American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation
  • DCDS-B Home
  • This Issue
  • Next Article
    Stochastic invariance for neutral functional differential equation with non-lipschitz coefficients
July  2019, 24(7): 3319-3334. doi: 10.3934/dcdsb.2018322

On the global convergence of frequency synchronization for Kuramoto and Winfree oscillators

Chun-Hsiung Hsia 1, , Chang-Yeol Jung 2, and  Bongsuk Kwon 2,

1. 

Department of Mathematics, Institute of Applied Mathematical Sciences and National Center for Theoretical Sciences, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., Taipei 10617, Taiwan
2. 

Department of Mathematical Sciences, School of Natural Science, Ulsan National Institute of Science and Technology, Ulsan 44919, Republic of Korea

Received  February 2018 Revised  August 2018 Published  January 2019

We investigate the collective behavior of synchrony for the Kuramoto and Winfree models. We first prove the global convergence of frequency synchronization for the non-identical Kuramoto system of three oscillators. It is shown that the uniform boundedness of the diameter of the phase functions implies complete frequency synchronization. In light of this, we show, under a suitable condition on the coupling strength and deviation of the intrinsic frequencies, that the diameter function of the phases is uniformly bounded. In a similar spirit, we also prove the global convergence of phase-locked synchronization for the Winfree model of $ N $ oscillators for $ N\ge2 $.

Keywords: Kuramoto model, Winfree model, synchronization.
Mathematics Subject Classification: Primary: 34D06; Secondary: 34D05.
Citation: Chun-Hsiung Hsia, Chang-Yeol Jung, Bongsuk Kwon. On the global convergence of frequency synchronization for Kuramoto and Winfree oscillators. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3319-3334. doi: 10.3934/dcdsb.2018322
References:
[1]

J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77, 137–185Google Scholar

[2]

Y.-P. Choi, S.-Y. Ha, S. Jung and Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Physica D, 241 (2012), 735-754. doi: 10.1016/j.physd.2011.11.011. Google Scholar

[3]

F. Dörfler and F. Bullo, Synchronization in Complex Networks of Phase Oscillators: A Survey, Automatica, 50 (2014), 1539-1564. doi: 10.1016/j.automatica.2014.04.012. Google Scholar

[4]

J.-G. Dong and X. Xue, Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci., 11 (2013), 465-480. doi: 10.4310/CMS.2013.v11.n2.a7. Google Scholar

[5]

B. Ermentrout, Synchronization in a pool of mutually coupled oscillators with random frequencies, J. Math. Biol., 22 (1985), 1-9. doi: 10.1007/BF00276542. Google Scholar

[6]

B. Ermentrout, An adaptive model for synchrony in the firefly Pteroptyx malaccae, J. Math. Biol., 29 (1991), 571-585. doi: 10.1007/BF00164052. Google Scholar

[7]

S.-Y. Ha, H. K. Kim and S. W. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci., 14 (2016), 1073-1091. doi: 10.4310/CMS.2016.v14.n4.a10. Google Scholar

[8]

S.-Y. Ha, D. Ko, J. Park and S. W. Ryoo, Emergent dynamics of Winfree oscillators on locally coupled networks, J. Differential Equations, 260 (2016), 4203-4236. doi: 10.1016/j.jde.2015.11.008. Google Scholar

[9]

C.-H. Hsia, C.-Y. Jung and B. Kwon, On the synchronization theory of Kuramoto oscillators under the effect of inertia, preprint, arXiv: 1712.10111Google Scholar

[10]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3. Google Scholar

[11]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Phys., 39, Springer, New York, 1975,420–422. Google Scholar

[12]

J. Lunz, Complete synchronization of Kuramoto oscillators, J. Phys. A: Math. Theor., 44 (2011), 425102, 14 pp. doi: 10.1088/1751-8113/44/42/425102. Google Scholar

[13]

S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D, 143 (2000), 1-20. doi: 10.1016/S0167-2789(00)00094-4. Google Scholar

[14]

J. L. van Hemmen and W. F. Wreszinski, Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators, J. Stat. Phys., 72 (1993), 145-166. Google Scholar

[15]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. Google Scholar

show all references

References:
[1]

J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77, 137–185Google Scholar

[2]

Y.-P. Choi, S.-Y. Ha, S. Jung and Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Physica D, 241 (2012), 735-754. doi: 10.1016/j.physd.2011.11.011. Google Scholar

[3]

