22 research outputs found
Resonance bifurcations of robust heteroclinic networks
Robust heteroclinic cycles are known to change stability in resonance
bifurcations, which occur when an algebraic condition on the eigenvalues of the
system is satisfied and which typically result in the creation or destruction
of a long-period periodic orbit. Resonance bifurcations for heteroclinic
networks are more complicated because different subcycles in the network can
undergo resonance at different parameter values, but have, until now, not been
systematically studied. In this article we present the first investigation of
resonance bifurcations in heteroclinic networks. Specifically, we study two
heteroclinic networks in and consider the dynamics that occurs as
various subcycles in each network change stability. The two cases are
distinguished by whether or not one of the equilibria in the network has real
or complex contracting eigenvalues. We construct two-dimensional Poincare
return maps and use these to investigate the dynamics of trajectories near the
network. At least one equilibrium solution in each network has a
two-dimensional unstable manifold, and we use the technique developed in [18]
to keep track of all trajectories within these manifolds. In the case with real
eigenvalues, we show that the asymptotically stable network loses stability
first when one of two distinguished cycles in the network goes through
resonance and two or six periodic orbits appear. In the complex case, we show
that an infinite number of stable and unstable periodic orbits are created at
resonance, and these may coexist with a chaotic attractor. There is a further
resonance, for which the eigenvalue combination is a property of the entire
network, after which the periodic orbits which originated from the individual
resonances may interact. We illustrate some of our results with a numerical
example.Comment: 46 pages, 20 figures. Supplementary material (two animated gifs) can
be found on
http://www.maths.leeds.ac.uk/~alastair/papers/KPR_res_net_abs.htm
Stability of cycling behaviour near a heteroclinic network model of Rock-Paper-Scissors-Lizard-Spock
The well-known game of Rock--Paper--Scissors can be used as a simple model of
competition between three species. When modelled in continuous time using
differential equations, the resulting system contains a heteroclinic cycle
between the three equilibrium solutions representing the existence of only a
single species. The game can be extended in a symmetric fashion by the addition
of two further strategies (`Lizard' and `Spock'): now each strategy is dominant
over two of the remaining four strategies, and is dominated by the remaining
two. The differential equation model contains a set of coupled heteroclinic
cycles forming a heteroclinic network. In this paper we carefully consider the
dynamics near this heteroclinic network. We are able to identify regions of
parameter space in which arbitrarily long periodic sequences of visits are made
to the neighbourhoods of the equilibria, which form a complicated pattern in
parameter space.Comment: Submitted to Nonlinearit
Travelling waves and heteroclinic networks in models of spatially-extended cyclic competition
Dynamical systems containing heteroclinic cycles and networks can be invoked
as models of intransitive competition between three or more species. When
populations are assumed to be well-mixed, a system of ordinary differential
equations (ODEs) describes the interaction model. Spatially extending these
equations with diffusion terms creates a system of partial differential
equations which captures both the spatial distribution and mobility of species.
In one spatial dimension, travelling wave solutions can be observed, which
correspond to periodic orbits in ODEs that describe the system in a
steady-state travelling frame of reference. These new ODEs also contain a
heteroclinic structure. For three species in cyclic competition, the topology
of the heteroclinic cycle in the well-mixed model is preserved in the
steady-state travelling frame of reference. We demonstrate that with four
species, the heteroclinic cycle which exists in the well-mixed system becomes a
heteroclinic network in the travelling frame of reference, with additional
heteroclinic orbits connecting equilibria not connected in the original cycle.
We find new types of travelling waves which are created in symmetry-breaking
bifurcations and destroyed in an orbit flip bifurcation with a cycle between
only two species. These new cycles explain the existence of "defensive
alliances" observed in previous numerical experiments. We further describe the
structure of the heteroclitic network for any number of species, and we
conjecture how these results may generalise to systems of any arbitrary number
of species in cyclic competition
Stabilizing unstable periodic orbits in the Lorenz equations using time-delayed feedback control
For many years it was believed that an unstable periodic orbit with an odd
number of real Floquet multipliers greater than unity cannot be stabilized by
the time-delayed feedback control mechanism of Pyragus. A recent paper by
Fiedler et al uses the normal form of a subcritical Hopf bifurcation to give a
counterexample to this theorem. Using the Lorenz equations as an example, we
demonstrate that the stabilization mechanism identified by Fiedler et al for
the Hopf normal form can also apply to unstable periodic orbits created by
subcritical Hopf bifurcations in higher-dimensional dynamical systems. Our
analysis focuses on a particular codimension-two bifurcation that captures the
stabilization mechanism in the Hopf normal form example, and we show that the
same codimension-two bifurcation is present in the Lorenz equations with
appropriately chosen Pyragus-type time-delayed feedback. This example suggests
a possible strategy for choosing the feedback gain matrix in Pyragus control of
unstable periodic orbits that arise from a subcritical Hopf bifurcation of a
stable equilibrium. In particular, our choice of feedback gain matrix is
informed by the Fiedler et al example, and it works over a broad range of
parameters, despite the fact that a center-manifold reduction of the
higher-dimensional problem does not lead to their model problem.Comment: 21 pages, 8 figures, to appear in PR
Resonance bifurcations from robust homoclinic cycles
We present two calculations for a class of robust homoclinic cycles with
symmetry Z_n x Z_2^n, for which the sufficient conditions for asymptotic
stability given by Krupa and Melbourne are not optimal.
Firstly, we compute optimal conditions for asymptotic stability using
transition matrix techniques which make explicit use of the geometry of the
group action.
Secondly, through an explicit computation of the global parts of the Poincare
map near the cycle we show that, generically, the resonance bifurcations from
the cycles are supercritical: a unique branch of asymptotically stable period
orbits emerges from the resonance bifurcation and exists for coefficient values
where the cycle has lost stability. This calculation is the first to explicitly
compute the criticality of a resonance bifurcation, and answers a conjecture of
Field and Swift in a particular limiting case. Moreover, we are able to obtain
an asymptotically-correct analytic expression for the period of the bifurcating
orbit, with no adjustable parameters, which has not proved possible previously.
We show that the asymptotic analysis compares very favourably with numerical
results.Comment: 24 pages, 3 figures, submitted to Nonlinearit
Effect of angiotensin-converting enzyme inhibitor and angiotensin receptor blocker initiation on organ support-free days in patients hospitalized with COVID-19
IMPORTANCE Overactivation of the renin-angiotensin system (RAS) may contribute to poor clinical outcomes in patients with COVID-19.
Objective To determine whether angiotensin-converting enzyme (ACE) inhibitor or angiotensin receptor blocker (ARB) initiation improves outcomes in patients hospitalized for COVID-19.
DESIGN, SETTING, AND PARTICIPANTS In an ongoing, adaptive platform randomized clinical trial, 721 critically ill and 58 nonâcritically ill hospitalized adults were randomized to receive an RAS inhibitor or control between March 16, 2021, and February 25, 2022, at 69 sites in 7 countries (final follow-up on June 1, 2022).
INTERVENTIONS Patients were randomized to receive open-label initiation of an ACE inhibitor (nâ=â257), ARB (nâ=â248), ARB in combination with DMX-200 (a chemokine receptor-2 inhibitor; nâ=â10), or no RAS inhibitor (control; nâ=â264) for up to 10 days.
MAIN OUTCOMES AND MEASURES The primary outcome was organ supportâfree days, a composite of hospital survival and days alive without cardiovascular or respiratory organ support through 21 days. The primary analysis was a bayesian cumulative logistic model. Odds ratios (ORs) greater than 1 represent improved outcomes.
RESULTS On February 25, 2022, enrollment was discontinued due to safety concerns. Among 679 critically ill patients with available primary outcome data, the median age was 56 years and 239 participants (35.2%) were women. Median (IQR) organ supportâfree days among critically ill patients was 10 (â1 to 16) in the ACE inhibitor group (nâ=â231), 8 (â1 to 17) in the ARB group (nâ=â217), and 12 (0 to 17) in the control group (nâ=â231) (median adjusted odds ratios of 0.77 [95% bayesian credible interval, 0.58-1.06] for improvement for ACE inhibitor and 0.76 [95% credible interval, 0.56-1.05] for ARB compared with control). The posterior probabilities that ACE inhibitors and ARBs worsened organ supportâfree days compared with control were 94.9% and 95.4%, respectively. Hospital survival occurred in 166 of 231 critically ill participants (71.9%) in the ACE inhibitor group, 152 of 217 (70.0%) in the ARB group, and 182 of 231 (78.8%) in the control group (posterior probabilities that ACE inhibitor and ARB worsened hospital survival compared with control were 95.3% and 98.1%, respectively).
CONCLUSIONS AND RELEVANCE In this trial, among critically ill adults with COVID-19, initiation of an ACE inhibitor or ARB did not improve, and likely worsened, clinical outcomes.
TRIAL REGISTRATION ClinicalTrials.gov Identifier: NCT0273570
Effects of Temporal Resolution on an Inferential Model of Animal Movement
<div><p>Recently, there has been much interest in describing the behaviour of animals by fitting various movement models to tracking data. Despite this interest, little is known about how the temporal âgrainâ of movement trajectories affects the outputs of such models, and how behaviours classified at one timescale may differ from those classified at other scales. Here, we present a study in which random-walk state-space models were fit both to nightly geospatial lifelines of common brushtail possums (<i>Trichosurus vulpecula</i>) and synthetic trajectories parameterised from empirical observations. Observed trajectories recorded by GPS collars at 5-min intervals were sub-sampled at periods varying between 10 and 60 min, to approximate the effect of collecting data at lower sampling frequencies. Markov-Chain Monte-Carlo fitting techniques, using information about movement rates and turning angles between sequential fixes, were employed using a Bayesian framework to assign distinct behavioural states to individual location estimates. We found that in trajectories with higher temporal granularities behaviours could be clearly differentiated into âslow-area-restrictedâ and âfast-transitingâ states, but for trajectories with longer inter-fix intervals this distinction was markedly less obvious. Specifically, turning-angle distributions varied from being highly peaked around either or at fine temporal scales, to being uniform across all angles at low sampling intervals. Our results highlight the difficulty of comparing model results amongst tracking-data sets that vary substantially in temporal grain, and demonstrate the importance of matching the observed temporal resolution of tracking devices to the timescales of behaviours of interest, otherwise inter-individual comparisons of inferred behaviours may be invalid, or important biological information may be obscured.</p> </div
ARTICLE IN PRESS
www.elsevier.com/locate/physd Spatial and temporal feedback control of traveling wave solutions of the two-dimensional complex GinzburgâLandau equatio