Research Abstracts

A two-parameter family of maps of the plane is generated by varying the forcing frequency omega and amplitude alpha of a planar oscillator with unforced frequency omega_0. For small forcing amplitude, resonance horns open into the first quadrant of the parameter plane from every point on the alpha=0 axis where omega_0/omega=p/q is rational. Inside the "p/q resonance horn" the corresponding phase portraits contain at least one period-q orbit, with rotation number p/q. In this thesis, we investigate the continuation of resonance horns which terminate at some higher forcing amplitude, where the small amplitude theory is no longer applicable.

By varying the forcing frequency and amplitude of a periodically forced planar oscillator, we can obtain a rich variety of responses. Whenever the resonance regions that are known to exist for small amplitudes of forcing terminate, we show that a fixed-point Hopf bifurcation must be involved. The main tool, whose properties we discuss in detail, is a self rotation number for orbits in the plane. We illustrate our theorems with a numerical model.

A family of local diffeomorphisms of n-dimensional Euclidean space can under go a period-doubling (flip) bifurcation as an eigenvalue of a fixed point passes through -1. This bifurcation is either supercritical or subcritical, depending on the sign of a coefficient determined by higher-order terms. If this coefficient is zero, the resulting bifurcation is "degenerate." The period doubling bifurcation with a single higher-order degeneracy is treated, as well as the more general degenerate period-doubling bifurcation where a fixed point has a -1 eigenvalue and any number of higher-order degeneracies. The main procedure is a Lyapunov-Schmidt reduction: period-2 orbits are shown to be in one-to-one correspondence with roots of the reduced "bifurcation function," which has Z_2 symmetry. Ilustrative examples of the occurrence of the singly degenerate period doubling in the context of periodically forced oscillators are also presented.

Maps of the plane can be generated by sampling the flow of periodically forced planar oscillators at the period of forcing. Numerical studies of the bifurcations present in a two-parameter family of such maps, obtained by varying the forcing frequency and amplitude, have revealed a rich structure. Resonance regions in the parameter space, corresponding to maps having periodic orbits of a certain period, are always a part of the bifurcation picture. Much insight has been gained into the bifurcation structure by viewing resonan ce regions as projections to the two-dimensional parameter space of ``resonance surfaces'' from the four-dimensional phase cross parameter space. Here we continue the study of these surfaces by presenting an algorithm to determine their global topology from the bifurcation diagram in the parameter plane and knowledge of generic codimension-one and -two bifurcations.

The set of Hopf bifurcations for a two-parameter family of maps is typically a curve in the parameter plane. The side of the curve on which the invariant circle exists is further divided by horn-shaped resonance regions, each region corresponding to maps having a periodic orbit of a certain period. With the presence of a parametric degeneracy, the resonance regions sometimes take the form of closed ``bananas", instead of open-ended horns. We investigate this local codimension-two bifurcation, emphasizing resonance regions as projections to the parameter plane of surfaces in phase $\times$ parameter space. We present scenarios where the degeneracy occurs ``naturally'', and illustrate them through an adaptive control application. We also discuss more global implications of the local study.

Periodically forced planar oscillators are often studied by varying the two parameters of forcing amplitude and forcing frequency. For low forcing amplitudes, the study of the essential oscillator dynamics can be reduced to the study of families of circle maps. The primary features of the resulting parameter plane bifurcation diagrams are ``(Arnold) resonance horns'' emanating from zero forcing amplitude. Each horn is characterized by the existence of a periodic orbit with a certain period and rotation number. In this paper we investigate divisions of these horns into subregions -- different subregions corresponding to maps having different numbers of periodic orbits. The existence of subregions having more than the ``usual'' one pair of attracting and repelling periodic orbits implies the existence of ``extra folds'' in the corresponding surface of periodic points in the cartesian product of the phase and parameter spaces. The existence of more than one attracting and one repelling periodic orbit is shown to be generic. For some of the families we create, the resulting parameter plane bifurcation pictures appear in shapes we call ``Arnold flames.'' Such an example is pictured here. The magenta curve of saddle-node bifurcations on the fixed-point surface projects to a flame in the parameter plane. Results apply both to circle maps and forced oscillator maps.