F. Dörfler and F. Bullo, Synchronization in Complex Networks of Phase Oscillators: A Survey, Automatica, 50 (2014), 1539-1564. doi: 10.1016/j.automatica.2014.04.012. Google Scholar

[4]

J.-G. Dong and X. Xue, Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci., 11 (2013), 465-480. doi: 10.4310/CMS.2013.v11.n2.a7. Google Scholar

[5]

B. Ermentrout, Synchronization in a pool of mutually coupled oscillators with random frequencies, J. Math. Biol., 22 (1985), 1-9. doi: 10.1007/BF00276542. Google Scholar

[6]

B. Ermentrout, An adaptive model for synchrony in the firefly Pteroptyx malaccae, J. Math. Biol., 29 (1991), 571-585. doi: 10.1007/BF00164052. Google Scholar

[7]

S.-Y. Ha, H. K. Kim and S. W. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci., 14 (2016), 1073-1091. doi: 10.4310/CMS.2016.v14.n4.a10. Google Scholar

[8]

S.-Y. Ha, D. Ko, J. Park and S. W. Ryoo, Emergent dynamics of Winfree oscillators on locally coupled networks, J. Differential Equations, 260 (2016), 4203-4236. doi: 10.1016/j.jde.2015.11.008. Google Scholar

[9]

C.-H. Hsia, C.-Y. Jung and B. Kwon, On the synchronization theory of Kuramoto oscillators under the effect of inertia, preprint, arXiv: 1712.10111Google Scholar

[10]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3. Google Scholar

[11]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Phys., 39, Springer, New York, 1975,420–422. Google Scholar

[12]

J. Lunz, Complete synchronization of Kuramoto oscillators, J. Phys. A: Math. Theor., 44 (2011), 425102, 14 pp. doi: 10.1088/1751-8113/44/42/425102. Google Scholar

[13]

S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D, 143 (2000), 1-20. doi: 10.1016/S0167-2789(00)00094-4. Google Scholar

[14]

J. L. van Hemmen and W. F. Wreszinski, Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators, J. Stat. Phys., 72 (1993), 145-166. Google Scholar

[15]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. Google Scholar

Figure 4.2.  The Kuramoto model (1.1) with $N = 3$, $K = 1$, $D(\Omega)/K = 1.23691$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.1.  The Kuramoto model (1.1) with $N = 3$, $K = 1$, $D(\Omega)/K = 0.0201916$. The plots are in log scale in $t$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.3.  The Winfree model (3.1) with $N = 5$, $\max_{1\leq i \leq N}\frac{|\omega_i|}{k_{ii}} = 1.15405$ where the matrix $K = K_1$ is given in (4.2). The plots are in log scale in $t$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.4.  The Winfree model (3.1) with $N = 5$, $K = $, $\max_{1\leq i \leq N}\frac{|\omega_i|}{k_{ii}} = 13.8456$ where the matrix $K = K_2$ is given in (4.2)
Figure Options

Download full-size image

Download as PowerPoint slide

Table 2.  Parameters for Kuramoto model experimented in Table 3. The notation $ U(a, b) $ is a uniform random distribution over $ [a, b] $
Case $N$ $K$ $\Omega$ $D(\Omega)/K$ $\Theta(0)$
(Ⅰ) $3$ $1$ $\{-0.1, 0.1, 0.0\}$ $0.2$ $\{1.5, -1.7, 2.1\}$
(Ⅱ) $5$ $2$ $\{0.9, 0.1, -1.1, -0.9, 0.0\}$ $1$ $\{-1.4, 2.3, -1.8, 0.5, -2.4\}$
(Ⅲ) $20$ $1$ $U(-0.123, 0.123)$ $0.214915$ $U(-\pi, \pi)$
(Ⅰ)' $3$ $1$ $\{-0.6, 0.9, 0.5\}$ $1.5$ $\{-3.0, -0.7, -2.0\}$
(Ⅱ)' $5$ $1$ $\{0.9, 0.1, -1.1, -0.9, 0.0\}$ $2$ $\{-1.4, 2.3, -1.8, 0.5, -2.4\}$
(Ⅲ)' $20$ $1.5$ $U(-1.23, 1.23)$ $1.59328$ $U(-\pi, \pi)$
Table Options

