2024-03-28T09:07:36Z
https://eprints.maths.manchester.ac.uk/cgi/oai2
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:170
2017-10-20T14:12:04Z
7374617475733D707562
7375626A656374733D50414353:504143535F3030:504143535F3035
74797065733D626F6F6B5F73656374696F6E
https://eprints.maths.manchester.ac.uk/170/
Fractals
Broomhead, D. S.
05 Statistical physics, thermodynamics, and nonlinear dynamical systems
The idea of a fractal as a mathematical object and as a model of natural phenomena is introduced by way of simple examples. The characterization of fractals both mathematically and experimentally is then discussed. Finally a model exemplifying the generation of fractal geometries in nature is discussed.
A.A. Balkema
Spencer, A. J. M.
1987
Book Section
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/170/1/1985ContModelsDiscSys.pdf
Broomhead, D. S. (1987) Fractals. In: Proceedings of the Fifth International Symposium on Continuum Models of Discrete Systems, Nottingham, 14-20 July 1985. A.A. Balkema, Rotterdam, pp. 27-34. ISBN 9061916828
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:261
2017-10-20T14:12:07Z
7374617475733D707562
7375626A656374733D50414353:504143535F3030:504143535F3032
7375626A656374733D50414353:504143535F3030:504143535F3035
74797065733D61727469636C65
https://eprints.maths.manchester.ac.uk/261/
Imperfect homoclinic bifurcations
Glendinning, Paul
Abshagen, Jan
Mullin, Tom
02 Mathematical methods in physics
05 Statistical physics, thermodynamics, and nonlinear dynamical systems
Experimental observations of an almost symmetric electronic circuit show complicated sequences of bifurcations. These results are discussed in the light of a theory of imperfect global bifurcations. It is shown that much of the dynamics observed in the circuit can be understood by reference to imperfect homoclinic bifurcations without constructing an explicit mathematical model of the system.
2001-09
Article
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/261/1/tom.pdf
Glendinning, Paul and Abshagen, Jan and Mullin, Tom (2001) Imperfect homoclinic bifurcations. Physical Review E, 64 (3). 036208. ISSN 1539-3755
http://link.aps.org/abstract/PRE/v64/e036208
10.1103/PhysRevE.64.036208
10.1103/PhysRevE.64.036208
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:628
2017-10-20T14:12:20Z
7374617475733D707562
7375626A656374733D50414353:504143535F3030:504143535F3035
7375626A656374733D50414353:504143535F3430:504143535F3432
74797065733D61727469636C65
https://eprints.maths.manchester.ac.uk/628/
When gap solitons become embedded solitons: a generic unfolding
Wagenknecht, T.
Champneys, A. R.
05 Statistical physics, thermodynamics, and nonlinear dynamical systems
42 Optics
A two-parameter unfolding is considered of single-pulsed homoclinic orbits to an equilibrium with two real and two zero eigenvalues in fourth-order reversible dynamical systems. One parameter controls the linearisation, with a transition occurring between a saddle-centre and a hyperbolic equilibrium. In the saddle-centre region, the homoclinic orbit is of codimension-one, which is controlled by the second generic parameter, whereas when the equilibrium is hyperbolic the homoclinic orbit is structurally stable. A geometric approach reveals the homoclinic orbits to the saddle to be generically destroyed either by developing an algebraically decaying tail or through a fold, depending on the sign of the perturbation of the second parameter. Special cases of different actions of Z2-symmetry are considered, as is the case of the system being Hamiltonian. Application of these results is considered to the transition between embedded solitons (corresponding to the codimension-one-homoclinic orbits) and gap solitons (the structurally stable ones) in nonlinear wave systems. The theory is shown to match numerical experiments on two models arising in nonlinear optics and on a form of fifth-order Korteweg de Vries equation.
