periodic orbits
in general position of a family of reversible 1:1 resonant Hamiltonian normal
forms invariant under symmetry. The rich structure of these
classical systems is investigated both with a singularity theory approach and
geometric methods. The geometric approach readily allows to find an
energy-momentum map describing the phase space structure of each member of the
family and a catastrophe map that captures its global features. Quadrature
formulas for the actions, periods and rotation number are also provided.Comment: 22 pages, 3 figures, 1 tabl
Anomalous scaling behavior in Takens-Bogdanov bifurcations
A general algorithm is presented for estimating the nonlinear instability
threshold, , for subcritical transitions in systems where the
linearized dynamics is significantly non-normal (i.e. subcritical bifurcations
of {\em Takens-Bogdanov} type). The -dimensional degenerate node is
presented as an example. The predictions are then compared to numerical studies
with excellent agreement.Comment: 6 page
- …