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- 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)

- Christos E Frouzakis, Ioannis G Kevrekidis, Bruce B Peckham
- 2008

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)

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)

- Richard P Mcgehee, Bruce B Peckham
- 2002

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)

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)

- Wendy Gruner Graves, Bruce B Peckham, John Pastor
- 2006

2 NOTE: This technical report is an expanded version of a paper by the same authors entitled " A Bifurcation Analysis of a Differential Equations Model for Mutualism. " The shorter version was accepted on August 9, 2005 and is expected to appear in the Journal of Mathematical Biology in 2006. Much of the work in both papers appeared first as part of a… (More)

- Bruce B Peckham
- 2000

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… (More)

We study doubly forced nonlinear planar oscillators: ˙ x = V(x) + α 1 W 1 (x, ω 1 t) + α 2 W 2 (x, ω 2 t), whose forcing frequencies have a fixed rational ratio: ω 1 = m n ω 2. After some changes of parameter, we arrive at the form we study: ˙ x = V(x) + α{(2 − γ)W 1 (x, m ω 0 β t) + (γ − 1)W 2 (x, n ω 0 β t)}. We assume ˙ x = V(x) has an attracting limit… (More)