2003-03-15
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/628/1/gapsolit_new.pdf
Wagenknecht, T. and Champneys, A. R. (2003) When gap solitons become embedded solitons: a generic unfolding. Physica D, 177. pp. 50-70. ISSN 0167-2789
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVK-47GJ15R-6&_coverDate=03%2F15%2F2003&_alid=468406024&_rdoc=1&_fmt=&_orig=search&_qd=1&_cdi=5537&_sort=d&view=c&_acct=C000010021&_version=1&_urlVersion=0&_userid=121749&md5=6efccc2b073ffc8475067
10.1016/S0167-2789(02)00773-X
10.1016/S0167-2789(02)00773-X
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:663
2017-10-20T14:12:21Z
7374617475733D707562
7375626A656374733D50414353:504143535F3030:504143535F3032
7375626A656374733D50414353:504143535F3030:504143535F3035
74797065733D61727469636C65
https://eprints.maths.manchester.ac.uk/663/
Homoclinic Snaking near a Heteroclinic Cycle in Reversible Systems
Knobloch, J.
Wagenknecht, T.
02 Mathematical methods in physics
05 Statistical physics, thermodynamics, and nonlinear dynamical systems
Snaking curves of homoclinic orbits have been found numerically in a number of ODE models from water wave theory and structural mechanics. Along such a curve infinitely many fold bifurcation of homoclinic orbits occur. Thereby the corresponding solutions spread out and develop more and more bumps (oscillations) about their own centre. A common feature of the examples is that the systems under consideration are reversible.
In this paper it is shown that such a homoclinic snaking can be caused by a heteroclinic cycle between two equilibria, one of which is a bi-focus. Using Lin’s method a snaking of 1-homoclinic orbits is proved to occur in an unfolding of such a cycle. Further dynamical consequences are discussed.
As an application a system of Boussinesq equations is considered, where numerically a homoclinic snaking curve is detected and it is shown that the homoclinic orbits accumulate along a heteroclinic cycle between a real saddle and a bi-focus equilibrium.
2005-06-15
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/663/1/snaking_prep.pdf
Knobloch, J. and Wagenknecht, T. (2005) Homoclinic Snaking near a Heteroclinic Cycle in Reversible Systems. Physica D, 206 (1-2). pp. 82-93. ISSN 0167-2789
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVK-4G7G48P-4&_coverDate=06%2F15%2F2005&_alid=503647922&_rdoc=1&_fmt=&_orig=search&_qd=1&_cdi=5537&_sort=d&view=c&_acct=C000010021&_version=1&_urlVersion=0&_userid=121749&md5=5d1688c5cd4ebe258f374
10.1016/j.physd.2005.04.018
10.1016/j.physd.2005.04.018
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:842
2018-09-17T14:46:22Z
7374617475733D707562
7375626A656374733D4D5343:4D53435F3334
7375626A656374733D4D5343:4D53435F3335
7375626A656374733D4D5343:4D53435F3932
7375626A656374733D50414353:504143535F3030:504143535F3035
7375626A656374733D50414353:504143535F3830:504143535F3832
7375626A656374733D50414353:504143535F3830:504143535F3837
74797065733D4D494D535F7072657072696E74
https://eprints.maths.manchester.ac.uk/842/
Does Reaction-diffusion Dynamics on a Fractal Space Imply Power Law Behaviour?
Riley, C. J.
Muldoon, M. R.
Huke, J. P.
Broomhead, D. S.
34 Ordinary differential equations
35 Partial differential equations
92 Biology and other natural sciences
05 Statistical physics, thermodynamics, and nonlinear dynamical systems
82 Physical chemistry and chemical physics molecular physics
87 Biological and medical physics
In biological systems, chemical reactions often take place in complex spatial environments. For example, the translation of m-RNA to produce protein within eukaryotic cells takes place within the extremely crowded cytoplasmic environment and appears to require the spatial coordination of many translation factors. It is important, therefore, to understand the transport processes within such an environment. While there is growing interest in both experimental and computational studies of such environments, it is also important to develop suitable mathematical models. Here, as an example of such a model, we study a reaction-diffusion equation defined on the Sierpinski gasket. Both experimental and computational studies of analogous systems have shown power law behaviour and associated deviations from mass action kinetics. The analysis presented here allows us to distinguish the roles of the fractal domain and of the discreteness of molecular interactions in producing this effect. Indeed, we show that the fractal domain alone is insufficient.
2007-09-04
MIMS Preprint
NonPeerReviewed
text
en
https://eprints.maths.manchester.ac.uk/842/1/fraclett2.pdf
Riley, C. J. and Muldoon, M. R. and Huke, J. P. and Broomhead, D. S. (2007) Does Reaction-diffusion Dynamics on a Fractal Space Imply Power Law Behaviour? [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:1234
2017-11-07T22:38:45Z
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1540
2017-10-20T14:12:52Z
7374617475733D7375626D6974746564
7375626A656374733D50414353:504143535F3030:504143535F3035
74797065733D61727469636C65
https://eprints.maths.manchester.ac.uk/1540/
Modelling human balance using switched systems
with linear feedback control
Kowalczyk, Piotr
Glendinning, Paul
Brown, Martin
Medrano-Cerda, Gustavo
Dallali, Houman
Shapiro, Jonathan
05 Statistical physics, thermodynamics, and nonlinear dynamical systems
We are interested in understanding the mechanisms behind and the
character of the sway motion of healthy human subjects during quiet
standing with eyes closed. We assume that a human body can be modelled
as a single-link inverted pendulum, and the balance is achieved using liner
feedback control. Using these assumptions we derive a switched model
which we then investigate. Stable periodic motions (limit cycles) about an
upright position are found. The existence of these limit cycles is studied
as a function of system parameters. The exploration of the parameter
space leads to the detection of multistability and homoclinic bifurcations
2011-03
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1540/1/Manuscript_FinShrt_RevFinV2.pdf
Kowalczyk, Piotr and Glendinning, Paul and Brown, Martin and Medrano-Cerda, Gustavo and Dallali, Houman and Shapiro, Jonathan (2011) Modelling human balance using switched systems with linear feedback control. Journal of the Royal Society Interface. ISSN 1742-5689 (Submitted)
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1544
2017-11-08T18:18:33Z
7374617475733D707562
7375626A656374733D4D5343:4D53435F3337
7375626A656374733D50414353:504143535F3030:504143535F3035
74797065733D4D494D535F7072657072696E74
https://eprints.maths.manchester.ac.uk/1544/
Two dimensional attractors in the border collision normal form
Glendinning, Paul
Wong, Chi Hong
37 Dynamical systems and ergodic theory
05 Statistical physics, thermodynamics, and nonlinear dynamical systems
New techniques are developed to show that the two-dimensional normal form for codimension one border collision bifurcations of fixed points of discrete time piecewise smooth dynamical systems has attractors which are themselves two dimensional. This makes it possible to prove the existence of these attractors for a countable set of parameter values which cannot be treated using the essentially one-dimensional methods in the literature.
2010-11-17
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1544/1/robust2d.pdf
Glendinning, Paul and Wong, Chi Hong (2010) Two dimensional attractors in the border collision normal form. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1602
2017-11-08T18:18:33Z
7374617475733D707562
7375626A656374733D50414353:504143535F3030:504143535F3032
7375626A656374733D50414353:504143535F3030:504143535F3035
74797065733D4D494D535F7072657072696E74
https://eprints.maths.manchester.ac.uk/1602/
Attractors near grazing-sliding bifurcations
Glendinning, P.
Kowalcyk, P.
Nordmark, A.B.
02 Mathematical methods in physics
05 Statistical physics, thermodynamics, and nonlinear dynamical systems
In this paper we prove, for the first time, that multistability can occur in 3-dimensional
Fillipov type flows due to grazing-sliding bifurcations. We do this by reducing the study of the
dynamics of Filippov type flows around a grazing-sliding bifurcation to the study of appropri-
ately defined one-dimensional maps. In particular, we prove the presence of three qualitatively
different types of multiple attractors born in grazing-sliding bifurcations. Namely, a period-two
orbit with a sliding segment may coexsist with a chaotic attractor, two stable, period-two and
period-three orbits with a segment of sliding each may coexist, or a non-sliding and period-three
orbit with two sliding segments may coexist.
2011-04-07
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1602/1/manuscriptFin.pdf
Glendinning, P. and Kowalcyk, P. and Nordmark, A.B. (2011) Attractors near grazing-sliding bifurcations. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1645
2017-11-08T18:18:33Z
7374617475733D707562
7375626A656374733D50414353:504143535F3030:504143535F3035
74797065733D4D494D535F7072657072696E74
https://eprints.maths.manchester.ac.uk/1645/
Global structure of periodicity hubs in Lyapunov phase diagrams of dissipative flows
Vitolo, Renato
Glendinning, Paul
Gallas, Jason A.C.
05 Statistical physics, thermodynamics, and nonlinear dynamical systems
Infinite cascades of periodicity hubs were predicted and very recently observed experimentally to organize stable oscillations of some dissipative flows. Here we describe the global mechanism underlying the genesis and organization of networks of periodicity hubs in control parameter space of a simple prototypical flow. We show that spirals associated with periodicity hubs emerge/accumulate
at the folding of certain fractal-like sheaves of Shilnikov homoclinic bifurcations of a common saddle-focus equilibrium. The specific organization of hub networks is found to depend strongly on the interaction between the homoclinic orbits and the global structure of the underlying attractor.
2011-07-04
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1645/1/pre_network.pdf
Vitolo, Renato and Glendinning, Paul and Gallas, Jason A.C. (2011) Global structure of periodicity hubs in Lyapunov phase diagrams of dissipative flows. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1653
2017-11-08T18:18:33Z
7374617475733D707562
7375626A656374733D50414353:504143535F3030:504143535F3035
74797065733D4D494D535F7072657072696E74
https://eprints.maths.manchester.ac.uk/1653/
Attractors near grazing-sliding bifurcations
Glendinning, Paul
Kowalczyk, Piotr
Nordmark, Arne
05 Statistical physics, thermodynamics, and nonlinear dynamical systems
In this paper we prove, for the first time, that multistability can occur in 3-dimensional Fillipov type flows due to grazing-sliding bifurcations. We do this by reducing the study of the dynamics of Filippov type flows around a grazing-sliding bifurcation to the study of appropriately defined one-dimensional maps. In particular, we prove the presence of three qualitatively different types of multiple attractors born in grazing-sliding bifurcations. Namely, a period-two orbit with a sliding segment may coexsist with a chaotic attractor, two stable, period-two and period-three orbits with a segment of sliding each may coexist, or a non-sliding and period-three
orbit with two sliding segments may coexist.
2011-07-15
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1653/1/manuscriptFin-1.pdf
Glendinning, Paul and Kowalczyk, Piotr and Nordmark, Arne (2011) Attractors near grazing-sliding bifurcations. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1691
2017-10-20T14:12:57Z
7374617475733D707562
7375626A656374733D4D5343:4D53435F3337
7375626A656374733D4D5343:4D53435F3533
7375626A656374733D50414353:504143535F3030:504143535F3035
74797065733D626F6F6B5F73656374696F6E
https://eprints.maths.manchester.ac.uk/1691/
Delay reconstruction for multiprobe signals
Muldoon, Mark R.
Broomhead, David S.
Huke, Jeremy P.
37 Dynamical systems and ergodic theory
53 Differential geometry
05 Statistical physics, thermodynamics, and nonlinear dynamical systems
A physical system governed by low-dimensional dynamics may be described completely with just a few measurements. Once one has such a description, any further measurements are redundant-one ought to be able to determine the results from what one already knows. Here we apply this idea to multivariate time series; we use the signal in one of the channels to build a model of the underlying system, then use the model to predict all the other channels. We demonstrate the method on a signal from a fluid-mechanical experiment, then discuss the implications for signal compression and for the secrecy of messages masked by chaotic noise
IEE
1994-06
Book Section
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1691/1/MuldoonHukeBroomhead_IEE_Digest143.pdf
Muldoon, Mark R. and Broomhead, David S. and Huke, Jeremy P. (1994) Delay reconstruction for multiprobe signals. In: IEE Colloquium on Exploiting Chaos in Signal Processing. IEE, London, 3/1-3/6.
http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=369883&tag=1
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1727
2017-11-08T18:18:34Z
7374617475733D707562
7375626A656374733D50414353:504143535F3030:504143535F3035
74797065733D4D494D535F7072657072696E74
https://eprints.maths.manchester.ac.uk/1727/
Emergence of hierarchical networks and polysynchronous behaviour in simple adaptive systems
Botella-Soler, V
Glendinning, Paul
05 Statistical physics, thermodynamics, and nonlinear dynamical systems
We describe the dynamics of a simple adaptive network. The network architecture evolves to a number of disconnected components on which the dynamics is characterized by the possibility of differently synchronized nodes within the same network (polysynchronous states). These systems may have implications for the evolutionary emergence of polysynchrony and hierarchical networks in physical or biological systems modeled by adaptive networks.
