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  1. Article ; Online: Unstable dimension variability and heterodimensional cycles in the border-collision normal form.

    Glendinning, P A / Simpson, D J W

    Physical review. E

    2023  Volume 108, Issue 2, Page(s) L022202

    Abstract: Chaotic attractors commonly contain periodic solutions with unstable manifolds of different dimensions. This allows for a zoo of dynamical phenomena not possible for hyperbolic attractors. The purpose of this Letter is to emphasize the existence of these ...

    Abstract Chaotic attractors commonly contain periodic solutions with unstable manifolds of different dimensions. This allows for a zoo of dynamical phenomena not possible for hyperbolic attractors. The purpose of this Letter is to emphasize the existence of these phenomena in the border-collision normal form. This is a continuous, piecewise-linear family of maps that is physically relevant as it captures the dynamics created in border-collision bifurcations in diverse applications. Since the maps are piecewise linear, they are relatively amenable to an exact analysis. We explicitly identify parameter values for heterodimensional cycles and argue that the existence of heterodimensional cycles between two given saddles can be dense in parameter space. We numerically identify key bifurcations associated with unstable dimension variability by studying a one-parameter subfamily that transitions continuously from where periodic solutions are all saddles to where they are all repellers. This is facilitated by fast and accurate computations of periodic solutions; indeed the piecewise-linear form should provide a useful testbed for further study.
    Language English
    Publishing date 2023-09-18
    Publishing country United States
    Document type Journal Article
    ZDB-ID 2844562-4
    ISSN 2470-0053 ; 2470-0045
    ISSN (online) 2470-0053
    ISSN 2470-0045
    DOI 10.1103/PhysRevE.108.L022202
    Database MEDical Literature Analysis and Retrieval System OnLINE

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  2. Article ; Online: Classification of boundary equilibrium bifurcations in planar Filippov systems.

    Glendinning, Paul

    Chaos (Woodbury, N.Y.)

    2016  Volume 26, Issue 1, Page(s) 13108

    Abstract: If a family of piecewise smooth systems depending on a real parameter is defined on two different regions of the plane separated by a switching surface, then a boundary equilibrium bifurcation occurs if a stationary point of one of the systems intersects ...

    Abstract If a family of piecewise smooth systems depending on a real parameter is defined on two different regions of the plane separated by a switching surface, then a boundary equilibrium bifurcation occurs if a stationary point of one of the systems intersects the switching surface at a critical value of the parameter. We derive the leading order terms of a normal form for boundary equilibrium bifurcations of planar systems. This makes it straightforward to derive a complete classification of the bifurcations that can occur. We are thus able to confirm classic results of Filippov [Differential Equations with Discontinuous Right Hand Sides (Kluwer, Dordrecht, 1988)] using different and more transparent methods, and explain why the 'missing' cases of Hogan et al. [Piecewise Smooth Dynamical Systems: The Case of the Missing Boundary Equilibrium Bifurcations (University of Bristol, 2015)] are the only cases omitted in more recent work.
    Language English
    Publishing date 2016-01
    Publishing country United States
    Document type Journal Article
    ZDB-ID 1472677-4
    ISSN 1089-7682 ; 1054-1500
    ISSN (online) 1089-7682
    ISSN 1054-1500
    DOI 10.1063/1.4940017
    Database MEDical Literature Analysis and Retrieval System OnLINE

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  3. Article ; Online: Dynamic switching of lateral inhibition spatial patterns.

    Hawley, Joshua / Manning, Cerys / Biga, Veronica / Glendinning, Paul / Papalopulu, Nancy

    Journal of the Royal Society, Interface

    2022  Volume 19, Issue 193, Page(s) 20220339

    Abstract: Hes genes are transcriptional repressors activated by Notch. In the developing mouse neural tissue, HES5 expression oscillates in neural progenitors ( ... ...

    Abstract Hes genes are transcriptional repressors activated by Notch. In the developing mouse neural tissue, HES5 expression oscillates in neural progenitors (Manning
    MeSH term(s) Animals ; Basic Helix-Loop-Helix Transcription Factors/genetics ; Basic Helix-Loop-Helix Transcription Factors/metabolism ; Cell Communication ; Cell Differentiation ; Mice ; Receptors, Notch/metabolism ; Repressor Proteins/genetics ; Repressor Proteins/metabolism ; Signal Transduction/physiology
    Chemical Substances Basic Helix-Loop-Helix Transcription Factors ; Receptors, Notch ; Repressor Proteins
    Language English
    Publishing date 2022-08-24
    Publishing country England
    Document type Journal Article ; Research Support, Non-U.S. Gov't
    ZDB-ID 2156283-0
    ISSN 1742-5662 ; 1742-5689
    ISSN (online) 1742-5662
    ISSN 1742-5689
    DOI 10.1098/rsif.2022.0339
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  4. Article: Creation of discontinuities in circle maps.

