American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2018061

Mechanism for the color transition of the Belousov-Zhabotinsky reaction catalyzed by cerium ions and ferroin

Chikahiro Egami

Laboratory of Mathematics, Faculty of Regional Environment Science, Tokyo University of Agriculture, 1-1-1 Sakuragaoka, Setagaya-ku, Tokyo 156-8502, Japan

Received  May 2017 Revised  September 2017 Published  February 2018

Fund Project: This work was supported by JSPS KAKENHI (Grant Number 22740114,26400183)

The oscillation property of the Belousov-Zhabotinsky reaction and the color transition of its solution depend on the catalytic action of the metal ions. The solution of the reaction system catalyzed by both cerium ions and ferroin has a more complicated effect on the color than either the cerium-catalyzed case or the ferroin-catalyzed case. To theoretically elucidate the color transition of the case catalyzed by these two ions, a reduced model consisting of three differential equations is proposed, incorporating both the Rovinsky-Zhabotinsky scheme and the Field-Körös-Noyes scheme simplified by Tyson [Ann. N.Y. Acad. Sci., 316 (1979), pp.279-295]. The presented model can have a limit cycle under reasonable conditions through a Hopf bifurcation, and its existence theorem is proven by employing the bifurcation criterion established by Liu [J. Math. Anal. Appl., 182 (1994), pp.250-256].

Keywords: Chemical oscillator, redox potential, limit cycle, Hopf bifurcation, Hurwitz determinant.
Mathematics Subject Classification: Primary: 34C26; Secondary: 92E20.
Citation: Chikahiro Egami. Mechanism for the color transition of the Belousov-Zhabotinsky reaction catalyzed by cerium ions and ferroin. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018061
References:
[1]

C. Egami, Bifurcation analysis of the Nowak-Bangham model in CTL dynamics, Math. Biosci., 221 (2009), 33-42. doi: 10.1016/j.mbs.2009.06.005.

[2]

R. J. Field and H.-D. Försterling, On the oxybromine chemistry rate constants with cerium ions in the Field-Körös-Noyes mechanism of the Belousov-Zhabotinskii reaction: The equilibrium $\rm HBrO_2 + BrO_3^{-} + H^{+} \rightleftharpoons 2BrO_2{·} + H_2O$, J. Phys. Chem., 90 (1986), 5400-5407.

[3]

R. J. Field, E. Körös and R. M. Noyes, Oscillations in chemical systems. Ⅱ. Thorough analysis of temporal oscillation in the Bromate-Cerium-Malonic acid system, J. Am. Chem. Soc., 94 (1972), 8649-8664. doi: 10.1021/ja00780a001.

[4]

R. J. Field and R. M. Noyes, Oscillation in chemical systems. Ⅳ. Limit cycle behavior in a model of a real chemical reaction, J. Chem. Phys., 60 (1974), 1877-1884. doi: 10.1063/1.1681288.

[5]

H.-D. Försterling, L. Stuk, A. Barr and W. D. McCormick, Stoichiometry of bromide production from ceric oxidation of bromomalonic acid in the Belousov-Zhabotinskii reaction, J. Phys. Chem., 97 (1993), 7578-7584.

[6]

L. Györgyi and R. J. Field, Aperiodicity resulting from two-cycle coupling in the Belousov-Zhabotinskii reaction. ⅲ. Analysis of a model of the effect of spatial inhomogeneities at the input ports of a continuous-flow, stirred tank reactor, J. Chem. Phys., 91 (1989), 6131-6141. doi: 10.1063/1.457432.

[7]

L. Györgyi and R. J. Field, Simple models of deterministic chaos in the Belousov-Zhabotinskii reaction, J. Phys. Chem., 95 (1991), 6594-6602.

[8]

L. Györgyi, T. Turányi and R. J. Field, Mechanistic details of the oscillatory Belousov-Zhabotinskii reaction, J. Phys. Chem., 94 (1990), 7162-7170.

[9]

J. P. Keener and J. J. Tyson, Target patterns in a realistic model of the Belousov-Zhabotinskii reaction, Phys. D, 21 (1986), 307-324. doi: 10.1016/0167-2789(86)90007-2.

[10]

G. Kshirsagar and R. J. Field, A kinetic and thermodynamic study of component processes in the equilibrium $\rm 5HOBr \rightleftharpoons 2BR_2+BrO_3^{-}+2H_2O+H^{+}$, J. Phys. Chem., 92 (1988), 7074-7079.

