2024-03-29T09:23:29Z
https://eprints.maths.manchester.ac.uk/cgi/oai2
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:136
2017-10-20T14:12:03Z
7374617475733D707562
7375626A656374733D4D5343:4D53435F3339
7375626A656374733D4D5343:4D53435F3630
7375626A656374733D4D5343:4D53435F3635
74797065733D4D494D535F7072657072696E74
https://eprints.maths.manchester.ac.uk/136/
The Weak Euler Scheme for Stochastic
Differential Delay Equations
Buckwar, Evelyn
Kuske, Rachel
Mohammed, Salah-Eldin
Shardlow, Tony
39 Difference and functional equations
60 Probability theory and stochastic processes
65 Numerical analysis
We develop a weak numerical Euler scheme for non-linear stochastic delay differential equations (SDDEs) driven by multidimensional Brownian motion. The weak Euler scheme has
order of convergence 1, as in the case of stochastic ordinary
differential equations (SODEs) (i.e., without delay).The result
holds for SDDEs with multiple finite fixed delays in the drift and
diffusion terms. Although the set-up is non-anticipating,
our approach uses the Malliavin calculus and the
anticipating stochastic analysis techniques of Nualart
and Pardoux.
2006-01-05
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/136/1/weak_mal_web.pdf
Buckwar, Evelyn and Kuske, Rachel and Mohammed, Salah-Eldin and Shardlow, Tony (2006) The Weak Euler Scheme for Stochastic Differential Delay Equations. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:149
2017-10-20T14:12:04Z
7374617475733D707562
7375626A656374733D4D5343:4D53435F3339
7375626A656374733D4D5343:4D53435F3630
7375626A656374733D4D5343:4D53435F3635
74797065733D4D494D535F7072657072696E74
https://eprints.maths.manchester.ac.uk/149/
The Weak Euler Scheme for Stochastic
Differential Delay Equations
Buckwar, Evelyn
Kuske, Rachel
Mohammed, Salah-Eldin
Shardlow, Tony
39 Difference and functional equations
60 Probability theory and stochastic processes
65 Numerical analysis
We develop a weak numerical Euler scheme for non-linear stochastic delay differential equations (SDDEs) driven by multidimensional Brownian motion. The weak Euler scheme has
order of convergence 1, as in the case of stochastic ordinary
differential equations (SODEs) (i.e., without delay).The result
holds for SDDEs with multiple finite fixed delays in the drift and
diffusion terms. Although the set-up is non-anticipating,
our approach uses the Malliavin calculus and the
anticipating stochastic analysis techniques of Nualart
and Pardoux.
2006-01-24
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/149/1/weak_mal_web.pdf
Buckwar, Evelyn and Kuske, Rachel and Mohammed, Salah-Eldin and Shardlow, Tony (2006) The Weak Euler Scheme for Stochastic Differential Delay Equations. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:235
2017-11-08T18:18:29Z
7374617475733D707562
7375626A656374733D4D5343:4D53435F3337
7375626A656374733D4D5343:4D53435F3339
74797065733D4D494D535F7072657072696E74
https://eprints.maths.manchester.ac.uk/235/
How chaotic are strange nonchaotic attractors?
Glendinning, Paul
Jager, Tobias
Keller, Gerhard
37 Dynamical systems and ergodic theory
39 Difference and functional equations
We show that the classic examples of quasi-periodically forced maps with strange nonchaotic
attractors described by Grebogi et al and Herman in the mid-1980s have some chaotic properties.
More precisely, we show that these systems exhibit sensitive dependence on initial conditions,
both on the whole phase space and restricted to the attractor. The results also remain valid
in more general classes of quasiperiodically forced systems. Further, we include an elementary
proof of a classic result by Glasner and Weiss on sensitive dependence, and we clarify the
structure of the attractor in an example with two-dimensional bers also introduced by Grebogi
et al.
2006-05-17
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/235/1/gjksdic-final.pdf
Glendinning, Paul and Jager, Tobias and Keller, Gerhard (2006) How chaotic are strange nonchaotic attractors? [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:296
2017-10-20T14:12:08Z
7374617475733D707562
7375626A656374733D4D5343:4D53435F3339
7375626A656374733D4D5343:4D53435F3932
74797065733D61727469636C65
https://eprints.maths.manchester.ac.uk/296/
Relating Individual Behaviour to Population Dynamics
Sumpter, D.J.T.
Broomhead, D.S.
39 Difference and functional equations
92 Biology and other natural sciences
How do the behavioural interactions between individuals in an ecological system produce the global population dynamics of that system? We present a stochastic individual-based model of the reproductive cycle of the mite Varroa jacobsoni, a parasite of honeybees. The model has the interesting property in that its population level behaviour is approximated extremely accurately by the exponential logistic equation or Ricker map. We demonstrated how this approximation is obtained mathematically and how the parameters of the exponential logistic equation can be written in terms of the parameters of the individual-based model. Our procedure demonstrates, in at least one case, how study of animal ecology at an individual level can be used to derive global models which predict population change over time.
2001
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/296/1/2001ProcRoySoc.pdf
Sumpter, D.J.T. and Broomhead, D.S. (2001) Relating Individual Behaviour to Population Dynamics. Proceedings of the Royal Society B: Biological Sciences, 268 (1470). pp. 925-932. ISSN 0962-8452
10.1098/rspb.2001.1604
10.1098/rspb.2001.1604
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1160
2017-11-08T18:18:31Z
7374617475733D707562
7375626A656374733D4D5343:4D53435F3337
7375626A656374733D4D5343:4D53435F3339
74797065733D4D494D535F7072657072696E74
https://eprints.maths.manchester.ac.uk/1160/
Renormalization for the boundary of chaos in piecewise monotonic maps with a single discontinuity
Glendinning, Paul
37 Dynamical systems and ergodic theory
39 Difference and functional equations
Monotonic maps with a single discontinuity arise in a variety of situations. We describe the infinite sets of periods for such maps on the boundary of chaos; this gives a sense of the routes to chaos in such maps. The description involves an explicit subshift of finite type which describes the sequences of different renormalizations possible in these maps.
2008-10-24
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1160/1/renchaosnew.pdf
Glendinning, Paul (2008) Renormalization for the boundary of chaos in piecewise monotonic maps with a single discontinuity. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1162
2017-11-08T18:18:31Z
7374617475733D707562
7375626A656374733D4D5343:4D53435F3337
7375626A656374733D4D5343:4D53435F3339
74797065733D4D494D535F7072657072696E74
https://eprints.maths.manchester.ac.uk/1162/
Renormalization for the boundary of chaos in piecewise monotonic maps with a single discontinuity
Glendinning, Paul
37 Dynamical systems and ergodic theory
39 Difference and functional equations
Monotonic maps with a single discontinuity arise in a variety of situations. We describe the infinite sets of periods for such maps on the boundary of chaos; this gives a sense of the routes to chaos in such maps. The description involves an explicit subshift of finite type which describes the sequences of different renormalizations possible in these maps.
2008-10-24
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1162/1/renchaosnew.pdf
Glendinning, Paul (2008) Renormalization for the boundary of chaos in piecewise monotonic maps with a single discontinuity. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1175
2017-11-08T18:18:31Z
7374617475733D707562
7375626A656374733D4D5343:4D53435F3339
7375626A656374733D50414353:504143535F3030:504143535F3035
74797065733D4D494D535F7072657072696E74
https://eprints.maths.manchester.ac.uk/1175/
Border collision bifurcations, snap-back repellers and chaos
Glendinning, Paul
Wong, Chi Hong
39 Difference and functional equations
05 Statistical physics, thermodynamics, and nonlinear dynamical systems
The normal form for codimension one border collision bifurcations of fixed points of discrete time piecewise smooth dynamical systems is considered in the unstable case. We show that in appropriate parameter regions there is a snap-back repeller immediately after the bifurcation, and hence that the bifurcation creates chaos. Although the chaotic solutions are repellers they may explain observations, and this is illustrated through an example.
2008-11-27
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1175/1/boundchaos.pdf
Glendinning, Paul and Wong, Chi Hong (2008) Border collision bifurcations, snap-back repellers and chaos. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1178
2017-11-08T18:18:31Z
7374617475733D707562
7375626A656374733D4D5343:4D53435F3337
7375626A656374733D4D5343:4D53435F3339
74797065733D4D494D535F7072657072696E74
https://eprints.maths.manchester.ac.uk/1178/
Strangely Dispersed Minimal Sets in the Quasiperiodically Forced Arnold Circle Map
Glendinning, Paul
Jager, Tobias
Stark, Jaroslav
37 Dynamical systems and ergodic theory
39 Difference and functional equations
We study quasiperiodically forced circle endomorphisms, homotopic to the identity, and show that under suitable conditions these exhibit uncountably many minimal sets with a complicated structure, to which we refer to as ‘strangely dispersed’. Along the way, we generalise some well-known results about circle endomorphisms to the uniquely ergodically forced case. Namely, all rotation numbers in the rotation interval of a uniquely ergodically forced circle endomorphism are realised on minimal sets, and if the rotation interval has non-empty interior then the topological entropy is strictly positive. The results apply in particular to the quasiperiodically forced Arnold circle map, which serves as a paradigm example.
2008-11-06
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1178/1/gjs-sdms.pdf
Glendinning, Paul and Jager, Tobias and Stark, Jaroslav (2008) Strangely Dispersed Minimal Sets in the Quasiperiodically Forced Arnold Circle Map. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1179
2017-11-08T18:18:31Z
7374617475733D707562
7375626A656374733D4D5343:4D53435F3337
7375626A656374733D4D5343:4D53435F3339
7375626A656374733D4D5343:4D53435F3933
74797065733D4D494D535F7072657072696E74
https://eprints.maths.manchester.ac.uk/1179/
Dynamics of a hybrid thermostat model with discrete sampling
time control
Glendinning, Paul
Kowalczyk, Piotr
37 Dynamical systems and ergodic theory
39 Difference and functional equations
93 Systems theory; control
The dynamics of a simple thermostat model is described. In the model the control system samples the temperature at regular but discrete time intervals rather than by continuous monitoring. The model exhibits quasi-periodic oscillations and banding, where the response falls into two or more bands of phase space representing either better or poorer control. A return circle map is derived which explains the observed dynamics. Some extensions of these results to the case where the flow is nonlinear are also given.
2008-11-06
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1179/1/gk-thermostat.pdf
Glendinning, Paul and Kowalczyk, Piotr (2008) Dynamics of a hybrid thermostat model with discrete sampling time control. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1234
2017-11-07T22:38:45Z
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1264
2017-11-08T18:18:31Z
7374617475733D707562
7375626A656374733D4D5343:4D53435F3337
7375626A656374733D4D5343:4D53435F3339
74797065733D4D494D535F7072657072696E74
https://eprints.maths.manchester.ac.uk/1264/
Bifurcations of Snap-back Repellers with application to Border-Collision Bifurcations
Glendinning, Paul
37 Dynamical systems and ergodic theory
39 Difference and functional equations
The bifurcation theory of snap-back repellers in hybrid dynamical systems is developed. Infinite sequences of bifurcations are shown to arise due to the creation of snap-back repellers in non-invertible maps. These are
analogous to the cascades of bifurcations known to occur close to homoclinic tangencies for diffeomorphisms. The theoretical results are illustrated with reference to bifurcations in the normal form for border-collision
bifurcations.
2009-05-11
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1264/1/sbrbif.pdf
Glendinning, Paul (2009) Bifurcations of Snap-back Repellers with application to Border-Collision Bifurcations. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1583
2017-11-08T18:18:33Z
7374617475733D707562
7375626A656374733D4D5343:4D53435F3339
7375626A656374733D4D5343:4D53435F3630
74797065733D4D494D535F7072657072696E74
https://eprints.maths.manchester.ac.uk/1583/
Hysteretic regime switching diffusions and resource
extraction
Moriarty, John
Evatt, Geoffrey W.
Johnson, Paul V.
Duck, Peter W.
39 Difference and functional equations
60 Probability theory and stochastic processes
We calculate the probability that an extraction project will be abandoned, directly from a real options
model closely related the seminal work of Brennan and Schwartz (1985). We assume that the resource is
extracted at two alternative rates, with a capital cost for switching, and with an option to abandon due
to unsatisfactory market prices. The abandonment probability is expressed as a hitting probability for a
regime switching diffusion with hysteresis, which is shown to be the unique solution of a system of coupled
boundary value problems. Our work lends itself to use as a quantitative and easily interpreted measure of
risk in the planning of extraction projects. Numerical results show that the abandonment probability may
be non-monotone with respect to the volatility of the price process, in contrast with project valuations. In
the one-dimensional stationary case, the stochastic process is a hysteretic system with noise in the sense of
Freidlin et al. (2000), and we obtain a closed-form expression for the hitting or abandonment probability in
this case.
2011-02-18
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1583/1/Moriarty_Evatt_hysteretic_resource.pdf
Moriarty, John and Evatt, Geoffrey W. and Johnson, Paul V. and Duck, Peter W. (2011) Hysteretic regime switching diffusions and resource extraction. [MIMS Preprint]