Download as excel

Case $N$ $K$ $\Omega$ $D(\Omega)/K$ $\Theta(0)$
(Ⅰ) $3$ $1$ $\{-0.1, 0.1, 0.0\}$ $0.2$ $\{1.5, -1.7, 2.1\}$
(Ⅱ) $5$ $2$ $\{0.9, 0.1, -1.1, -0.9, 0.0\}$ $1$ $\{-1.4, 2.3, -1.8, 0.5, -2.4\}$
(Ⅲ) $20$ $1$ $U(-0.123, 0.123)$ $0.214915$ $U(-\pi, \pi)$
(Ⅰ)' $3$ $1$ $\{-0.6, 0.9, 0.5\}$ $1.5$ $\{-3.0, -0.7, -2.0\}$
(Ⅱ)' $5$ $1$ $\{0.9, 0.1, -1.1, -0.9, 0.0\}$ $2$ $\{-1.4, 2.3, -1.8, 0.5, -2.4\}$
(Ⅲ)' $20$ $1.5$ $U(-1.23, 1.23)$ $1.59328$ $U(-\pi, \pi)$
Table 3.  The Kuramoto phases $ \Theta(t) $ and the modulus of the order parameter, $ |r| $, given in (4.1)
$t$(Ⅰ)(Ⅱ)(Ⅲ)(Ⅰ)'(Ⅱ)'(Ⅲ)'
$0$$D(\Theta(t))$3.800004.700004.601702.300004.700005.67130
$D(\dot{\Theta}(t))$0.231081.932600.678780.611641.886702.41560
$|r|$0.345150.305110.272910.603970.305110.03716
$5$$D(\Theta(t))$6.051706.961000.485077.563208.1075013.97020
$D(\dot{\Theta}(t))$0.156160.003750.415490.577951.226803.03850
$|r|$0.988820.914980.992690.868060.758480.72047
$20$$D(\Theta(t))$6.182706.962200.2157820.3800028.0432022.71800
$D(\dot{\Theta}(t))$0.000000.000000.000000.385121.013600.94404
$|r|$0.996640.914830.998140.808570.616950.60180
$150$$D(\Theta(t))$6.182706.962200.21578133.80190228.90490128.11660
$D(\dot{\Theta}(t))$0.000000.000000.000000.363090.839002.09790
$|r|$0.996640.914830.998140.720700.603480.76208
$500$$D(\Theta(t))$6.182706.962200.21578441.10500782.61200405.01650
$D(\dot{\Theta}(t))$0.000000.000000.000000.493831.247700.46759
$|r|$0.996640.914830.998140.866420.408410.68921
Table Options

Download as excel

$t$(Ⅰ)(Ⅱ)(Ⅲ)(Ⅰ)'(Ⅱ)'(Ⅲ)'
$0$$D(\Theta(t))$3.800004.700004.601702.300004.700005.67130
$D(\dot{\Theta}(t))$0.231081.932600.678780.611641.886702.41560
$|r|$0.345150.305110.272910.603970.305110.03716
$5$$D(\Theta(t))$6.051706.961000.485077.563208.1075013.97020
$D(\dot{\Theta}(t))$0.156160.003750.415490.577951.226803.03850
$|r|$0.988820.914980.992690.868060.758480.72047
$20$$D(\Theta(t))$6.182706.962200.2157820.3800028.0432022.71800
$D(\dot{\Theta}(t))$0.000000.000000.000000.385121.013600.94404
$|r|$0.996640.914830.998140.808570.616950.60180
$150$$D(\Theta(t))$6.182706.962200.21578133.80190228.90490128.11660
$D(\dot{\Theta}(t))$0.000000.000000.000000.363090.839002.09790
$|r|$0.996640.914830.998140.720700.603480.76208
$500$$D(\Theta(t))$6.182706.962200.21578441.10500782.61200405.01650
$D(\dot{\Theta}(t))$0.000000.000000.000000.493831.247700.46759
$|r|$0.996640.914830.998140.866420.408410.68921
Table 1.  Parameters for Winfree model experimented in Table 4. The upper triangular entries of matrices $ K_i $, $ i = 3, \cdots, 8 $ are generated from a uniform random distribution over $ [0.5, 1.0] $ and the lower triangular entries by a symmetry. $ U(a, b) $ denotes a uniform random distribution over $ [a, b] $.
Case $N$ $K$ $\Omega$ $\displaystyle\max_{1\leq i \leq N}\frac{|\omega_i|}{k_{ii}}$ $\Theta(0)$
(Ⅰ) $3$ $K_3$ $\{1.7, 1.1, -1.7\}$ $2.56732$ $\{-0.9, 2.7, -3.0\}$
(Ⅱ) $5$ $K_4$ $\{-1.1, -1.7, 0.9, 1.4, -0.4\}$ $2.52008$ $\{-0.8, 1.8, -0.2, 2.6, -1.4\}$
(Ⅲ) $20$ $K_5$ $U(-28, 28)$ $37.4299$ $U(-\pi, \pi)$
(Ⅰ)' $3$ $K_6$ $\{5.0, 2.1, -3.7\}$ $7.01639$ $\{-0.9, 2.7, -3.0\}$
(Ⅱ)' $5$ $K_7$ $\{-2.1, -1.7, 0.9, 10.4, -8.4\}$ $12.6565$ $\{-0.8, 1.8, -0.2, 2.6, -1.4\}$
(Ⅲ)' $20$ $K_8$ $U(-28, 28)$ $49.8966$ $U(-\pi, \pi)$
Table Options

