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Finding non-polynomial positive invariants and lyapunov functions for polynomial systems through Darboux polynomials

Eric Goubault, Jacques-Henri Jourdan, Sylvie Putot, Sriram Sankaranarayanan
2014 2014 American Control Conference  
As a result, numerous techniques have been developed to search for polynomial differential variants that yield positive invariants and polynomial Lyapunov functions that prove stability, for systems defined  ...  In this paper, we focus on finding positive invariants and Lyapunov functions to establish reachability and stability properties, respectively, of polynomial ordinary differential equations (ODEs).  ...  Acknowledgments The authors would like to thank the anonymous reviewers for their careful reading and insightful comments.  ... 
doi:10.1109/acc.2014.6859330 dblp:conf/amcc/GoubaultJPS14 fatcat:6t5lgh2pjbd5dkpr6ppo3pruei

Pegasus: sound continuous invariant generation

Andrew Sogokon, Stefan Mitsch, Yong Kiam Tan, Katherine Cordwell, André Platzer
2021 Formal methods in system design  
for hybrid systems.  ...  AbstractContinuous invariants are an important component in deductive verification of hybrid and continuous systems.  ...  Acknowledgements The authors would like to thank the guest editor for handling this article, the anonymous reviewers for providing value feedback, and FM 2019 for the special issue invitation.  ... 
doi:10.1007/s10703-020-00355-z fatcat:6bzcbvszcfbedifbtuezu6xzuq

Pegasus: Sound Continuous Invariant Generation [article]

Andrew Sogokon, Stefan Mitsch, Yong Kiam Tan, Katherine Cordwell, André Platzer
2020 arXiv   pre-print
for hybrid systems.  ...  Continuous invariants are an important component in deductive verification of hybrid and continuous systems.  ...  Acknowledgements The authors would like to thank the anonymous reviewers for providing valuable feedback and FM 2019 for the special issue invitation.  ... 
arXiv:2005.09348v2 fatcat:4bwfemvuu5g7ln25bcnn74yslq

Automated and Formal Synthesis of Neural Barrier Certificates for Dynamical Models [article]

Andrea Peruffo, Daniele Ahmed, Alessandro Abate
2020 arXiv   pre-print
We compare the approach against state-of-the-art techniques, over polynomial and non-polynomial dynamical models: the outcomes show that we can synthesise sound BCs up to two orders of magnitude faster  ...  We introduce an automated, formal, counterexample-based approach to synthesise Barrier Certificates (BC) for the safety verification of continuous and hybrid dynamical models.  ...  and sound Lyapunov function synthesis: in [26] Lyapunov functions are soundly found within parametric templates, by constructing a system of linear inequality constraints over unknown coefficients  ... 
arXiv:2007.03251v2 fatcat:voq7prf6fzhslodibxnt3httqq

On the Dynamics of a Model with Coexistence of Three Attractors: A Point, a Periodic Orbit and a Strange Attractor

Jaume Llibre, Claudia Valls
2017 Mathematical physics, analysis and geometry  
We study using analytic tools the dynamics of such system. We describe its global dynamics at infinity, and show that it has no Darboux first integrals.  ...  For a dynamical system described by a set of autonomous differential equations, an attractor can be either a point, or a periodic orbit, or even a strange attractor.  ...  The second author is partially supported by FCT/Portugal through UID/MAT/04459/2013.  ... 
doi:10.1007/s11040-017-9240-6 fatcat:suduzsle4fc35d53te2rehtzzy

Automated and Formal Synthesis of Neural Barrier Certificates for Dynamical Models [chapter]

Andrea Peruffo, Daniele Ahmed, Alessandro Abate
2021 Lecture Notes in Computer Science  
We compare the approach against state-of-the-art techniques, over polynomial and non-polynomial dynamical models: the outcomes show that we can synthesise sound BCs up to two orders of magnitude faster  ...  AbstractWe introduce an automated, formal, counterexample-based approach to synthesise Barrier Certificates (BC) for the safety verification of continuous and hybrid dynamical models.  ...  and sound Lyapunov function synthesis: in [27] Lyapunov functions are soundly found within parametric templates, by constructing a system of linear inequality constraints over unknown coefficients  ... 
doi:10.1007/978-3-030-72016-2_20 fatcat:bpbllc6omzedhowy3gvwextkjq

Integrability of complex planar systems with homogeneous nonlinearities

Brigita Ferčec, Jaume Giné, Valery G. Romanovski, Victor F. Edneral
2016 Journal of Mathematical Analysis and Applications  
q y p+1 ) = Q, we can always find function Ψ(x, y; a pq , b qp ) = xy + ∞ s=3 s j=0 v j,s−j x j y s−j such that ∂Ψ ∂x P + ∂Ψ ∂y Q = g 11 (xy) 2 + g 22 (xy) 3 + · · · , and g 11 , g 22 , . . . are polynomials  ...  Ferčec (FE,CAMTP) Integrability of complex planar systems 20 / 50 We now briefly present the method of Darboux integration for proving the existence of first integrals and integrating factors for polynomial  ...  Since k 1 and k 2 should be different from 0, we add to the polynomials defining the system given above the polynomials 1 − αk 1 and 1 − βk 2 obtaining an ideal which we denote by I.  ... 
doi:10.1016/j.jmaa.2015.09.037 fatcat:s7upfbq2yjfkjhq6cwrms3m434

