메뉴 건너뛰기




Volumn 1, Issue 2, 2014, Pages 91-106

Synchronization of diffusively-connected nonlinear systems: Results based on contractions with respect to general norms

Author keywords

Behavioral science; Consensus; Contraction of nonlinear systems; Convergence; Eigenvalues and eigenfunctions; Jacobian matrices; Laplace equations; Linear matrix inequalities; Nonlinear systems; Synchronization

Indexed keywords

BEHAVIORAL RESEARCH; DIFFERENTIAL EQUATIONS; DYNAMICAL SYSTEMS; EIGENVALUES AND EIGENFUNCTIONS; JACOBIAN MATRICES; LAPLACE EQUATION; LINEAR MATRIX INEQUALITIES; NONLINEAR DYNAMICAL SYSTEMS; NONLINEAR SYSTEMS; SYNCHRONIZATION;

EID: 84922746020     PISSN: None     EISSN: 23274697     Source Type: Journal    
DOI: 10.1109/TNSE.2015.2395075     Document Type: Article
Times cited : (50)

References (63)
  • 1
    • 0028457501 scopus 로고
    • Synchronous oscillations in neuronal systems: Mechanisms and functions
    • C. M. Gray, "Synchronous oscillations in neuronal systems: Mechanisms and functions," J. Comput. Neurosci., vol. 1, pp. 11-38, 1994.
    • (1994) J. Comput. Neurosci. , vol.1 , pp. 11-38
    • Gray, C.M.1
  • 2
    • 0031796185 scopus 로고    scopus 로고
    • Diffusively coupled bursters: Effects of cell heterogeneity
    • G. de Vries, A. Sherman, and H.-R. Zhu, "Diffusively coupled bursters: Effects of cell heterogeneity," Bull. Math. Biol., vol. 60, pp. 1167-1199, 1998.
    • (1998) Bull. Math. Biol. , vol.60 , pp. 1167-1199
    • De Vries, G.1    Sherman, A.2    Zhu, H.-R.3
  • 3
    • 0019156722 scopus 로고
    • Human sleep: Its duration and organization depend on its circadian phase
    • C. A. Czeisler, E. D. Weitzman, M. C. Moore-Ede, J. C. Zimmerman, and R. S. Knauer, "Human sleep: Its duration and organization depend on its circadian phase," Science, vol. 210, pp. 1264-1267, 1980.
    • (1980) Science , vol.210 , pp. 1264-1267
    • Czeisler, C.A.1    Weitzman, E.D.2    Moore-Ede, M.C.3    Zimmerman, J.C.4    Knauer, R.S.5
  • 4
    • 0023791327 scopus 로고
    • Emergence of organized bursting in clusters of pancreatic beta-cells by channel sharing
    • A. Sherman, J. Rinzel, and J. Keizer, "Emergence of organized bursting in clusters of pancreatic beta-cells by channel sharing," Biophys. J., vol. 54, pp. 411-425, 1988.
    • (1988) Biophys. J. , vol.54 , pp. 411-425
    • Sherman, A.1    Rinzel, J.2    Keizer, J.3
  • 5
    • 0034223131 scopus 로고    scopus 로고
    • Dynamics of spiking neurons with electrical coupling
    • C. C. Chow and N. Kopell, "Dynamics of spiking neurons with electrical coupling," Neural Comput., vol. 12, pp. 1643-1678, 2000.
    • (2000) Neural Comput. , vol.12 , pp. 1643-1678
    • Chow, C.C.1    Kopell, N.2
  • 6
    • 0038701723 scopus 로고    scopus 로고
    • Dynamics of spiking neurons connected by both inhibitory and electrical coupling
    • T. J. Lewis and J. Rinzel, "Dynamics of spiking neurons connected by both inhibitory and electrical coupling," J. Comput. Neurosci., vol. 14, pp. 283-309, 2003.
    • (2003) J. Comput. Neurosci. , vol.14 , pp. 283-309
    • Lewis, T.J.1    Rinzel, J.2
  • 7
    • 0010160412 scopus 로고
    • Synchronous flashing of fireflies
    • H. M. Smith, "Synchronous flashing of fireflies," Science, vol. 82, pp. 151-152, 1935.