[11]

W. M. Liu, Criterion of Hopf bifurcations without using eigenvalues, J. Math. Anal. Appl., 182 (1994), 250-256. doi: 10.1006/jmaa.1994.1079.

[12]

A. B. Rovinsky and A. M. Zhabotinsky, Mechanism and mathematical model of the oscillating Bromate-Ferroin-Bromomalonic acid reaction, J. Phys. Chem.(25), 88 (), 6081-6084,1984. doi: 10.1021/j150669a001.

[13]

C. E. Sjogren, S. Kolboe and P. Ruoff, Transitions between two oscillatory states in a closed malonic acid Belousov-Zhabotinsky reaction simultaneously catalyzed by ferroin and cerium ions, Chem. Phys. Lett., 130 (1986), 72-75.

[14]

A. F. Taylor, V. Gáspár, B. R. Johnson and S. K. Scott, Analysis of reaction-diffusion waves in the ferroin-catalysed Belousov-Zhabotinsky reaction, Phys. Chem. Chem. Phys., 1 (1999), 4595-4599. doi: 10.1039/a904994k.

[15]

J. J. Tyson, Oscillations, bistability, and echo waves in models of the Belusov-Zhabotinskii reaction, Ann. N.Y. Acad. Sci., 316 (1979), 279-295.

[16]

J. J. Tyson, Relaxation oscillations in the revised Oregonator, J. Chem. Phys., 80 (1984), 6079-6082. doi: 10.1063/1.446690.

[17]

J. J. Tyson and P. C. Fife, Target patterns in a realistic model of the Belousov-Zhabotinskii reaction, J. Chem. Phys., 73 (1980), 2224-2237. doi: 10.1063/1.440418.

[18]

A. M. Zhabotinsky, F. Buchholtz, A. B. Kiyatkin and I. R. Epstein, Oscillations and waves in metal-ion-catalyzed Bromate oscillating reactions in highly oxidized states, J. Phys. Chem., 97 (1993), 7578-7584. doi: 10.1021/j100131a030.

show all references

References:
[1]

C. Egami, Bifurcation analysis of the Nowak-Bangham model in CTL dynamics, Math. Biosci., 221 (2009), 33-42. doi: 10.1016/j.mbs.2009.06.005.

[2]

R. J. Field and H.-D. Försterling, On the oxybromine chemistry rate constants with cerium ions in the Field-Körös-Noyes mechanism of the Belousov-Zhabotinskii reaction: The equilibrium $\rm HBrO_2 + BrO_3^{-} + H^{+} \rightleftharpoons 2BrO_2{·} + H_2O$, J. Phys. Chem., 90 (1986), 5400-5407.

[3]

R. J. Field, E. Körös and R. M. Noyes, Oscillations in chemical systems. Ⅱ. Thorough analysis of temporal oscillation in the Bromate-Cerium-Malonic acid system, J. Am. Chem. Soc., 94 (1972), 8649-8664. doi: 10.1021/ja00780a001.

[4]

R. J. Field and R. M. Noyes, Oscillation in chemical systems. Ⅳ. Limit cycle behavior in a model of a real chemical reaction, J. Chem. Phys., 60 (1974), 1877-1884. doi: 10.1063/1.1681288.

[5]

H.-D. Försterling, L. Stuk, A. Barr and W. D. McCormick, Stoichiometry of bromide production from ceric oxidation of bromomalonic acid in the Belousov-Zhabotinskii reaction, J. Phys. Chem., 97 (1993), 7578-7584.

[6]

L. Györgyi and R. J. Field, Aperiodicity resulting from two-cycle coupling in the Belousov-Zhabotinskii reaction. ⅲ. Analysis of a model of the effect of spatial inhomogeneities at the input ports of a continuous-flow, stirred tank reactor, J. Chem. Phys., 91 (1989), 6131-6141. doi: 10.1063/1.457432.

[7]

L. Györgyi and R. J. Field, Simple models of deterministic chaos in the Belousov-Zhabotinskii reaction, J. Phys. Chem., 95 (1991), 6594-6602.

[8]

L. Györgyi, T. Turányi and R. J. Field, Mechanistic details of the oscillatory Belousov-Zhabotinskii reaction, J. Phys. Chem., 94 (1990), 7162-7170.