We investigate the two-real-parameter bifurcation diagram in the (complex) C plane for the family: z -> z^2 + C + A*zbar. A is treated as a real auxiliary parameter. For A near zero, the result is a distorted Mandlebrot set which has grown small "Arnold tongues."

Periodically forced planar oscillators are typically studied by varying the two parameters of forcing amplitude and forcing frequency. Such differential equations can be reduced via stroboscopic sampling to a two-parameter family of diffeomorphisms of the plane. A bifurcation analysis of this family almost always includes a study of the birth and death of periodic orbits. For low forcing amplitudes, this leads to a now-classic picture of Arnold resonance tongues. Studying these resonance tongues for higher forcing amplitudes requires numerical continuation. Previous work has revealed the usefulness of considering these tongues as projections of surfaces of periodic points from the cartesian product of the phase and parameter planes to the parameter plane. Many surfaces were displayed and described in [MP 1994], but their parametrization and computation was not discussed. In this paper, we do discuss their parametrization and computation. Especially useful are global parametrizations which allow automatic computation of the surfaces. We argue that parametrization by "f_p(x) - x" is both more likely to be a global parametrization and more ``dynamically natural" than two more obvious parametrizations. As a side benefit, f_p(x) - x parametrization leads to a computable way of establishing the nonorientablility of period-two surfaces.

We investigate the two-real-parameter bifurcation diagram in the (complex) C plane for the family: z -> z^2 + C + A*zbar. A is treated as a complex auxiliary parameter.

We study doubly forced nonlinear planar oscillators: $$\dot{\bfx} = \bfV(\bfx) + \alpha_1 \boldW_1(\bfx,\omega_1 t) + \alpha_2 \boldW_2(\bfx,\omega_2 t),$$ whose forcing frequencies have a fixed rational ratio: $\omega_1={m \over n}\omega_2$. After some changes of parameter, we arrive at the form we study: $$\dot{\bfx} = \bfV(\bfx) + \alpha \{( 2-\gamma)\boldW_1(\bfx,{m \:\omega_0\over \beta} t) + (\gamma-1) \boldW_2(\bfx,{n \:\omega_0\over \beta} t)\}.$$ We assume $\dot{\bfx} = \bfV(\bfx)$ has an attracting limit cycle --- the unforced planar oscillator --- with frequency $\omega_0$, and the two forcing functions $\boldW_1$ and $\boldW_2$ are period one in their second variables. We consider two parameters as primary: $\beta$, an appropriate multiple of the forcing period, and $\alpha$, the forcing amplitude. The relative forcing amplitude $\gamma \in [1,2]$ is treated as an auxiliary parameter. The dynamics are studied by considering the stroboscopic maps induced by sampling the solutions of the differential equations at time intervals equal to the period of forcing, $T={\beta \over \omega_0}$. For any fixed $\gamma$, these oscillators have a standard form of a periodically forced oscillator, and thus exhibit the Arnold resonance tongues in the primary parameter plane. The special forms at $\gamma=1$ and $\gamma=2$ can introduce certain symmetries into the problem. One effect of these symmetries is to provide a relatively natural example of oscillators with multiple attractors. Such oscillators typically have interesting bifurcation features {\it within} corresponding resonance regions --- features we call ``Arnold flames" because of their flamelike appearance in the corresponding bifurcation diagrams. By changing the auxiliary parameter $\gamma$ we ``melt'' one singly forced oscillator bifurcation diagram into another, and in the process we control certain of these ``intraresonance region" bifurcation features.

Peatlands store one-third of the world's soil carbon and nitrogen pools and are located almost entirely in northern regions where climatic warming is expected to be greatest in the coming decades. We construct and analyze a model of peatlands which sheds some light on this ecosystem. The model is a set of six coupled differential equations that define the flow of nutrients from moss and vascular plants to their litters, then to peat, and finally to an inorganic nutrient resource pool. Analytic and numerical solutions of the model mimic many dynamics of peatland development. In particular, the model allows for the possibility of a stable moss monoculture or of a stable coexistence equilibrium. Which one is stable depends on system parameters. A transcritical bifurcation forms the boundary between the two cases.