    • (1935) Science , vol.82 , pp. 151-152
    • Smith, H.M.1
  • 8
    • 0027519363 scopus 로고
    • Coupled oscillators and biological synchronization
    • S. H. Strogatz and I. Stewart, "Coupled oscillators and biological synchronization," Sci. Am., vol. 269, pp. 102-109, 1993.
    • (1993) Sci. Am. , vol.269 , pp. 102-109
    • Strogatz, S.H.1    Stewart, I.2
  • 11
    • 34247397566 scopus 로고    scopus 로고
    • Flocking in fixed and switching networks
    • May
    • H. Tanner, A. Jadbabaie, and G. J. Pappas, "Flocking in fixed and switching networks," IEEE Trans. Autom. Control, vol. 52, no. 5, pp. 863-868, May 2007.
    • (2007) IEEE Trans. Autom. Control , vol.52 , Issue.5 , pp. 863-868
    • Tanner, H.1    Jadbabaie, A.2    Pappas, G.J.3
  • 12
    • 70349863068 scopus 로고    scopus 로고
    • Synchronization of complex dynamical networks with switching topology: A switched system point of view
    • J. Zhao, D. J. Hill, and T. Liu, "Synchronization of complex dynamical networks with switching topology: A switched system point of view," Automatica, vol. 45, no. 11, pp. 2502-2511, 2009.
    • (2009) Automatica , vol.45 , Issue.11 , pp. 2502-2511
    • Zhao, J.1    Hill, D.J.2    Liu, T.3
  • 13
    • 80053619900 scopus 로고    scopus 로고
    • Clustering in diffusively coupled networks
    • W. Xia and M. Cao, "Clustering in diffusively coupled networks," Automatica, vol. 47, no. 11, pp. 2395-2405, 2011.
    • (2011) Automatica , vol.47 , Issue.11 , pp. 2395-2405
    • Xia, W.1    Cao, M.2
  • 14
    • 84902182078 scopus 로고    scopus 로고
    • Synchronization in complex networks of phase oscillators: A survey
    • F. Dorfler, and F. Bullo, "Synchronization in complex networks of phase oscillators: A survey," Automatica, vol. 50, no. 6, pp. 1539-1564, 2014.
    • (2014) Automatica , vol.50 , Issue.6 , pp. 1539-1564
    • Dorfler, F.1    Bullo, F.2
  • 17
    • 12244287257 scopus 로고
    • Metric properties of differential equations
    • D. C. Lewis, "Metric properties of differential equations," Amer. J. Math., vol. 71, pp. 294-312, 1949.
    • (1949) Amer. J. Math. , vol.71 , pp. 294-312
    • Lewis, D.C.1
  • 18
    • 0012184302 scopus 로고
    • On stability in the large for systems of ordinary differential equations
    • P. Hartman, "On stability in the large for systems of ordinary differential equations," Can. J. Math., vol. 13, pp. 480-492, 1961.
    • (1961) Can. J. Math. , vol.13 , pp. 480-492
    • Hartman, P.1
  • 19
    • 0003304056 scopus 로고
    • Stability and error bounds in the numerical integration of ordinary differential equations
    • G. Dahlquist, "Stability and error bounds in the numerical integration of ordinary differential equations," Kungl. Tekn. H €ogsk. Handl. Stockholm. No., vol. 130, p. 87, 1959.
    • (1959) Kungl. Tekn. H [[Ampi]]euro;ogsk. Handl. Stockholm. No. , vol.130 , pp. 87
    • Dahlquist, G.1
  • 20
    • 0041048294 scopus 로고
    • On the dissipativity of a certain non-linear system of differential equations I
    • B. P. Demidovič, "On the dissipativity of a certain non-linear system of differential equations. I," Vestnik Moskov. Univ. Ser. I Mat. Meh., vol. 1961, no. 6, pp. 19-27, 1961.