[9]

J. P. Keener and J. J. Tyson, Target patterns in a realistic model of the Belousov-Zhabotinskii reaction, Phys. D, 21 (1986), 307-324. doi: 10.1016/0167-2789(86)90007-2.

[10]

G. Kshirsagar and R. J. Field, A kinetic and thermodynamic study of component processes in the equilibrium $\rm 5HOBr \rightleftharpoons 2BR_2+BrO_3^{-}+2H_2O+H^{+}$, J. Phys. Chem., 92 (1988), 7074-7079.

[11]

W. M. Liu, Criterion of Hopf bifurcations without using eigenvalues, J. Math. Anal. Appl., 182 (1994), 250-256. doi: 10.1006/jmaa.1994.1079.

[12]

A. B. Rovinsky and A. M. Zhabotinsky, Mechanism and mathematical model of the oscillating Bromate-Ferroin-Bromomalonic acid reaction, J. Phys. Chem.(25), 88 (), 6081-6084,1984. doi: 10.1021/j150669a001.

[13]

C. E. Sjogren, S. Kolboe and P. Ruoff, Transitions between two oscillatory states in a closed malonic acid Belousov-Zhabotinsky reaction simultaneously catalyzed by ferroin and cerium ions, Chem. Phys. Lett., 130 (1986), 72-75.

[14]

A. F. Taylor, V. Gáspár, B. R. Johnson and S. K. Scott, Analysis of reaction-diffusion waves in the ferroin-catalysed Belousov-Zhabotinsky reaction, Phys. Chem. Chem. Phys., 1 (1999), 4595-4599. doi: 10.1039/a904994k.

[15]

J. J. Tyson, Oscillations, bistability, and echo waves in models of the Belusov-Zhabotinskii reaction, Ann. N.Y. Acad. Sci., 316 (1979), 279-295.

[16]

J. J. Tyson, Relaxation oscillations in the revised Oregonator, J. Chem. Phys., 80 (1984), 6079-6082. doi: 10.1063/1.446690.

[17]

J. J. Tyson and P. C. Fife, Target patterns in a realistic model of the Belousov-Zhabotinskii reaction, J. Chem. Phys., 73 (1980), 2224-2237. doi: 10.1063/1.440418.

[18]

A. M. Zhabotinsky, F. Buchholtz, A. B. Kiyatkin and I. R. Epstein, Oscillations and waves in metal-ion-catalyzed Bromate oscillating reactions in highly oxidized states, J. Phys. Chem., 97 (1993), 7578-7584. doi: 10.1021/j100131a030.

Figure 2.  Periodic color transition of the BZ reaction solution (Exp. Ⅲ) catalyzed by cerium ions and ferroin. The value of $E_{\rm ORP}$ shows the redox potential of solution at the moment the image was captured
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 1.  Redox potential curves recorded by the three catalytic types of BZ reactions
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Redox potential curves represented by the solutions of (T), (RZ), and (CF)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  The rule of color transition for the BZ reaction catalyzed by cerium ions and ferroin: (a) the color sequence of the reaction solution, (b) the redox potential curve generated by (CF), (c) the time sequences of the ratios of $\rm Ce^{4+}$ and $\rm Fe(phen)_3^{3+}$ to the total cerium ion and ferroin concentrations, (d) the color phases exhibited by the cerium ions and ferroin. The solution color changes in six phases of four colors because of the simultaneous continuous fluctuation of the ratios between the oxidant and reductant of each ion
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Bifurcation diagram obtained by varying $\varepsilon$: {(a)} The state-space plot of the $\omega$-limit sets by setting $\sigma$ ($ = \varepsilon^{-1}$) between 1.0 and 20.0 every 1.0; the orbits of (CF) approach the equilibrium $\bar{E}$ for every $\sigma<\sigma^*$, whereas a limit cycle appears around $\bar{E}$ for each $\sigma>\sigma^*$. {(b)} The time sequence $y(\tau)$, $z(\tau)$ of the solution for the initial condition $(x_0, y_0, z_0) = (0.5, 0.1, 0.3)$ at $\sigma = 2.7, 2.8, 5.0, 10.0$. An increase in $\sigma$ enlarges the frequency and the amplitude of oscillation
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  $\omega$-limit sets obtained by varying $\xi_1$: (a) The phase-space plots for $\kappa$ ($ = \xi_1^{-1}$) between 0.1 and 10.1 every 0.5; the equilibrium $\bar{E}$ is always unstable. (b) The time-sequence $y(\tau)$, $z(\tau)$ of the solution for the initial condition $(x_0, y_0, z_0) = (0.5, 0.1, 0.3)$ at $\kappa = 0.1, 0.3, 1.0, 10.0$; the period of the limit cycle is almost proportional to $\kappa$ as long as $\kappa$ is small
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  $\omega$-limit sets obtained by varying $\xi_2$: (a) The phase-space plots for $\kappa$ ($ = \xi_2^{-1}$) between 0.1 and 50.1 every 2.5; the equilibrium $\bar{E}$ is always unstable. (b) The time-sequence $y(\tau)$, $z(\tau)$ of the solution for the initial condition $(x_0, y_0, z_0) = (0.5, 0.1, 0.3)$ at $\kappa = 0.1, 3.0, 10.0, 50.0$; the period of limit cycle is almost proportional to $\kappa$ as long as $\kappa$ is small
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Solution preparation. (M = mol/L)
No. Contents and concentrations
(S1) $\rm KBrO_3$ 0.23M
(S2) $\rm CH_2(COOH)_2$ 0.31M, $\rm KBr$ 0.059M
(S3) $\rm H_2SO_4$ 3M, $\rm Ce(NH_4)_2(NO_3)_6$ 0.019M
(S4) $\rm H_2SO_4$ 4M
(S5) $\rm FeSO_4\!\cdot\!7H_2O$ 7.6mM, $\rm C_{12}H_8N_2$ 22.8mM
Table Options