Download as excel

Case $N$ $K$ $\Omega$ $\displaystyle\max_{1\leq i \leq N}\frac{|\omega_i|}{k_{ii}}$ $\Theta(0)$
(Ⅰ) $3$ $K_3$ $\{1.7, 1.1, -1.7\}$ $2.56732$ $\{-0.9, 2.7, -3.0\}$
(Ⅱ) $5$ $K_4$ $\{-1.1, -1.7, 0.9, 1.4, -0.4\}$ $2.52008$ $\{-0.8, 1.8, -0.2, 2.6, -1.4\}$
(Ⅲ) $20$ $K_5$ $U(-28, 28)$ $37.4299$ $U(-\pi, \pi)$
(Ⅰ)' $3$ $K_6$ $\{5.0, 2.1, -3.7\}$ $7.01639$ $\{-0.9, 2.7, -3.0\}$
(Ⅱ)' $5$ $K_7$ $\{-2.1, -1.7, 0.9, 10.4, -8.4\}$ $12.6565$ $\{-0.8, 1.8, -0.2, 2.6, -1.4\}$
(Ⅲ)' $20$ $K_8$ $U(-28, 28)$ $49.8966$ $U(-\pi, \pi)$
Table 4.  The Winfree phases $ \Theta(t) $ and the modulus of the order parameter, $ |r| $, given in (4.1).
$t$(Ⅰ)(Ⅱ)(Ⅲ)(Ⅰ)'(Ⅱ)'(Ⅲ)'
$0$$D(\Theta(t))$5.700004.000005.740005.700004.000005.74710
$D(\dot{\Theta}(t))$4.277409.3906054.673309.4228015.1192076.17700
$|r|$0.455370.173380.032570.455370.173380.24487
$5$$D(\Theta(t))$13.186400.4028714.9570030.7802081.8434027.60650
$D(\dot{\Theta}(t))$0.000000.000000.0000011.3926019.274900.98300
$|r|$0.945440.987660.776020.909940.421640.80234
$20$$D(\Theta(t))$13.186400.4028714.95700109.18220316.9740078.33540
$D(\dot{\Theta}(t))$0.000000.000000.000003.667109.381800.88459
$|r|$0.945440.987660.776020.425670.648360.77224
$150$$D(\Theta(t))$13.186400.4028714.95700818.106002347.20000549.54340
$D(\dot{\Theta}(t))$0.000000.000000.0000010.2044016.589800.72854
$|r|$0.945440.987660.776020.780810.558720.77419
$500$$D(\Theta(t))$13.186400.4028714.957002725.000007813.200001818.60000
$D(\dot{\Theta}(t))$0.000000.000000.000005.5728011.352200.35697
$|r|$0.945440.987660.776020.190580.577540.78170
Table Options

Download as excel

$t$(Ⅰ)(Ⅱ)(Ⅲ)(Ⅰ)'(Ⅱ)'(Ⅲ)'
$0$$D(\Theta(t))$5.700004.000005.740005.700004.000005.74710
$D(\dot{\Theta}(t))$4.277409.3906054.673309.4228015.1192076.17700
$|r|$0.455370.173380.032570.455370.173380.24487
$5$$D(\Theta(t))$13.186400.4028714.9570030.7802081.8434027.60650
$D(\dot{\Theta}(t))$0.000000.000000.0000011.3926019.274900.98300
$|r|$0.945440.987660.776020.909940.421640.80234
$20$$D(\Theta(t))$13.186400.4028714.95700109.18220316.9740078.33540
$D(\dot{\Theta}(t))$0.000000.000000.000003.667109.381800.88459
$|r|$0.945440.987660.776020.425670.648360.77224
$150$$D(\Theta(t))$13.186400.4028714.95700818.106002347.20000549.54340
$D(\dot{\Theta}(t))$0.000000.000000.0000010.2044016.589800.72854
$|r|$0.945440.987660.776020.780810.558720.77419
$500$$D(\Theta(t))$13.186400.4028714.957002725.000007813.200001818.60000
$D(\dot{\Theta}(t))$0.000000.000000.000005.5728011.352200.35697
$|r|$0.945440.987660.776020.190580.577540.78170
[1]