    Derks, G / Glendinning, P A / Skeldon, A C

    Proceedings. Mathematical, physical, and engineering sciences

    2021  Volume 477, Issue 2251, Page(s) 20200872

    Abstract: Circle maps frequently arise in mathematical models of physical or biological systems. Motivated by Cherry flows and 'threshold' systems such as integrate and fire neuronal models, models of cardiac arrhythmias, and models of sleep/wake regulation, we ... ...

    Abstract Circle maps frequently arise in mathematical models of physical or biological systems. Motivated by Cherry flows and 'threshold' systems such as integrate and fire neuronal models, models of cardiac arrhythmias, and models of sleep/wake regulation, we consider how structural transitions in circle maps occur. In particular, we describe how maps evolve near the creation of a discontinuity. We show that the natural way to create discontinuities in the maps associated with both threshold systems and Cherry flows results in a singularity in the derivative of the map as the discontinuity is approached from either one or both sides. For the threshold systems, the associated maps have square root singularities and we analyse the generic properties of such maps with gaps, showing how border collisions and saddle-node bifurcations are interspersed. This highlights how the Arnold tongue picture for tongues bordered by saddle-node bifurcations is amended once gaps are present. We also show that a loss of injectivity naturally results in the creation of multiple gaps giving rise to a novel codimension two bifurcation.
    Language English
    Publishing date 2021-07-07
    Publishing country England
    Document type Journal Article
    ZDB-ID 209241-4
    ISSN 1471-2946 ; 1364-5021 ; 0962-8444 ; 0080-4630 ; 0950-1207
    ISSN (online) 1471-2946
    ISSN 1364-5021 ; 0962-8444 ; 0080-4630 ; 0950-1207
    DOI 10.1098/rspa.2020.0872
    Database MEDical Literature Analysis and Retrieval System OnLINE

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    In stock of ZB MED Bonn / Germany

    Z 46/95: Show issues

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  5. Article ; Online: Hierarchy and polysynchrony in an adaptive network.

    Botella-Soler, V / Glendinning, P

    Physical review. E, Statistical, nonlinear, and soft matter physics

    2014  Volume 89, Issue 6, Page(s) 62809

    Abstract: We describe a simple adaptive network of coupled chaotic maps. The network reaches a stationary state (frozen topology) for all values of the coupling parameter, although the dynamics of the maps at the nodes of the network can be nontrivial. The ... ...

    Abstract We describe a simple adaptive network of coupled chaotic maps. The network reaches a stationary state (frozen topology) for all values of the coupling parameter, although the dynamics of the maps at the nodes of the network can be nontrivial. The structure of the network shows interesting hierarchical properties and in certain parameter regions the dynamics is polysynchronous: Nodes can be divided in differently synchronized classes but, contrary to cluster synchronization, nodes in the same class need not be connected to each other. These complicated synchrony patterns have been conjectured to play roles in systems biology and circuits. The adaptive system we study describes ways whereby this behavior can evolve from undifferentiated nodes.
    MeSH term(s) Adaptation, Physiological/physiology ; Animals ; Computer Simulation ; Humans ; Models, Neurological ; Models, Statistical ; Nerve Net/physiology ; Neuronal Plasticity/physiology
    Language English
    Publishing date 2014-06
    Publishing country United States
    Document type Journal Article ; Research Support, Non-U.S. Gov't
    ISSN 1550-2376
    ISSN (online) 1550-2376
    DOI 10.1103/PhysRevE.89.062809
    Database MEDical Literature Analysis and Retrieval System OnLINE

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  6. Article ; Online: Two-ball Newton's cradle.

    Glendinning, Paul

    Physical review. E, Statistical, nonlinear, and soft matter physics

    2011  Volume 84, Issue 6 Pt 2, Page(s) 67201

    Abstract: Newton's cradle for two balls with Hertzian interactions is considered as a hybrid system, and this makes it possible to derive return maps for the motion between collisions in an exact form despite the fact that the three-halves interaction law cannot ... ...

    Abstract Newton's cradle for two balls with Hertzian interactions is considered as a hybrid system, and this makes it possible to derive return maps for the motion between collisions in an exact form despite the fact that the three-halves interaction law cannot be solved in closed form. The return maps depend on a constant whose value can only be determined numerically, but solutions can be written down explicitly in terms of this parameter, and we compare this with the results of simulations. The results are in fact independent of the details of the interaction potential.
    Language English
    Publishing date 2011-12
    Publishing country United States
    Document type Journal Article
    ISSN 1550-2376
    ISSN (online) 1550-2376
    DOI 10.1103/PhysRevE.84.067201
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  7. Book: Math in minutes

    Glendinning, Paul

    [200 key concepts explained in an instant]

    (Quercus mathematics)

    2013  

    Abstract: Glendinning provides a visually enhanced introduction to key mathematical concepts. Each idea is quickly and clearly explained, and easy to remember thanks to the simple yet essential illustrations that complement each ... ...

