Bifurcations of Random Differential Equations with Bounded Noise

Homburg, Ale Jan; Young, Todd; Gharaei, Masoumeh

doi:10.1007/978-1-4614-7385-5_9

Cited by 8 publications

(8 citation statements)

References 38 publications

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“…In [10] the authors showed that the boundary of an MFI set consists of solutions of the extremal vector fields defined by the bounded noise differential equations. Observe that for ε = 0, (8) reads θ̇ = 1 and

\dot{\overset{r}{true}} = 0

and its right hand varies continuously with ε.…”

Section: Random Perturbations Of a Planar Hopf-andronov Bifurcationmentioning

confidence: 99%

“…We adopt from [10] the following assumptions on (1) and its flow: H1 . The set Δ is a closed disk with smooth boundary.…”

Section: Introductionmentioning

confidence: 99%

“…In [10], under these noise conditions the authors provided a complete classification of bounded noise co-dimension one hard bifurcations in phase space dimensions 1 and 2. We call a set F ⊂ X forward invariant if

{normalΦ}_{λ}^{t} (F, 𝒰) \subset F

for all t ∈ ℝ + .…”

Section: Introductionmentioning

confidence: 99%

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The Hopf bifurcation with bounded noise

Botts

Homburg

Young

2012

Discrete &Amp; Continuous Dynamical Systems - A

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\dot{\overset{r}{true}} = 0

and its right hand varies continuously with ε.…”

Section: Random Perturbations Of a Planar Hopf-andronov Bifurcationmentioning

confidence: 99%

“…We adopt from [10] the following assumptions on (1) and its flow: H1 . The set Δ is a closed disk with smooth boundary.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Hopf bifurcation with bounded noise

Botts

Homburg

Young

2012

Discrete &Amp; Continuous Dynamical Systems - A

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“…Note that asymmetric division and variable rate models are both examples of perturbations by bounded noise (see [11, 12, 27]). It is known that random differential equations with bounded noise may undergo discontinuous bifurcations of their minimal forward invariant (MFI) sets.…”

Section: Introductionmentioning

confidence: 99%

“…It is known that random differential equations with bounded noise may undergo discontinuous bifurcations of their minimal forward invariant (MFI) sets. MFI sets are the bounded noise analogue of attractors [12]. Namely, if there is a topological change in the collection of MFI sets, this change must be discontinuous (with respect to the Hausdorff metric on sets) [14].…”

Section: Introductionmentioning

confidence: 99%

Noise-induced dispersion and breakup of clusters in cell cycle dynamics

Gong

Moses

Neiman

et al. 2014

Journal of Theoretical Biology

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We study the effects of random perturbations on collective dynamics of a large ensemble of interacting cells in a model of the cell division cycle. We consider a parameter region for which the unperturbed model possesses asymptotically stable two-cluster periodic solutions. Two biologically motivated forms of random perturbations are considered: bounded variations in growth rate and asymmetric division. We compare the effects of these two dispersive mechanisms with additive Gaussian white noise perturbations. We observe three distinct phases of the response to noise in the model. First, for weak noise there is a linear relationship between the applied noise strength and the dispersion of the clusters. Second, for moderate noise strengths the clusters begin to mix, i.e. individual cells move between clusters, yet the population distribution clearly continues to maintain a two-cluster structure. Third, for strong noise the clusters are destroyed and the population is characterized by a uniform distribution. The second and third phases are separated by an order - disorder phase transition that has the characteristics of a Hopf bifurcation. Furthermore, we show that for the cell cycle model studied, the effects of bounded random perturbations are virtually indistinguishable from those induced by additive Gaussian noise, after appropriate scaling of the variance of noise strength. We then use the model to predict the strength of coupling among the cells from experimental data. In particular, we show that coupling must be rather strong to account for the observed clustering of cells given experimentally estimated noise variance.

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Bounded noises as a natural tool to model extrinsic fluctuations in biomolecular networks

2014

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Chemical reaction networks (CRNs) are a versatile language for describing the dynamical behaviour of chemical kinetics, capable of modelling a variety of digital and analogue processes. While CRN designs for synchronous sequential logic circuits have been proposed and their implementation in DNA demonstrated, a physical realisation of these devices is difficult because of their reliance on a clock. Asynchronous sequential logic, on the other hand, does not require a clock, and instead relies on handshaking protocols to ensure the temporal ordering of different phases of the computation. This paper provides novel CRN designs for the construction of asynchronous logic, arithmetic and control flow elements based on a bi-molecular reaction motif with catalytic reactions and uniform reaction rates. We model and validate the designs for the deterministic and stochastic semantics using Microsoft's GEC tool and the probabilistic model checker PRISM, demonstrating their ability to emulate the function of asynchronous components under low molecular count.

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Bifurcations of Random Differential Equations with Bounded Noise

Cited by 8 publications

References 38 publications

The Hopf bifurcation with bounded noise

The Hopf bifurcation with bounded noise

Noise-induced dispersion and breakup of clusters in cell cycle dynamics

Bounded noises as a natural tool to model extrinsic fluctuations in biomolecular networks

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