Seung-Yeal Ha, Jinyeong Park, Sang Woo Ryoo. Emergence of phase-locked states for the Winfree model in a large coupling regime. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3417-3436. doi: 10.3934/dcds.2015.35.3417
[2]

Seung-Yeal Ha, Se Eun Noh, Jinyeong Park. Practical synchronization of generalized Kuramoto systems with an intrinsic dynamics. Networks & Heterogeneous Media, 2015, 10 (4) : 787-807. doi: 10.3934/nhm.2015.10.787
[3]

Seung-Yeal Ha, Jaeseung Lee, Zhuchun Li. Emergence of local synchronization in an ensemble of heterogeneous Kuramoto oscillators. Networks & Heterogeneous Media, 2017, 12 (1) : 1-24. doi: 10.3934/nhm.2017001
[4]

Seung-Yeal Ha, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. Uniform stability and mean-field limit for the augmented Kuramoto model. Networks & Heterogeneous Media, 2018, 13 (2) : 297-322. doi: 10.3934/nhm.2018013
[5]

Paolo Antonelli, Seung-Yeal Ha, Dohyun Kim, Pierangelo Marcati. The Wigner-Lohe model for quantum synchronization and its emergent dynamics. Networks & Heterogeneous Media, 2017, 12 (3) : 403-416. doi: 10.3934/nhm.2017018
[6]

Igor Chueshov, Peter E. Kloeden, Meihua Yang. Synchronization in coupled stochastic sine-Gordon wave model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2969-2990. doi: 10.3934/dcdsb.2016082
[7]

Young-Pil Choi, Seung-Yeal Ha, Seok-Bae Yun. Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto--Daido model with inertia. Networks & Heterogeneous Media, 2013, 8 (4) : 943-968. doi: 10.3934/nhm.2013.8.943
[8]

Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the Kuramoto model on graphs Ⅰ. The mean field equation and transition point formulas. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 131-155. doi: 10.3934/dcds.2019006
[9]

Zhuchun Li, Yi Liu, Xiaoping Xue. Convergence and stability of generalized gradient systems by Łojasiewicz inequality with application in continuum Kuramoto model. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 345-367. doi: 10.3934/dcds.2019014
[10]

Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3897-3921. doi: 10.3934/dcds.2019157
[11]

Michael Cranston, Benjamin Gess, Michael Scheutzow. Weak synchronization for isotropic flows. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3003-3014. doi: 10.3934/dcdsb.2016084
[12]

Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225
[13]

Johannes Kellendonk, Lorenzo Sadun. Conjugacies of model sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3805-3830. doi: 10.3934/dcds.2017161
[14]

Joachim Escher, Boris Kolev, Marcus Wunsch. The geometry of a vorticity model equation. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1407-1419. doi: 10.3934/cpaa.2012.11.1407
[15]

Paolo Secchi. An alpha model for compressible fluids. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 351-359. doi: 10.3934/dcdss.2010.3.351
[16]

Monica De Angelis, Gaetano Fiore. Diffusion effects in a superconductive model. Communications on Pure & Applied Analysis, 2014, 13 (1) : 217-223. doi: 10.3934/cpaa.2014.13.217
[17]

Avner Friedman, Chuan Xue. A mathematical model for chronic wounds. Mathematical Biosciences & Engineering, 2011, 8 (2) : 253-261. doi: 10.3934/mbe.2011.8.253
[18]

J. R. Fernández, R. Martínez, J. M. Viaño. Analysis of a bone remodeling model. Communications on Pure & Applied Analysis, 2009, 8 (1) : 255-274. doi: 10.3934/cpaa.2009.8.255
[19]

Ali Ashher Zaidi, Bruce Van Brunt, Graeme Charles Wake. A model for asymmetrical cell division. Mathematical Biosciences & Engineering, 2015, 12 (3) : 491-501. doi: 10.3934/mbe.2015.12.491
[20]

Fritz Gesztesy, Rudi Weikard, Maxim Zinchenko. On a class of model Hilbert spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5067-5088. doi: 10.3934/dcds.2013.33.5067

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (71)
  • HTML views (393)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences