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In a one-parameter study of a noninvertible family of maps of the plane arising in the context of a numerical integration scheme, Lorenz studied a sequence of transitions from an attracting fixed point to " computational chaos. " As part of the transition sequence, he proposed the following as a possible scenario for the breakup of an invariant circle: the… (More)

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… (More)

A famous phenomenon in circle-maps and synchronisation problems leads to a two-parameter bifurcation diagram commonly referred to as the Arnol ′ d 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… (More)

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)… (More)

The study of resonances in systems such as periodically forced 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… (More)

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, A.M., Am. Nat. 121(3), 409-417 (1983)], but corrects problems due to singularities in that model. In addition, we investigate our model in more depth by varying the… (More)

- Bruce B. Peckham
- 1998

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 C k or C ∞) maps of the real plane R 2. We perform a numerical study of local bifurcations in the families of maps of the plane given by… (More)

This paper is primarily a numerical study of the xed-point bifurcation loci { saddle-node, period-doubling and Hopf bifurcations { present in the family: z ! f (C;A) (z; z) z + z 2 + C + Az where z is a complex dynamic (phase) variable, z its complex conjugate, and C and A are complex parameters. We treat the parameter C as a primary parameter and A as a… (More)

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… (More)

Predator-prey systems have been studied for many years in terms of energy flow or mass balance of the system. Loladze, Kuang, and Elser (2000) gave a model using stoichiometry which yielded different dynamics than the predator-prey model without stoichiometry. We generalize their model and give a discussion of the dynamics for several parameter values.