    • (1961) Vestnik Moskov. Univ. Ser. I Mat. Meh. , vol.1961 , Issue.6 , pp. 19-27
    • Demidovič, B.P.1
  • 21
    • 84922702286 scopus 로고
    • Lektsii po matematicheskoi teorii ustoichivosti
    • Nauka, Moscow
    • B. P. Demidovič, Lektsii po matematicheskoi teorii ustoichivosti, Izdat. Nauka, Moscow, 1967.
    • (1967) Izdat
    • Demidovič, B.P.1
  • 22
    • 0003842823 scopus 로고
    • (series Publications of the Mathematical Society of Japan, No. 9).Tokyo, Japan: The Mathematical Society of Japan
    • T. Yoshizawa, Stability Theory by Liapunov's Second Method (series Publications of the Mathematical Society of Japan, No. 9). Tokyo, Japan: The Mathematical Society of Japan, 1966.
    • (1966) Stability Theory by Liapunov's Second Method
    • Yoshizawa, T.1
  • 24
    • 0032094219 scopus 로고    scopus 로고
    • On contraction analysis for nonlinear systems
    • W. Lohmiller and J. J. E. Slotine, "On contraction analysis for nonlinear systems," Automatica, vol. 34, pp. 683-696, 1998.
    • (1998) Automatica , vol.34 , pp. 683-696
    • Lohmiller, W.1    Slotine, J.J.E.2
  • 25
    • 0034013767 scopus 로고    scopus 로고
    • Nonlinear process control using contraction theory
    • W. Lohmiller and J. Slotine, "Nonlinear process control using contraction theory," AIChe J., vol. 46, pp. 588-596, 2000.
    • (2000) AIChe J. , vol.46 , pp. 588-596
    • Lohmiller, W.1    Slotine, J.2
  • 26
    • 12344327544 scopus 로고    scopus 로고
    • On partial contraction analysis for coupled nonlinear oscillators
    • W. Wang and J. J. E. Slotine, "On partial contraction analysis for coupled nonlinear oscillators," Biol. Cybern., vol. 92, pp. 38-53, 2005.
    • (2005) Biol. Cybern. , vol.92 , pp. 38-53
    • Wang, W.1    Slotine, J.J.E.2
  • 27
    • 67349182327 scopus 로고    scopus 로고
    • A contraction theory approach to stochastic incremental stability
    • Apr.
    • Q. C. Pham, N. Tabareau, and J. Slotine, "A contraction theory approach to stochastic incremental stability," IEEE Trans. Autom. Control, vol. 54, no. 4, pp. 816-820, Apr. 2009.
    • (2009) IEEE Trans. Autom. Control , vol.54 , Issue.4 , pp. 816-820
    • Pham, Q.C.1    Tabareau, N.2    Slotine, J.3
  • 28
    • 80054992489 scopus 로고    scopus 로고
    • Symmetries, stability, and control in nonlinear systems and networks
    • Oct. [Online]
    • G. Russo and J.-J. E. Slotine. (2011, Oct.). Symmetries, stability, and control in nonlinear systems and networks. Phys. Rev. E [Online]. 84, p. 041929. Available: http://link.aps.org/doi/ 10.1103/PhysRevE.84.041929
    • (2011) Phys. Rev. E , vol.84 , pp. 041929
    • Russo, G.1    Slotine, J.-J.E.2
  • 32
    • 33749001694 scopus 로고    scopus 로고
    • The logarithmic norm. History and modern theory
    • G. Soderlind, "The logarithmic norm. history and modern theory," BIT, vol. 46, no. 3, pp. 631-652, 2006.
    • (2006) BIT , vol.46 , Issue.3 , pp. 631-652
    • Soderlind, G.1
  • 33
    • 84988221155 scopus 로고    scopus 로고
    • Contraction methods for nonlinear systems: A brief introduction and some open problems
    • Z. Aminzare, and E. Sontag, "Contraction methods for nonlinear systems: A brief introduction and some open problems," in Proc. IEEE Conf. Dec. Control, 2014, pp. 3835-3847.
    • (2014) Proc. IEEE Conf. Dec. Control , pp. 3835-3847
    • Aminzare, Z.1    Sontag, E.2
  • 34
    • 79956192798 scopus 로고    scopus 로고
    • Certifying spatially uniform behavior in reaction-diffusion pde and compartmental ode systems
    • M. Arcak, "Certifying spatially uniform behavior in reactiondiffusion pde and compartmental ode systems," Automatica, vol. 47, no. 6, pp. 1219-1229, 2011.
