Research Abstracts

Bruce B. Peckham


  • Go to the Math/Stats Dept. Home Page
  • Go to Bruce Peckham's Home Page
    1. "The Closing of Resonance Horns for Periodically Forced Oscillators," Thesis, University of Minnesota, 1988.
    2. 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continuation of resonance horns which terminate at some higher forcing amplitude, where the small amplitude theory is no longer applicable.


    3. "The Necessity of the Hopf Bifurcation in Families of Periodically Forced Oscillators," Nonlinearity (3) , pp. 261-280, 1990.
    4. 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.


    5. "Period Doubling with Higher Order Degeneracies ," with I. G. Kevrekidis, SIAM Journal of Mathematical Analysis, Vol. 22, No. 6, pp. 1552-1574, 1991.
    6. 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.


    7. "Resonance Surfaces for Forced Oscillators," with R. P. McGehee, Geometry Center Research Report GCG70, Feb. 1995, and Experimental Mathematics, Vol.3, No. 3, pp. 221-244, 1994.
      Download from Geometry Center. Look for GCG70.

      The study of resonances in systems such as periodically forces oscillators has traditionally focused on understanding the regions in the parameter plane where these resonances occur. Resonance regions can also be viewed as projections to the parameter plane of resonance surfaces in the four-dimensional Cartesian product of the state space with the parameter space. This paper reports on a computer study of resonance surfaces for a particular family and illustrates some advantages of viewing resonance regions in this light.
      Picture Resonance Surfaces




      The picture at right is one which represents several "resonance surfaces," or sets of periodic points of a given period as they appear in a (projection to three dimensions of a) four-dimensional "phase cross parameter" space. From right to left are surfaces of period 2 (rotation number 1/2), period 5 (2/5), period 3 (1/3), period 4 (1/4), and period 5 (1/5). They are "threaded" together at the top by the blue Hopf bifurcation curve. The green triangular curve is a curve of fixed-point saddle-node bifurcations.











    8. "Bananas and banana splits: A parametric degeneracy in the Hopf bifurcation for maps," with C. E. Frouzakis and I. G. Kevrekidis, SIAM J Math Analysis, Vol. 26, No. 1, pp. 190-217, 1995.
    9. 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.


    10. "Determining the global topology of resonance surfaces for periodically forced oscillator families" with R. P. McGehee 1992 Normal Forms and Homoclinic Chaos Conference Proceedings, Fields Institute Communications, Vol. 4, American Mathematical Society, pp. 233-251, 1995.
    11. 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.


    12. "Arnold Flames and Resonance Surface Folds" with R. P. McGehee, Geometry Center Research Report GCG84, July 1995 (Download from Geometry Center; look for GCG84.) and International Journal of Bifurcation and Chaos, Vol. 6, No. 2 (1996) 315-336.
    13. Arnold Flame 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.


    14. "Real Perturbation of Complex Analytic Maps: Points to Regions" International Journal of Bifurcation and Chaos, Vol. 8, No. 1 (1998) 73-93.
    15. This study provides some connections between bifurcations of one-complex-parameter complex analytic families of maps of the complex plane C and bifurcations of more general two-real-parameter families of real analytic (or Ck or C-infinity) maps of the real plane R2. We perform a numerical study of local bifurcations in the families of maps of the plane given by z --> F(C,\alpha)(z,zbar) = z2 + C + alpha zbar where z is a complex dynamic (phase) variable, zbar its complex conjugate, C is a complex parameter, and alpha is a real parameter. For alpha=0, the resulting family is the familiar complex quadratic family. For alpha nonzero, the map fails to be complex analytic, but is still analytic (quadratic) when viewed as a map of R2. We treat alpha in this family as a perturbing parameter and ask how the two-parameter bifurcation diagrams in the C parameter plane change as the perturbing parameter alpha is varied. The most striking phenomenon that appears as alpha is varied is that bifurcation points in the C plane for the quadratic family (alpha=0) evolve into fascinating bifurcation regions in the C plane for nonzero alpha. Such points are the cusp of the main cardioid of the Mandelbrot set and contact points between ``bulbs" of the Mandelbrot set. Arnold resonance tongues are part of the evolved scenario. We also provide sufficient conditions for more general perturbations of complex analytic maps of the plane of the form: z --> F(C,alpha)(z,zbar) = fC(z) + alpha galpha(z,zbar) to have bifurcation points for alpha=0 which evolve into nontrivial bifurcation regions as alpha grows from zero.