Integrability and Limit Cycles via First Integrals

Jaume Llibre
2021 Symmetry  
First, we study how to compute first integrals for polynomial differential systems using the so-called Darboux theory of integrability.  ...  Furthermore, second, we show how to use the existence of first integrals for finding limit cycles in piecewise differential systems.  ...  Now, in order to find a first integral for the differential system (14) we applied the Darboux theory of integrability.  ... 
doi:10.3390/sym13091736 fatcat:vyyyq7irknhkng6x4kjfidq6g4

A survey on the inverse integrating factor [article]

Isaac A. García, Maite Grau
2009 arXiv   pre-print
The relation between limit cycles of planar differential systems and the inverse integrating factor was first shown in an article of Giacomini, Llibre and Viano appeared in 1996.  ...  From that moment on, many research articles are devoted to the study of the properties of the inverse integrating factor and its relation with limit cycles and their bifurcations.  ...  Acknowledgements We would like to thank Professor Héctor Giacomini, from Université de Tours (France), for his useful comments on this survey and for encouraging us to study the inverse integrating factor  ... 
arXiv:0903.0941v1 fatcat:xd2yk6ulonanpiylsow7bhkyou

A Survey on the Inverse Integrating Factor

Isaac A. García, Maite Grau
2010 Qualitative Theory of Dynamical Systems  
The relation between limit cycles of planar differential systems and the inverse integrating factor was first shown in an article of Giacomini, Llibre and Viano appeared in 1996.  ...  From that moment on, many research articles are devoted to the study of the properties of the inverse integrating factor and its relation with limit cycles and their bifurcations.  ...  Acknowledgements We would like to thank Professor Héctor Giacomini, from Université de Tours (France), for his useful comments on this survey and for encouraging us to study the inverse integrating factor  ... 
doi:10.1007/s12346-010-0023-8 fatcat:mcnonpkmgrc2paxqqezxewmziy

On the Dynamics of the Unified Chaotic System Between Lorenz and Chen Systems

Jaume Llibre, Ana Rodrigues
2015 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering  
We shall describe its global dynamics at infinity, and for two special values of the parameter a we also can describe the global dynamics in the whole R 3 using the invariant algebraic surfaces of the  ...  A one-parameter family of differential systems that bridges the gap between the Lorenz and the Chen systems was proposed by Lu, Chen, Cheng and Celikovsy.  ...  318999 and 316338, FEDER-UNAB 10-4E-378.  ... 
doi:10.1142/s0218127415501229 fatcat:46hbytwnwzgbtp7dymnb7rncsq

The problem of center for resonant singular points of polynomial vector fields

Henryk Żołądek
1997 Journal of Differential Equations  
We generalize the notion of"center" to the case of a p : &q resonant singular point of a polynomial vector field in C 2 and to some other situations (resonant node, saddle-node, non-elementary singular  ...  point of vector field and resonant fixed point of one-dimensional complex diffeomorphism).  ...  He thanks the University for its hospitality and support. Discussions with J.-P. Franc oise and A. Fronville were very fruitful.  ... 
doi:10.1006/jdeq.1997.3260 fatcat:gvrzjvh7jnhchaarz2njwbwhuq

Nondegenerate linearizable centres of complex planar quadratic and symmetric cubic systems in $\mathbb{C}^2$

C. Christopher, C. Rousseau
2001 Publicacions matemàtiques  
We find necessary and sufficient conditions for linearizability for the class of complex quadratic systems and for the class of complex cubic systems symmetric with respect to a centre.  ...  The sufficiency of these conditions is shown by exhibiting explicitly a linearizing change of coordinates, either of Darboux type or a generalization of it.  ...  The authors are grateful to Dana Schlomiuk for helpful discussions. They thank the referee for pointing to them the theorem of Cartan-CarathThetaodory.  ... 
doi:10.5565/publmat_45101_04 fatcat:i77ri5xd6zg2nhd44yzsmxmoye

Linearizability conditions of a time-reversible quartic-like system

Xingwu Chen, Wentao Huang, Valery G. Romanovski, Weinian Zhang
2011 Journal of Mathematical Analysis and Applications  
Applying this reduction, we find some linearizability conditions for a time-reversible quartic-like complex system and derive from them conditions of isochronous center for the corresponding real system  ...  We give a reduction of linearizability problem of such non-polynomial systems to the problem of polynomial systems.  ...  Acknowledgments The authors are grateful to the Associated Editor and the two Reviewers for their helpful suggestions and comments. This work was supported by the  ... 
doi:10.1016/j.jmaa.2011.05.018 fatcat:uf7dwdaazvci7ouvr6opefpmsi

Convex computation of extremal invariant measures of nonlinear dynamical systems and Markov processes [article]

Milan Korda, Igor Mezic
2020 arXiv   pre-print
We propose a convex-optimization-based framework for computation of invariant measures of polynomial dynamical systems and Markov processes, in discrete and continuous time.  ...  The presented framework, where a convex functional is minimized or maximized among all invariant measures, can be seen as a generalization of and a computational method to carry out the so called ergodic  ...  method for computation of invariant measures for continuous and discrete time deterministic and stochastic systems.  ... 
arXiv:1807.08956v2 fatcat:fcplvfltpfhfbhggvfmli7el7i
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