    • (2011) Automatica , vol.47 , Issue.6 , pp. 1219-1229
    • Arcak, M.1
  • 35
    • 27844596643 scopus 로고    scopus 로고
    • Contraction analysis of nonlinear distributed systems
    • W. Lohmiller and J. Slotine, "Contraction analysis of nonlinear distributed systems," Int. J. Control, vol. 78, pp. 678-688, 2005.
    • (2005) Int. J. Control , vol.78 , pp. 678-688
    • Lohmiller, W.1    Slotine, J.2
  • 36
    • 33845599202 scopus 로고    scopus 로고
    • Stable concurrent synchronization in dynamic system networks
    • Jan. [Online]
    • Q.-C. Pham and J.-J. Slotine. (2007, Jan.). Stable concurrent synchronization in dynamic system networks. Neural Netw. [Online] 20(1), pp. 62-77. Available: http://dx.doi.org/10.1016/j.neunet. 2006.07.008
    • (2007) Neural Netw , vol.20 , Issue.1 , pp. 62-77
    • Pham, Q.-C.1    Slotine, J.-J.2
  • 37
    • 62749205064 scopus 로고    scopus 로고
    • Contraction theory and master stability function: Linking two approaches to study synchronization of complex networks
    • Feb.
    • G. Russo and M. di Bernardo, "Contraction theory and master stability function: Linking two approaches to study synchronization of complex networks," IEEE Trans. Circuits Syst. II, Exp. Briefs., vol. 56, no. 2, pp. 177-181, Feb. 2009.
    • (2009) IEEE Trans. Circuits Syst. II, Exp. Briefs. , vol.56 , Issue.2 , pp. 177-181
    • Russo, G.1    Di Bernardo, M.2
  • 38
    • 84922706457 scopus 로고    scopus 로고
    • Guaranteeing spatial uniformity in reaction-diffusion systems using weighted L2 -norm contractions
    • V. Kulkarni, G.-B. Stan, and K. Raman, Eds. New York, NY, USA: Springer-Verlag
    • Z. Aminzare, Y. Shafi, M. Arcak, and E. Sontag, "Guaranteeing spatial uniformity in reaction-diffusion systems using weighted l2 -norm contractions," in A Systems Theoretic Approach to Systems and Synthetic Biology I: Models and System Characterizations, V. Kulkarni, G.-B. Stan, and K. Raman, Eds. New York, NY, USA: Springer-Verlag, 2014, pp. 73-101.
    • (2014) A Systems Theoretic Approach to Systems and Synthetic Biology I: Models and System Characterizations , pp. 73-101
    • Aminzare, Z.1    Shafi, Y.2    Arcak, M.3    Sontag, E.4
  • 39
    • 34547182357 scopus 로고    scopus 로고
    • Synchronization in time-varying networks: A matrix measure approach
    • Jul. [Online]
    • M. Chen. (2007, Jul.). Synchronization in time-varying networks: A matrix measure approach," Phys. Rev. E [Online]. 76, p. 016104. Available: http://link.aps.org/doi/10.1103/PhysRevE.76.016104
    • (2007) Phys. Rev. E , vol.76 , pp. 016104
    • Chen, M.1
  • 40
    • 84866747290 scopus 로고    scopus 로고
    • On contraction of piecewise smooth dynamical systems
    • G. Russo and M. di Bernardo, "On contraction of piecewise smooth dynamical systems," in Proc. IFAC World Congress, vol. 18, 2011, pp. 13 299-13 304.
    • (2011) Proc. IFAC World Congress , vol.18 , pp. 13299-13304
    • Russo, G.1    Di Bernardo, M.2
  • 41
    • 84911091329 scopus 로고    scopus 로고
    • Contraction analysis for a class of non differentiable systems with applications to stability and setwork synchronization
    • M. di Bernardo, D. Liuzza, and G. Russo, "Contraction analysis for a class of non differentiable systems with applications to stability and setwork synchronization," SIAM J. Control Optim, vol. 52, pp. 3203-3227, 2014.