2011-12-11
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1727/1/botellaglendinningnote.pdf
Botella-Soler, V and Glendinning, Paul (2011) Emergence of hierarchical networks and polysynchronous behaviour in simple adaptive systems. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1847
2017-11-08T18:18:34Z
7374617475733D707562
7375626A656374733D50414353:504143535F3030:504143535F3035
74797065733D4D494D535F7072657072696E74
https://eprints.maths.manchester.ac.uk/1847/
Grazing-sliding bifurcations, the border collision normal form, and the curse of dimensionality for nonsmooth bifurcation theory
Glendinning, Paul
Jeffrey, Mike
05 Statistical physics, thermodynamics, and nonlinear dynamical systems
In this paper we show that the border collision normal form of continuous but non-differentiable discrete time maps
is affected by a curse of dimensionality: it is impossible to reduce the study
of the general case to low dimensions, since in every dimension the bifurcation produces fundamentally different attractors (contrary to the case of smooth systems). In particular we show that the $n$-dimensional border collision normal form can have invariant sets of dimension $k$ for
integer $k$ from $0$ to $n$. We also show that the border collision normal form is related to grazing-sliding bifurcations of
switching dynamical systems. This implies that the dynamics of these
two apparently distinct bifurcations (one for discrete time dynamics, the other for continuous time dynamics) are closely related
and hence that a similar curse of dimensionality holds for this bifurcation.
2012-07-03
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1847/1/glendinning1.pdf
Glendinning, Paul and Jeffrey, Mike (2012) Grazing-sliding bifurcations, the border collision normal form, and the curse of dimensionality for nonsmooth bifurcation theory. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1856
2017-11-08T18:18:34Z
7374617475733D707562
7375626A656374733D50414353:504143535F3030:504143535F3035
74797065733D4D494D535F7072657072696E74
https://eprints.maths.manchester.ac.uk/1856/
The border collision normal form with stochastic switching surface
Glendinning, Paul
05 Statistical physics, thermodynamics, and nonlinear dynamical systems
The deterministic border collision normal form describes the bifurcations of a discrete time dynamical system as a fixed point moves across the switching surface with changing parameter. If the position of the switching surface varies randomly, but within some bounded region, we give conditions which imply that the attractor close to the bifurcation point is the attractor of an Iterated Function System. The proof uses an equivalent metric to the Euclidean metric because the functions involved are never contractions in the Euclidean metric. If the conditions do not hold then a range of possibilities may be realized, including local instability, and some examples are investigated numerically.
2012-08-08
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1856/1/noisyswitch.pdf
Glendinning, Paul (2012) The border collision normal form with stochastic switching surface. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:2250
2017-11-08T18:18:37Z
7374617475733D707562
7375626A656374733D4D5343:4D53435F3337
7375626A656374733D50414353:504143535F3030:504143535F3035
74797065733D4D494D535F7072657072696E74
https://eprints.maths.manchester.ac.uk/2250/
Bifurcation from stable fixed point to $N$-dimensional attractor in the border collision normal form
Glendinning, Paul
37 Dynamical systems and ergodic theory
05 Statistical physics, thermodynamics, and nonlinear dynamical systems
The $N$-dimensional border collision normal form describes bifurcations of piecewise smooth systems. It is shown that there is an open set of parameters such that on one side of the bifurcation the map has a stable fixed point and on the other an attractor with Hausdorff dimension $N$. For generic
parameters this attractor contains open sets and hence has
topological dimension equal to $N$.
2015-02-24
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/2250/1/bifntondim.pdf
Glendinning, Paul (2015) Bifurcation from stable fixed point to $N$-dimensional attractor in the border collision normal form. [MIMS Preprint]