    16. "Global parametrization and computation of resonance surfaces for periodically forced oscillators" Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems, E. Doedel and L. Tuckerman, Editors, IMA Volumes in Mathematics and its Applications, Vol. 119, Springer-Verlag (2000) 385-406.
    17. 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 "fp(x) - x" is both more likely to be a global parametrization and more ``dynamically natural" than two more obvious parametrizations. As a side benefit, fp(x) - x parametrization leads to a computable way of establishing the nonorientablility of period-two surfaces.


    18. "Real Continuation from the Complex Quadratic Family: Fixed Point Bifurcation Sets," with James Montaldi, International Journal of Bifurcation and Chaos, Vol. 10, No. 2, (2000) 391-414.
    19. This paper is primarily a numerical study of the fixed-point bifurcation loci -- saddle-node, period-doubling and Hopf bifurcations -- present in the family: z --> z + z^2 + C + A*zbar where z is a complex dynamic (phase) variable, zbar its complex conjugate, and C and A are complex parameters. We treat the parameter C as a primary parameter and A as a secondary parameter, asking how the bifurcation loci projected to the $C$ plane change as the auxiliary parameter A is varied. For A=0, the resulting two-real-parameter family is a familiar one-complex-parameter quadratic family, and the local fixed-point bifurcation locus is the main cardioid of the Mandlebrot set. For A nonzero, the resulting two-real-parameter families are not complex analytic, but are still analytic (quadratic) when viewed as a map of R2. Saddle-node and period-doubling loci evolve from points on the main cardioid for A=0 into closed curves for A nonzero. As A is varied further from 0 in the complex plane, the three sets interact in a variety of interesting ways. More generally, we discuss bifurcations of families of maps with some parameters designated as primary and the rest as auxiliary. The auxiliary parameter space is then divided into equivalence classes with respect to a specified set of bifurcation loci. This equivalence is defined by the existence of a diffeomorphism of corresponding primary parameter spaces which preserves the specified set of specified bifurcation loci. In our study there is a surprising amount of complexity added by specifying the three fixed-point bifurcation loci together, rather than one at a time. We also provide a preliminary classification of the types of codimension-one bifurcations one should expect in general studies of families of two-parameter families of maps of the plane. Comments on numerical continuation techniques are provided as well.


    20. "Lighting Arnold Flames: Resonance in Doubly Forced Periodic Oscillators," with I. G. Kevrekidis, Nonlinearity , Jan. 2001.
    21. 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.


    22. "Plant community dynamics, nutrient cycling and alternate stable equilibria in peatlands," with John Pastor, Scott Bridgham, Jake Weltzin, and Jiquan Chen, The American Naturalist , Feb. 2001.
    23. 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.


    24. "A route to computational chaos revisited: noninvertibility and the breakup of an invariant circle," with C. E. Frouzakis and I. G. Kevrekidis, Physica D , June 2002.
    25. 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.


    26. "A Bifurcation Analysis of a Differential Equations Model for Mutualism," with Wendy Graves and John Pastor, Journal of Mathematical Biology , Vol. 68, No. 8, 2006, 1837-1850.
    27. 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


    28. "Computing Arnold Tongue Scenarios," with Frank Schilder, Journal of Computational Physics 220, 932-951, 2007.
    29. 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.


    30. "Unfolding the cusp-cusp bifurcation of planar endomorphisms," with Bernd Krauskopf and Hinke Osinga, SIAM Journal on Dynamical Systems (SIADS), 2007
    31. 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 J0 where the Jacobian of the map is zero. The image J1 of J0 may contain a cusp point, which persists under perturbation; the literature also speaks of a map of type Z3 < Z1. The second type of cusp occurs when a forward invariant curve W, such as a segment of an unstable manifold, crosses J0 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 J0 at the pre-image of the cusp point on J1.

      We study the bifurcations in the images of J0 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 J1. 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.


    32. "A Consumer-Producer model with Stoichiometric Elimination Mechanisms," with Laura Zimmermamm, John Pastor, and Harlan Stech, UMD Technical Report , 2008.
    33. Producer-consumer (predator-prey) systems have been studied for many years in terms of energy flow or mass balance of the system. In recent years related models have been adjusted to take into account not only food quantity, but also food quality. In other words, the nutrient content, or equivalently, the stoichiometric ratio of nutrient to biomass, as well as the biomass, is of interest. In this paper we start from a version of the Rosenzweig-MacArthur model of a producer-consumer system and modify it by introducing stoichiometry. The model considered here includes a sediment class in addition to the producer and consumer classes. The model is open for both carbon and the nutrient. It sets ``target'' structural ratios for both the producer and consumer, who eliminate carbon or nutrient, whichever appears in excess. This introduction of stoichiometry allows for different bifurcation sequences and corresponding dynamics than those that appear in the Rosenzweig-MacArthur model. The stoichiometric mechanisms we use are also in contrast to those presented in the Loladze, Kuang, Elser model which has become a standard starting point for stoichiometric models. Especially notable is a parameter range where we observe the coexistence of an attracting equilibrium and an attracting periodic limit cycle.


    34. Two related papers:
      • "A Stoichiometric Model of two Producers and one Consumer," with Laurence Lin, John Pastor, and Harlan Stech, UMD Technical Report , 2008. (Extended version of JBD paper listed below.)
      • In this paper, we consider a stoichiometric population model of two producers and one consumer. It is a generalization of the Rosenzweig-MacArthur population growth model, which is a one-producer, one-consumer population model without stoichiometry. The generalization involves two steps: 1) adding a second producer which competes with the first, and 2) introducing stoichiometry into the system. Both generalizations introduce additional equilibria and bifurcations to the Rosenzweig-MacArthur model without stoichiometry.

        The primary focus of this paper is to study the equilibria and bifurcations of the two-producer, one-consumer model with stoichiometry. The nutrient cycle in this model is closed. The primary parameters are the growth rates of both producers, and the secondary parameter is the total nutrient in the system. Depending on the parameters, the possible equilibria are: no-life, one-producer, coexistence of both producers, the consumer coexisting with either producer, and the consumer coexisting with both producers. Limit cycles exist in the latter three coexistence combinations. Bifurcation diagrams, along with corresponding representative time series, summarize the results presented in this paper.


      • ``Enrichment in a stoichiometric model of two producers and one consumer," with Laurence Lin, Harlan Stech and John Pastor, Journal of Biological Dynamics, 2012 (online January 2011).
      • We consider a stoichiometric population model of two producers and one consumer. Stoichiometry can be thought of as the tracking of food quality in addition to food quantity. Our model assumes a reduced rate of conversion of biomass from producer to consumer when food quality is low. The model is open for carbon but closed for nutrient. The introduction of the second producer, which competes with the first, leads to new equilibria, new limit cycles, and new bifurcations. The focus of this paper is on the bifurcations which are the result of enrichment. The primary parameters we vary are the growth rates of both producers. Secondary variable parameters are the total nutrients in the system, and the producer nutrient uptake rates. The possible equilibria are: no-life, one-producer, coexistence of both producers, the consumer coexisting with either producer, and the consumer coexisting with both producers. We observe limit cycles in the latter three coexistence combinations. Bifurcation diagrams along with corresponding representative time series summarize the behaviours observed for this model.


    35. ``Enrichment Effects in a Simple Stoichiometric Producer-Consumer Population Growth Model,'' Harlan Stech, Bruce Peckham and John Pastor, Communications in Applied Analysis, 2012.
    36. This paper presents the derivation and partial analysis of a general producer–consumer model. The model is stoichiometric in that it includes the growth constraints imposed by species-specific biomass carbon to nutrient ratios. The model unifies the approaches of other studies in recent years, and is calibrated from an extensive review of the algae–Daphnia literature. Numerical simulations and bifurcation analysis are used to examine the impact ∧ energy enrichment under nutrient and stoichiometric constraints. Our results suggest that the variety of system responses previously cited for related models can be attributed to the size of the total system nutrient pool, which is here assumed fixed. New, more complicated sequences, such as multiple homoclinic bifurcations, are demonstrated as well. The mechanistic basis of the model permits us to show the robustness of the system’s dynamics subject to alternate approaches to modeling producer and consumer biomass production.


    37. ``Enrichment Effects in a Simple Stoichiometric Producer-Consumer Population Growth Model,'' Harlan Stech, Bruce Peckham and John Pastor, Communications in Applied Analysis, 2012.
    38. A rudimentary predator-prey model is considered that is stoichometric, in that the nutrient content of the producer species affects the ability of the consumer to produce biomass. We show that the model supports biologically important dynamical properties that differ from the corresponding non-stoichiometric model. Specifically, under the assumption of Holling II-type functional response, for all sufficiently high system nutrient levels energy enrichment of the producer induces the loss of stability for the consumer-free (producer ``monoculture'') system and the transcritical creation of a non-trivial coexistence equilibrium. Under further energy enrichment, this equilibrium undergoes a loss of stability (generically via Hopf bifurcation.) The model then supports a nontrivial periodic coexistence solution. In contrast to the non-stoichiometric case, here further energy enrichment induces restabilization of the monoculture equilibrium. Moreover, under sufficiently high energy enrichment, the system supports no nontrivial periodic solutions. The details of the bifurcation structure are computed for a simple case. Our results suggest that the energy-induced loss of periodic coexistence state can be attributed to the dilution of a consumer-limiting nutrient when the producer population is large, resulting in a carbon-rich / nutrient-poor food source that cannot sustain the consumer's nutrient needs.


    39. ``Quasi-equilibrium Reduction in a General Class of Stoichiometric Producer-Consumer Models,'' Harlan Stech, Bruce Peckham and John Pastor, Journal of Biological Dynamics, 2012
    40. This paper compares a general closed nutrient, stoichiometric producer-consumer model to a two-dimensional ``quasi-equilibrium'' approximation. We demonstrate that the quasi-equilibrium system can be rigorously analyzed, resulting in nullcline-based criteria for the local stability of system equilibria and for the non-existence of periodic orbits. These results are applied to a study of the dependence of the reduced system on nutrient and energy enrichment. When energy and nutrient enrichment are considered together, the associated bifurcation structures of the two models are seen to share the same essential qualitative characteristices. In contrast, numerical simulations of the three-dimensional parent model show highly complex domains of persistence and extinction that by Pointare-Bendixon theory are not possible for the two-dimensional reduction. This complexity demonstrates a major differnce between the two models, and suggests potential challenges in the use of either model for predicting the long-term behavior of real-world systems at specific nutrient and energy levels.


    41. ``Nonholomorphic singular continuation: a case with radial symmetry,'' Brett D Bozyk and Bruce B Peckham, International Journal of Bifurcation and Chaos. Submitted October 2012.
    42. This paper is a study of special families of rational maps of the real plane of the form: z -> zn + c + beta/zbard, where the dynamic variable z is complex and the complex plane is identified with R2. The parameters c and beta are complex; n and d are positive integers. For beta small, this family can be considered a non-holomorphic singular perturbation of the holomorphic family z -> n + c, although we will consider large values of beta as well. We focus on the special case where n=d and c=0 because the radial component of these maps in polar coordinates decouples from the angular component. This reduces a significant part of the analysis to the study of a family of one-dimensional unimodal maps. For each fixed n, the \beta parameter plane separates into three major regions, corresponding to maps which have one of the following behaviors: (i) all orbits go off to infinity, (ii) only an annulus of points stays bounded, and (iii) only a Cantor set of circles stays bounded. In cases (ii) and (iii), there is a transitive invariant set; this set is an attractor in case (ii). The dynamics of z -> zn+beta/zbarn is compared and contrasted to the (holomorphic) singularly perturbed maps: z -> zn + lambda/zn, studied by Devaney and coauthors over the last decade. Additional observations, mostly numerical, are made about the cases where c is not zero and n does not equal d.



    This page (http://www.d.umn.edu/~bpeckham/www) is maintained by Bruce Peckham (bpeckham@d.umn.edu) and was last modified on Monday, 28-Jan-2013 04:28:51 CST.