    • (2014) SIAM J. Control Optim , vol.52 , pp. 3203-3227
    • Di Bernardo, M.1    Liuzza, D.2    Russo, G.3
  • 42
    • 84988042225 scopus 로고    scopus 로고
    • Using different logarithmic norms to show synchronization of diffusively connected systems
    • Z. Aminzare and E. Sontag, "Using different logarithmic norms to show synchronization of diffusively connected systems," in Proc. IEEE Conf. Dec. Control, 2014, pp. 6086-6091.
    • (2014) Proc. IEEE Conf. Dec. Control , pp. 6086-6091
    • Aminzare, Z.1    Sontag, E.2
  • 43
    • 39149113411 scopus 로고    scopus 로고
    • Modular cell biology: Retroactivity and insulation
    • D. Del Vecchio, A.Ninfa, and E. Sontag, "Modular cell biology: Retroactivity and insulation," NatureMol. Syst. Biol., vol. 4, p. 161, 2008.
    • (2008) Nature Mol.Syst. Biol. , vol.4 , pp. 161
    • Del Vecchio, D.1    Ninfa, A.2    Sontag, E.3
  • 44
    • 77954596479 scopus 로고    scopus 로고
    • Global entrainment of transcriptional systems to periodic inputs
    • G. Russo, M. di Bernardo, and E. Sontag, "Global entrainment of transcriptional systems to periodic inputs," PLoS Comput. Biol., vol. 6, p. e1000739, 2010.
    • (2010) PLoS Comput. Biol. , vol.6
    • Russo, G.1    Di Bernardo, M.2    Sontag, E.3
  • 46
    • 84873692291 scopus 로고    scopus 로고
    • Logarithmic Lipschitz norms and diffusion-induced instability
    • Z. Aminzare and E. D. Sontag, "Logarithmic Lipschitz norms and diffusion-induced instability," Nonlinear Anal.: Theory, Meth. Appl., vol. 83, pp. 31-49, 2013.
    • (2013) Nonlinear Anal.: Theory, Meth. Appl. , vol.83 , pp. 31-49
    • Aminzare, Z.1    Sontag, E.D.2
  • 47
    • 84883932589 scopus 로고    scopus 로고
    • (series Princeton Series in Applied Mathematics).Princeton, NJ, USA: Princeton Univ. Press
    • M. Mesbahi and M. Egerstedt, Graph Theoretic Methods in Multiagent Networks (series Princeton Series in Applied Mathematics). Princeton, NJ, USA: Princeton Univ. Press, 2010.
    • (2010) Graph Theoretic Methods in Multiagent Networks
    • Mesbahi, M.1    Egerstedt, M.2
  • 48
    • 4644317383 scopus 로고    scopus 로고
    • Information flow and cooperative control of vehicle formations
    • Sep.
    • J. Fax and R. Murray, "Information flow and cooperative control of vehicle formations," IEEE Trans. Autom. Control, vol. 49, no. 9, pp. 1465-1476, Sep. 2004.
    • (2004) IEEE Trans. Autom. Control , vol.49 , Issue.9 , pp. 1465-1476
    • Fax, J.1    Murray, R.2
  • 49
    • 84897695659 scopus 로고    scopus 로고
    • A differential Lyapunov framework for contraction analysis
    • Mar.
    • F. Forni and R. Sepulchre, "A differential Lyapunov framework for contraction analysis," IEEE Trans. Automat. Control, vol. 59, no. 3, pp. 614-628, Mar. 2014.
    • (2014) IEEE Trans. Automat. Control , vol.59 , Issue.3 , pp. 614-628
    • Forni, F.1    Sepulchre, R.2
  • 50
    • 77955716982 scopus 로고    scopus 로고
    • Heterogeneity and scalability in group agreement protocols: Beyond small gain and passivity approaches
    • Online
    • I. Lestas and G. Vinnicombe. (2010). Heterogeneity and scalability in group agreement protocols: Beyond small gain and passivity approaches. Automatica [Online]. 46(7), pp. 1141-1151. Available: http://www.sciencedirect.com/science/article/pii/ S000510981000138X
    • (2010) Automatica , vol.46 , Issue.7 , pp. 1141-1151
    • Lestas, I.1    Vinnicombe, G.2
  • 52
    • 27244462819 scopus 로고
    • The secant condition for instability in biochemical feedback control-parts i and ii
    • C. Thron, "The secant condition for instability in biochemical feedback control-parts i and ii," Bull. Math. Biol., vol. 53, pp. 383-424, 1991.