In a one-parameter study of a noninvertible map of the plane arising in the context of a numerical integration scheme, Lorenz proposed the following as a possible scenario for the breakup of an invariant circle: the invariant circle develops regions of increasingly sharper curvature until at a critical parameter value it develops cusps; beyond this parameter value, the invariant circle fails to persist, and the system exhibits chaotic behavior on an invariant set with loops [Lorenz, 1989]. We investigate this problem in more detail and show that the invariant circle is actually destroyed in a global bifurcation before it has a chance to develop cusps. Instead, the global unstable manifolds of saddle-type periodic points are the objects which develop cusps and subsequently "loops" or "antennae." The one-parameter study is better understood when embedded in the full two-parameter space and viewed in the context of the two-parameter Arnold horn structure. Certain elements of the interplay of noninvertibility with this structure, the associated invariant circles, periodic points and global bifurcations are examined.

We develop from basic principles a two-species differential equations model which exhibits mutualistic population interactions. The model is similar in spirit to a commonly cited model (Dean 1983), but corrects problems due to singularities in that model. In addition, we investigate our model in more depth by varying the intrinsic growth rate for each of the species and analyzing the resulting bifurcations in system behavior. We are especially interested in transitions between facultative and obligate mutualism. The model reduces to the familiar Lotka-Volterra model locally, but is more realistic for large populations in the case where mutualist interaction is strong. In particular, our model supports population thresholds necessary for survival in certain cases, but does this without allowing unbounded population growth. Experimental implications are discussed for a lichen population

A famous phenomenon in circle-maps and synchronisation problems leads to a two-parameter bifurcation diagram commonly referred to as the Arnold tongue scenario. One considers a perturbation of a rigid rotation of a circle, or a system of coupled oscillators. In both cases we have two natural parameters, the coupling strength and a detuning parameter that controls the rotation number/frequency ratio. The typical parameter plane of such systems has Arnold tongues with their tips on the decoupling line, opening up into the region where coupling is enabled, and in between these Arnold tongues, quasiperiodic arcs. In this paper we present unified algorithms for computing both Arnold tongues and quasiperiodic arcs for both maps and ODEs. The algorithms generalise and improve on the standard methods for computing these objects. We illustrate our methods by numerically investigating the Arnold tongue scenario for representative examples, including the well-known Arnold circle map family, a periodically forced oscillator caricature, and a system of coupled Van der Pol oscillators.

In many applications of practical interest, for example, in control theory, economics, electronics and neural networks, the dynamics of the system under consideration can be modelled by an endomorphism, which is a discrete smooth map that does not have a uniquely defined inverse; one also speaks simply of a noninvertible map. In contrast to the better known case of a dynamical system given by a planar diffeomorphism, many questions concerning the possible dynamics and bifurcations of planar endomorphisms remain open.

In this paper we make a contribution to the bifurcation theory of planar endomorphisms. Namely we present the unfoldings of a codimension-two bifurcation, which we call the cusp-cusp bifurcation, that occurs generically in families of endomorphisms of the plane. The cusp-cusp bifurcation acts as an organising center that involves the relevant codimension-one bifurcations. The central singularity involves an interaction of two different types of cusps. Firstly, an endomorphism typically folds the phase space along curves $J_0$ where the Jacobian of the map is zero. The image $J_1$ of $J_0$ may contain a cusp point, which persists under perturbation; the literature also speaks of a map of type $Z_3 < Z_1$. The second type of cusp occurs when a forward invariant curve $W$, such as a segment of an unstable manifold, crosses $J_0$ in a direction tangent to the zero eigenvector. Then the image of $W$ will typically contain a cusp. This situation is of codimension one and generically leads to a loop in the unfolding. The central singularity that defines the cusp-cusp bifurcation is, hence, defined by a tangency of an invariant curve $W$ with $J_0$ at the pre-image of the cusp point on $J_1$.

We study the bifurcations in the images of $J_0$ and the curve $W$ in a neighborhood of the parameter space of the organizing center --- where both images have a cusp at the same point in the phase space. To this end, we define a suitable notion of equivalence that distinguishes between the different possible local phase portraits of the invariant curve relative to the cusp on $J_1$. Our approach makes use of local singularity theory to derive and analyze completely a normal form of the cusp-cusp bifurcation. In total we find eight different two-parameter unfoldings of the central singularity. We illustrate how our results can be applied by showing the existence of a cusp-cusp bifurcation point in an adaptive control system. We are able to identify the associated two-parameter unfolding for this example and provide all different phase portraits.