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Motivation Synchronization using a reachability method Symbolic reachability using Euler’s method Brusselator example Biped example Conclusion and P Guaranteed phase synchronization of hybrid oscillators using symbolic Euler’s method Jawher Jerray 1 Laurent Fribourg 2 Étienne André 3 1Université Sorbonne Paris Nord, LIPN, CNRS, UMR 7030, F-93430, Villetaneuse, France and 2Université Paris-Saclay, LSV, CNRS, ENS Paris-Saclay and 3Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France Tuesday 23 rd June, 2020 Jawher Jerray (LIPN) Guaranteed phase synchronization of hybrid oscillators using symbolic Euler’s method 1 / 23

Motivation Synchronization using a reachability method Symbolic reachability using Euler’s method Brusselator example Biped example Conclusion and P Outline a Motivation 1 Synchronization using a reachability method 2 3 Symbolic reachability using Euler’s method Brusselator example 4 Biped example 5 Conclusion and Perspectives 6 Jawher Jerray (LIPN) Guaranteed phase synchronization of hybrid oscillators using symbolic Euler’s method 2 / 23

Motivation Synchronization using a reachability method Symbolic reachability using Euler’s method Brusselator example Biped example Conclusion and P Motivation Dynamical systems: in which a function describes the time dependence of a point in a geometrical space. we only know certain observed or calculated states of its past or present state (causality). dynamical systems are everywhere. dynamical systems have a direct impact on human development. ⇒ The importance of studying: stability compared to the initial conditions behavior synchronization Jawher Jerray (LIPN) Guaranteed phase synchronization of hybrid oscillators using symbolic Euler’s method 3 / 23

Motivation Synchronization using a reachability method Symbolic reachability using Euler’s method Brusselator example Biped example Conclusion and P Motivation Dynamical systems: in which a function describes the time dependence of a point in a geometrical space. we only know certain observed or calculated states of its past or present state (causality). dynamical systems are everywhere. dynamical systems have a direct impact on human development. ⇒ The importance of studying: stability compared to the initial conditions behavior synchronization Solar System A flock of birds Schooling fish Jawher Jerray (LIPN) Guaranteed phase synchronization of hybrid oscillators using symbolic Euler’s method 3 / 23

Motivation Synchronization using a reachability method Symbolic reachability using Euler’s method Brusselator example Biped example Conclusion and P Synchronization Coordination of multiple events. Done within an acceptably brief period of time. The example of two suspended mechanical clocks done by Huygens. Two oscillators in phase after a lapse of time Original drawing of Christian Huygens in which he observed synchro- nization Jawher Jerray (LIPN) Guaranteed phase synchronization of hybrid oscillators using symbolic Euler’s method 4 / 23

Motivation Synchronization using a reachability method Symbolic reachability using Euler’s method Brusselator example Biped example Conclusion and P how to highlight the synchronization of dynamical system formally? Challenge of describing such systems because their equations are non-linear. To study non-linear systems, we often visualize them in a space of configurations (position and speed). Jawher Jerray (LIPN) Guaranteed phase synchronization of hybrid oscillators using symbolic Euler’s method 5 / 23

Motivation Synchronization using a reachability method Symbolic reachability using Euler’s method Brusselator example Biped example Conclusion and P Synchronization using a reachability method We consider a system composed of 2 subsystems governed by a system of differential equations (ODEs) of the form ˙ x ( t ) = f ( x ( t )) . The system of ODEs is thus of the form: � ˙ x 1 ( t ) = f 1 ( x 1 ( t ) , x 2 ( t )) (1) x 2 ( t ) = f 2 ( x 1 ( t ) , x 2 ( t )) ˙ with x ( t ) = ( x 1 ( t ) , x 2 ( t )) ∈ R m × R m , where m is the dimension of the state space of each subsystem. Jawher Jerray (LIPN) Guaranteed phase synchronization of hybrid oscillators using symbolic Euler’s method 6 / 23