    • (1991) Bull. Math. Biol. , vol.53 , pp. 383-424
    • Thron, C.1
  • 53
    • 0013823048 scopus 로고
    • Oscillatory behavior in enzymatic control processes
    • B. Goodwin, "Oscillatory behavior in enzymatic control processes," Adv. Enzyme Regulation, vol. 3, pp. 425-439, 1965.
    • (1965) Adv. Enzyme Regulation , vol.3 , pp. 425-439
    • Goodwin, B.1
  • 54
    • 4243241129 scopus 로고    scopus 로고
    • Master stability functions for synchronized coupled systems
    • Mar. [Online]
    • L. M. Pecora and T. L. Carroll. (1998, Mar.). Master stability functions for synchronized coupled systems. Phys. Rev. Lett. [Online]. 80, pp. 2109-2112. Available: http://link.aps.org/doi/10.1103/ PhysRevLett.80.2109
    • (1998) Phys. Rev. Lett. , vol.80 , pp. 2109-2112
    • Pecora, L.M.1    Carroll, T.L.2
  • 55
    • 3343010976 scopus 로고    scopus 로고
    • Synchronization in small-world systems
    • Jul. [Online]
    • M. Barahona and L. M. Pecora. (2002, Jul.). Synchronization in small-world systems. Phys. Rev. Lett. [Online]. 89, pp. 054 101- 054 101. Available: http://link.aps.org/doi/10.1103/PhysRev Lett.89.054101
    • (2002) Phys. Rev. Lett. , vol.89 , pp. 054101
    • Barahona, M.1    Pecora, L.M.2
  • 56
    • 0000265121 scopus 로고
    • Synchronous chaos in coupled oscillator systems
    • Sep. [Online]
    • J. F. Heagy, T. L. Carroll, and L. M. Pecora. (1994, Sep.). Synchronous chaos in coupled oscillator systems. Phys. Rev. E [Online]. 50, pp. 1874-1885. Available: http://link.aps.org/doi/10.1103/ PhysRevE.50.1874
    • (1994) Phys. Rev. E , vol.50 , pp. 1874-1885
    • Heagy, J.F.1    Carroll, T.L.2    Pecora, L.M.3
  • 57
    • 84877693252 scopus 로고    scopus 로고
    • A contraction approach to the hierarchical analysis and design of networked systems
    • May
    • G. Russo, M. di Bernardo, and E. Sontag, "A contraction approach to the hierarchical analysis and design of networked systems," IEEE Trans. Autom. Control, vol. 58, no. 5, pp. 1328-1331, May 2013.
    • (2013) IEEE Trans. Autom. Control , vol.58 , Issue.5 , pp. 1328-1331
    • Russo, G.1    Di Bernardo, M.2    Sontag, E.3
  • 59
    • 0038538982 scopus 로고
    • Bounds on nonlinear operators in finite-dimensional Banach spaces
    • G. Soderlind, "Bounds on nonlinear operators in finite-dimensional Banach spaces," Numerische Mathematik, vol. 50, no. 1, pp. 27-44, 1986.
    • (1986) Numerische Mathematik , vol.50 , Issue.1 , pp. 27-44
    • Soderlind, G.1
  • 61
    • 55649094865 scopus 로고    scopus 로고
    • (ser. Classics in Applied Mathematics, SIAM).Philadelphia, PA, USA: SIAM
    • M. Vidyasagar, Nonlinear Systems Analysis vol. 42, (ser. Classics in Applied Mathematics, SIAM). Philadelphia, PA, USA: SIAM, 2002.
    • (2002) Nonlinear Systems Analysis , vol.42
    • Vidyasagar, M.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.