218 research outputs found
Computation of normal form coefficients of cycle bifurcations of maps by algorithmic differentiation
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
Finding first foliation tangencies in the Lorenz system
This is the final version of the article. Available from SIAM via the DOI in this record.Classical studies of chaos in the well-known Lorenz system are based on reduction to the
one-dimensional Lorenz map, which captures the full behavior of the dynamics of the chaotic
Lorenz attractor. This reduction requires that the stable and unstable foliations on a particular
Poincar e section are transverse locally near the chaotic Lorenz attractor. We study when this
so-called foliation condition fails for the rst time and the classic Lorenz attractor becomes
a quasi-attractor. This transition is characterized by the creation of tangencies between the
stable and unstable foliations and the appearance of hooked horseshoes in the Poincar e return
map. We consider how the three-dimensional phase space is organized by the global invariant
manifolds of saddle equilibria and saddle periodic orbits | before and after the loss of the
foliation condition. We compute these global objects as families of orbit segments, which are
found by setting up a suitable two-point boundary value problem (BVP). We then formulate a
multi-segment BVP to nd the rst tangency between the stable foliation and the intersection
curves in the Poincar e section of the two-dimensional unstable manifold of a periodic orbit.
It is a distinct advantage of our BVP set-up that we are able to detect and readily continue
the locus of rst foliation tangency in any plane of two parameters as part of the overall
bifurcation diagram. Our computations show that the region of existence of the classic Lorenz
attractor is bounded in each parameter plane. It forms a slanted (unbounded) cone in the
three-parameter space with a curve of terminal-point or T-point bifurcations on the locus of
rst foliation tangency; we identify the tip of this cone as a codimension-three T-point-Hopf
bifurcation point, where the curve of T-point bifurcations meets a surface of Hopf bifurcation.
Moreover, we are able to nd other rst foliation tangencies for larger values of the parameters
that are associated with additional T-point bifurcations: each tangency adds an extra twist to
the central region of the quasi-attractor
Numerical bifurcation analysis of homoclinic orbits embedded in one-dimensional manifolds of maps
We describe new methods for initializing the computation of homoclinic orbits for maps in a state space with arbitrary dimension and for detecting their bifurcations. The initialization methods build on known and improved methods for computing one-dimensional stable and unstable manifolds. The methods are implemented in MatContM, a freely available toolbox in Matlab for numerical analysis of bifurcations of fixed points, periodic orbits, and connecting orbits of smooth nonlinear maps. The bifurcation analysis of homoclinic connections under variation of one parameter is based on continuation methods and allows us to detect all known codimension 1 and 2 bifurcations in three-dimensional (3D) maps, including tangencies and generalized tangencies. MatContM provides a graphical user interface, enabling interactive control for all computations. As the prime new feature, we discuss an algorithm for initializing connecting orbits in the important special case where either the stable or unstable manifold is one-dimensional, allowing us to compute all homoclinic orbits to saddle points in 3D maps. We illustrate this algorithm in the study of the adaptive control map, a 3D map introduced in 1991 by Frouzakis, Adomaitis, and Kevrekidis, to obtain a rather complete bifurcation diagram of the resonance horn in a 1:5 Neimark-Sacker bifurcation point, revealing new features
Centre-Manifold Reduction of Bifurcating Flows
In this paper we describe a general and systematic approach to the centre-manifold reduction and normal form computation of flows undergoing complicated bifurcations. The proposed algorithm is based on the theoretical work of Coullet & Spiegel (SIAM J. Appl. Maths, vol. 43(4), 1983, pp. 776821) and can be used to approximate centre manifolds of arbitrary dimension for large-scale dynamical systems depending on a scalar parameter. Compared with the classical multiple-scale technique frequently employed in hydrodynamic stability, the proposed method can be coded in a rather general way without any need to resort to the introduction and tuning of additional time scales. The method is applied to the dynamical system described by the incompressible NavierStokes equations showing that high-order, weakly nonlinear models of bifurcating flows can be derived automatically, even for multiple codimension bifurcations. We first validate the method on the primary Hopf bifurcation of the flow past a circular cylinderand after we illustrate its application to a codimension-two bifurcation arising in the flow past two side-by-side circular cylinder
Discreet dynamical population models : higher dimensional pioneer-climax models
There are many population models in the literature for both continuous and discrete systems. This investigation begins with a general discrete model that subsumes almost all of the discrete population models currently in use. Some results related to the existence of fixed points are proved. Before launching into a mathematical analysis of the primary discrete dynamical model investigated in this dissertation, the basic elements of the model - pioneer and climax species - are described and discussed from an ecological as well as a dynamical systems perspective. An attempt is made to explain why the chosen hierarchical form of the model to explain why the chosen hierarchical form of the model can be expected to follow the real-world evolution of pioneer-climax species. Following the discussion of the discrete dynamical model from the applications viewpoint, an extensive dynamical systems investigation is conducted using analytical and simulation tools. Fixed and periodic points are found and their stability is investigated. Sufficient conditions for the existence and stability of n -cycles are proved and illustrated for several values of n. For eample, the existence of a stable, attracting 3-cycle is proved for a certain range of parameters for an all-pioneer model. It is also observed that the hierarchical system has a predisposition to period-doubling behavior.
Bifurcations of the hierarchical model are studied in considerable depth. It is proved, for example, that the model cannot exhibit a Hopf Bifurcation. However, in a series of theorems, it is shown that the system can exhibit a very rich array of flip (period-doubling) bifurcations, which are of codimension one, two or three. A key to proving this result is that the hierarchical nature of the system makes it essentially equivalent to a sequence of one-dimensional systems when it comes to several properties of the dynamics. This hierarchical principle is then used to prove chaos for the system in the limit of a period-doubling cascade, and also in terms of shift map behavior on an invariant two-component Cantor set for systems containing a climax component. Bifurcation diagrams and Lyapunov diagrams are computed to further illustrate the chaotic dynamics. Finally, the concept of a 3-dimensional horseshoe type map is also used to prove the existence of chaos in an approximate graphical manner
Computation of normal form coefficients of cycle bifurcations of maps by algorithmic differentiation
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