Motivation Synchronization using a reachability method Symbolic reachability using Euler’s method Brusselator example Biped example Conclusion and P The set S i is thus char- acterized by a triple v i ( a i , b i , e i ) where a i and b i are the end points of its ord ( b i ) main diagonal, and e i the ord ( x i ) size of its horizontal base. We assume that the paral- lelogram S i is “long”, i.e.: (H) The width e i of S i is “small” w.r.t. ord ( a i ) f i = | ord ( b i ) − ord ( a i ) | . u i where (resp. ord ( a i ) ord ( b i ) ) denotes the ordinate of a i (resp. b i ). Given a point of x i ( s ) of S i ≡ ( a i , b i , e i ) at time s ( i = 1 , 2), we can thus define its phase φ [ x i ( s )] (in a “linearized” and “normalized” man- ner w.r.t. S i ) by: φ [ x i ( s )] = ( ord ( x i ( s )) − ord ( a i )) / ( ord ( b i ) − ord ( a i )) Jawher Jerray (LIPN) Guaranteed phase synchronization of hybrid oscillators using symbolic Euler’s method 7 / 23

Motivation Synchronization using a reachability method Symbolic reachability using Euler’s method Brusselator example Biped example Conclusion and P Synchronization using a reachability method v 1 v 2 ord ( b 1 ) ord ( b 2 ) ord ( x 1 ) ord ( x 2 ) ord ( a 1 ) ord ( a 2 ) u 1 u 2 at t = 0 v 1 v 2 ord ( b 1 ) ord ( b 2 ) ′ ′ ord ( x 1 ) ord ( x 2 ) ord ( a 1 ) ord ( a 2 ) u 1 u 2 at t ∈ [ kT , ( k + 1 ) T ] Scheme of S 1 (left) and S 2 (right) at t = 0 (top) and for some t ∈ [ kT , ( k + 1 ) T ) (bottom) Jawher Jerray (LIPN) Guaranteed phase synchronization of hybrid oscillators using symbolic Euler’s method 8 / 23

Motivation Synchronization using a reachability method Symbolic reachability using Euler’s method Brusselator example Biped example Conclusion and P Symbolic reachability using Euler’s method As a symbolic method, we use here the symbolic Euler’s method [LCDVCF17,Fri17] and we consider a subset under the form of “(double) ball” of the form B = B 1 × B 2 , where B i ⊂ R m ( i = 1 , 2) is a ball of the form B ( c i , r ) with c i ∈ R m ( centre ) and r a positive real ( radius ). In order to compute (an overapproximation of) the set of solutions starting at B 0 . We define for t ≥ 0: B euler ( t ) = B ( c 1 ( t ) , r ( t )) × B ( c 2 ( t ) , r ( t )) , where ( c 1 ( t ) , c 2 ( t )) ∈ R m × R m is the approximated value of solution x ( t ) of ˙ x = f ( x ) with initial condition x ( 0 ) = ( c 0 1 , c 0 2 ) given by Euler’s explicit method , and r ( t ) ≈ r 0 e λ t is the expanded radius using the one-sided Lipschitz constant λ . [LCDVCF17] A. Le Coënt et al. , “Control synthesis of nonlinear sampled switched systems using Euler’s method,” in SNR , (Apr. 22, 2017), ser. EPTCS, vol. 247, Uppsala, Sweden, 2017, pp. 18–33. DOI : ✶✵✳✹✷✵✹✴❊P❚❈❙✳✷✹✼✳✷ . [Fri17] L. Fribourg, “Euler’s method applied to the control of switched systems,” in FORMATS , (Sep. 5, 2017–Sep. 7, 2017), ser. LNCS, vol. 10419, Berlin, Germany: Springer, Sep. 2017, pp. 3–21. DOI : ✶✵✳✶✵✵✼✴✾✼✽✲✸✲✸✶✾✲✻✺✼✻✺✲✸❴✶ . [Online]. Available: ❤tt♣s✿✴✴❞♦✐✳♦r❣✴✶✵✳✶✵✵✼✴✾✼✽✲✸✲✸✶✾✲✻✺✼✻✺✲✸❴✶ . Jawher Jerray (LIPN) Guaranteed phase synchronization of hybrid oscillators using symbolic Euler’s method 9 / 23

Motivation Synchronization using a reachability method Symbolic reachability using Euler’s method Brusselator example Biped example Conclusion and P One-Sided Lipschitz (OSL) constant Definition The one-sided Lipschitz (OSL) constant for f on D , denoted by λ , is defined by � f ( y 1 ) − f ( y 2 ) , y 1 − y 2 � λ := sup , � y 1 − y 2 � 2 y 1 � = y 2 ∈ D where �· , ·� denotes the scalar product of two vectors of R n × R n , and � · � the Euclidean norm. Value of λ when λ ≤ 0 locally, indicates contractive zone when λ ≥ 0 locally, indicates expansive zone Jawher Jerray (LIPN) Guaranteed phase synchronization of hybrid oscillators using symbolic Euler’s method 10 / 23