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Geometric Properties of Isostables and Basins of Attraction of Monotone Systems [article]

Aivar Sootla, Alexandre Mauroy
2017 arXiv   pre-print
In this paper, we study geometric properties of basins of attraction of monotone systems. Our results are based on a combination of monotone systems theory and spectral operator theory.  ...  This is a strong geometric property which simplifies the computation of isostables.  ...  Geometric Properties of Isostables and Basins of Attraction of Monotone Systems Aivar Sootla and Alexandre Mauroy Abstract-In this paper, we study geometric properties of basins of attraction of monotone  ... 
arXiv:1705.02853v1 fatcat:vfgxfalkcfcdrniv7oxpafuxay

Properties of Isostables and Basins of Attraction of Monotone Systems [article]

Aivar Sootla, Alexandre Mauroy
2016 arXiv   pre-print
In this paper, we investigate geometric properties of monotone systems by studying their isostables and basins of attraction.  ...  Our results generalize the celebrated Perron-Frobenius theorem to the nonlinear case and allow us to derive geometric properties of isostables and basins of attraction.  ...  CONCLUSION In this paper, we study geometric properties of monotone systems such as properties of isostables and basins of attraction.  ... 
arXiv:1510.01153v2 fatcat:zd6nvcvuxrhnjiiaeerjlcuh6m

Optimal control formulation of pulse-based control using Koopman operator

Aivar Sootla, Alexandre Mauroy, Damien Ernst
2018 Automatica  
It was recently shown that, for monotone bistable systems, this problem admits easy-to-implement open-loop solutions in terms of temporal pulses (i.e., step functions of fixed length and fixed magnitude  ...  We show that this problem can be solved using a static optimization problem in the case of monotone systems.  ...  Assume now that the system is monotone with two exponentially stable equilibria x * and x • and basins of attraction B(x * ) and B(x • ), respectively.  ... 
doi:10.1016/j.automatica.2018.01.036 fatcat:36avoe4ixfcyzlsrteypdcvmka

Operator-Theoretic Characterization of Eventually Monotone Systems [article]

Aivar Sootla, Alexandre Mauroy
2017 arXiv   pre-print
In order to provide a characterization, we drew inspiration from the results for linear systems, where eventually monotone (positive) systems are studied using the spectral properties of the system (i.e  ...  Perron-Frobenius property).  ...  We conclude this subsection by a corollary of Theorem 1, which describes the geometric properties of the system in the basin of attraction.  ... 
arXiv:1510.01149v3 fatcat:jr6wriw7sbftliffy24m7f2xrm

Global computation of phase-amplitude reduction for limit-cycle dynamics

A. Mauroy, I. Mezić
2018 Chaos  
basin of attraction of the limit cycle.  ...  We illustrate the method by computing the so-called isostables of limit cycles in two, three, and four-dimensional state spaces, as well as their responses to strong external inputs.  ...  We note that, in the case of planar systems, this is an exact (non-reduced) representation of the system in the basin of attraction of the limit cycle. C.  ... 
doi:10.1063/1.5030175 pmid:30070491 fatcat:mqfchfnoibfgxnrizvppqodody

An operator-theoretic viewpoint to non-smooth dynamical systems: Koopman analysis of a hybrid pendulum [article]

Nithin Govindarajan, Hassan Arbabi, Louis van Blargian, Timothy Matchen, Emma Tegling, Igor Mezić
2016 arXiv   pre-print
A connection is drawn between these two interpretations of a dynamical system by means of establishing a link between the spectral properties of the Koopman operator and the geometric properties in the  ...  The pendulum is modeled as a hybrid automaton and is analyzed from both a geometric perspective and the formalism carried out by Koopman operator theory.  ...  Basin of attraction The basin of a specific limit cycle p 1 can be found by repeatedly computing the pre-images of the map (13) .  ... 
arXiv:1608.08734v1 fatcat:a6nommyskbeztcledufejdg2bu

An operator-theoretic approach to differential positivity

A. Mauroy, F. Forni, R. Sepulchre
2015 2015 54th IEEE Conference on Decision and Control (CDC)  
In this paper, we investigate the general connections between the (geometric) properties of differentially positive systems and the (spectral) properties of the Koopman operator.  ...  In particular, we obtain converse results for differential positivity, showing for instance that any hyperbolic limit cycle is differentially positive in its basin of attraction.  ...  is strictly differentially positive in the basin of attraction B(Γ) ⊆ R n of Γ.  ... 
doi:10.1109/cdc.2015.7403327 dblp:conf/cdc/MauroyFS15 fatcat:7pvofe6jpjgu5edv5xmheemveu

An operator-theoretic approach to differential positivity [article]

A. Mauroy, F. Forni, R. Sepulchre
2015 arXiv   pre-print
In this paper, we investigate the general connections between the (geometric) properties of differentially positive systems and the (spectral) properties of the Koopman operator.  ...  In particular, we obtain converse results for differential positivity, showing for instance that any hyperbolic limit cycle is differentially positive in its basin of attraction.  ...  is strictly differentially positive in the basin of attraction B(Γ) ⊆ R n of Γ.  ... 
arXiv:1504.01548v1 fatcat:yngajdzecjdbhfndzziancnuj4

A geometric approach to Phase Response Curves and its numerical computation through the parameterization method [article]

Gemma Huguet, Alberto Pérez-Cervera, Tere M-Seara
2018 arXiv   pre-print
Moreover, we link the existence properties of this invariant curve as the amplitude of the perturbation is increased with changes in the PRC waveform and with the geometry of isochrons.  ...  The underlying idea relies on the construction of a periodic system whose corresponding stroboscopic map has an invariant curve.  ...  We also acknowledge the use of the UPC Dynamical Systems group's cluster for research computing (https://dynamicalsystems.upc.edu/en/computing/).  ... 
arXiv:1809.07318v1 fatcat:xh236iot2nb2po4opyufdrw5fe

Sensitivity analysis of oscillator models in the space of phase-response curves: Oscillators as open systems [article]

Pierre Sacré, Rodolphe Sepulchre
2014 arXiv   pre-print
Oscillator models are central to the study of system properties such as entrainment or synchronization. Due to their nonlinear nature, few system-theoretic tools exist to analyze those models.  ...  The proposed theoretical and numerical analysis tools are illustrated on several system-theoretic questions and models arising in the biology of cellular rhythms.  ...  The notion of the asymptotic phase variable can be extended to any nonzero-input trajectory φ(·, x0 , u(·)) in the basin of attraction of γ.  ... 
arXiv:1206.4144v3 fatcat:54pqyy7m5fdavlrbe7kgcqa7zi

Koopman Operator Spectrum for Random Dynamical Systems [article]

Nelida Črnjarić-Žic, Senka Maćešić, Igor Mezić
2019 arXiv   pre-print
In this paper we consider the Koopman operator associated with the discrete and the continuous time random dynamical system (RDS).  ...  We provide results that characterize the spectrum and the eigenfunctions of the stochastic Koopman operator associated with different types of linear RDS.  ...  Acknowledgment This research has been supported by the DARPA Contract HR0011-16-C-0116 "On A Data-Driven, Operator-Theoretic Framework for Space-Time Analysis of Process Dynamics" and AFOSR grants FA9550  ... 
arXiv:1711.03146v2 fatcat:2yoj2kuw7jh6zne7gr33u4tsdm

27th Annual Computational Neuroscience Meeting (CNS*2018): Part One

2018 BMC Neuroscience  
Acknowledgement Neural Engineering System Design (NESD) program from the Defense Advanced Research Projects Agency (DARPA).  ...  Acknowledgement Central fund of Finnish Cultural Foundation and Finnish Brain Foundation sr. Acknowledgement German Ministry of Research and Education Grant 01GQ1508.  ...  We show that this phenomenon always occurs when the initial conditions lie in the basin of attraction of the stable limit cycle.  ... 
doi:10.1186/s12868-018-0452-x pmid:30373544 pmcid:PMC6205781 fatcat:xv7pgbp76zbdfksl545xof2vzy

Frustration in biomolecules

Diego U. Ferreiro, Elizabeth A. Komives, Peter G. Wolynes
2014 Quarterly Reviews of Biophysics (print)  
We first review the notion of frustration from the areas of abstract logic and its uses in simple condensed matter systems.  ...  Most of these inanimate systems are polymers that compute their own structures and dynamics using as input seemingly random character strings of their sequence, following which they coalesce and perform  ...  The correlations between energy levels predicted by the polymer theory certainly exist as they allow states to be grouped together into basins of attraction which can interconvert without significant barriers  ... 
doi:10.1017/s0033583514000092 pmid:25225856 pmcid:PMC4256721 fatcat:xgc25cb5abfitf3k67mwzspxnq