    Author's details Paul Glendinning
    Series title Quercus mathematics
    Abstract Glendinning provides a visually enhanced introduction to key mathematical concepts. Each idea is quickly and clearly explained, and easy to remember thanks to the simple yet essential illustrations that complement each description
    Keywords Mathematics
    Language English
    Size 415 S., zahlr. Ill., graph. Darst., 13 cm
    Publisher Quercus
    Publishing place New York, NY
    Document type Book
    Note Includes index
    ISBN 1623650089 ; 9781623650087
    Database Library catalogue of the German National Library of Science and Technology (TIB), Hannover

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  8. Article ; Online: Boundary-equilibrium bifurcations in piecewise-smooth slow-fast systems.

    Kowalczyk, P / Glendinning, P

    Chaos (Woodbury, N.Y.)

    2011  Volume 21, Issue 2, Page(s) 23126

    Abstract: In this paper we study the qualitative dynamics of piecewise-smooth slow-fast systems (singularly perturbed systems) which are everywhere continuous. We consider phase space topology of systems with one-dimensional slow dynamics and one-dimensional fast ... ...

    Abstract In this paper we study the qualitative dynamics of piecewise-smooth slow-fast systems (singularly perturbed systems) which are everywhere continuous. We consider phase space topology of systems with one-dimensional slow dynamics and one-dimensional fast dynamics. The slow manifold of the reduced system is formed by a piecewise-continuous curve, and the differentiability is lost across the switching surface. In the full system the slow manifold is no longer continuous, and there is an O(ɛ) discontinuity across the switching manifold, but the discontinuity cannot qualitatively alter system dynamics. Revealed phase space topology is used to unfold qualitative dynamics of planar slow-fast systems with an equilibrium point on the switching surface. In this case the local dynamics corresponds to so-called boundary-equilibrium bifurcations, and four qualitative phase portraits are uncovered. Our results are then used to investigate the dynamics of a box model of a thermohaline circulation, and the presence of a boundary-equilibrium bifurcation of a fold type is shown.
    Language English
    Publishing date 2011-06
    Publishing country United States
    Document type Journal Article
    ZDB-ID 1472677-4
    ISSN 1089-7682 ; 1054-1500
    ISSN (online) 1089-7682
    ISSN 1054-1500
    DOI 10.1063/1.3596708
    Database MEDical Literature Analysis and Retrieval System OnLINE

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  9. Article: The mathematics of motion camouflage.

    Glendinning, Paul

    Proceedings. Biological sciences

    2004  Volume 271, Issue 1538, Page(s) 477–481

    Abstract: Motion camouflage is a strategy whereby an aggressor moves towards a target while appearing stationary to the target except for the inevitable change in perceived size of the aggressor as it approaches. The strategy has been observed in insects, and ... ...

    Abstract Motion camouflage is a strategy whereby an aggressor moves towards a target while appearing stationary to the target except for the inevitable change in perceived size of the aggressor as it approaches. The strategy has been observed in insects, and mathematical models using discrete time or neural-network control have been used to simulate the behaviour. Here, the differential equations for motion camouflage are derived and some simple cases are analysed. These equations are easy to simulate numerically, and simulations indicate that motion camouflage is more efficient than the classical pursuit strategy ('move directly towards the target').
    MeSH term(s) Aggression/physiology ; Animals ; Models, Theoretical ; Movement/physiology ; Predatory Behavior
    Language English
    Publishing date 2004-03-07
    Publishing country England
    Document type Journal Article
    ZDB-ID 209242-6
    ISSN 1471-2954 ; 0962-8452 ; 0080-4649 ; 0950-1193
    ISSN (online) 1471-2954
    ISSN 0962-8452 ; 0080-4649 ; 0950-1193
    DOI 10.1098/rspb.2003.2622
    Database MEDical Literature Analysis and Retrieval System OnLINE

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    In stock of ZB MED Cologne/Königswinter

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  10. Article: Mechanisms of intermittent state transitions in a coupled heterogeneous oscillator model of epilepsy.

    Goodfellow, Marc / Glendinning, Paul

    Journal of mathematical neuroscience

    2013  Volume 3, Issue 1, Page(s) 17

    Abstract: We investigate the dynamic mechanisms underlying intermittent state transitions in a recently proposed neural mass model of epilepsy. A low dimensional model is constructed, which preserves two key features of the neural mass model, namely (i) coupling ... ...

    Abstract We investigate the dynamic mechanisms underlying intermittent state transitions in a recently proposed neural mass model of epilepsy. A low dimensional model is constructed, which preserves two key features of the neural mass model, namely (i) coupling between oscillators and (ii) heterogeneous proximity of these oscillators to a bifurcation between distinct limit cycles. We demonstrate that state transitions due to intermittency occur in the abstract model. This suggests that there is a general bifurcation mechanism responsible for this behaviour and that this is independent of the precise form of the evolution equations. Such abstractions of neural mass models allow a deeper insight into underlying dynamic and physiological mechanisms, and also allow the more efficient exploration of large scale brain dynamics in disease.
    Language English
    Publishing date 2013-08-14
    Publishing country Germany
    Document type Journal Article
    ISSN 2190-8567
    ISSN 2190-8567
    DOI 10.1186/2190-8567-3-17
    Database MEDical Literature Analysis and Retrieval System OnLINE

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