Download as excel

No. Contents and concentrations
(S1) $\rm KBrO_3$ 0.23M
(S2) $\rm CH_2(COOH)_2$ 0.31M, $\rm KBr$ 0.059M
(S3) $\rm H_2SO_4$ 3M, $\rm Ce(NH_4)_2(NO_3)_6$ 0.019M
(S4) $\rm H_2SO_4$ 4M
(S5) $\rm FeSO_4\!\cdot\!7H_2O$ 7.6mM, $\rm C_{12}H_8N_2$ 22.8mM
Table 2.  Mixture quantities for the three kinds of BZ reactions
Exp.Solution number and volume [mL]Type of catalyst
Ⅰ(S1) 20, (S2) 20, (S3) 20cerium-catalyzed
Ⅱ(S1) 20, (S2) 20, (S4) 15, (S5) 5.0ferroin-catalyzed
Ⅲ(S1) 20, (S2) 20, (S3) 20, (S5) 1.0cerium-ferroin-catalyzed
Table Options

Download as excel

Exp.Solution number and volume [mL]Type of catalyst
Ⅰ(S1) 20, (S2) 20, (S3) 20cerium-catalyzed
Ⅱ(S1) 20, (S2) 20, (S4) 15, (S5) 5.0ferroin-catalyzed
Ⅲ(S1) 20, (S2) 20, (S3) 20, (S5) 1.0cerium-ferroin-catalyzed
Table 3.  Fixed concentrations and rate constants for (T), (RZ), and (CF)
Concentration Rate constant
$A$ 0.04 M $k_1$ $100$ $\rm {M^{-2}s^{-1}}$ $k_6$ $10^7$ $\rm {M^{-2}s^{-1}}$
$B$ 0.10 M $k_3$ $10^7$ $\rm {M^{-1}s^{-1}}$ $k_8$ $15$ $\rm {M^{-2}s^{-1}}$
$C$ (Exp. Ⅰ, Ⅲ) 6.3 mM $k_{-3}$ $10^3$ $\rm {M^{-1}s^{-1}}$ $k_9$ $2$ $\rm {M^{-1}s^{-1}}$
$F$ (Exp. Ⅲ) 0.13 mM $k_4$ $8.0\times 10^4$ $\rm {M^{-2}s^{-1}}$ $k_{-9}$ $2.0\times 10^8$ $\rm {M^{-2}s^{-1}}$
$F$ (Exp. Ⅱ) 0.65 mM $k_{-4}$ $8.9\times 10^3$ $\rm {M^{-1}s^{-1}}$ $k_{10}$ $2.0 \times 10^3$ $\rm s^{-1}$
$H$ 1.29 M $k_5$ $1.7\times 10^4$ $\rm {M^{-1}s^{-1}}$ $k_{11}$ $0.8$ $\rm {M^{-1}s^{-1}}$
Table Options

Download as excel

Concentration Rate constant
$A$ 0.04 M $k_1$ $100$ $\rm {M^{-2}s^{-1}}$ $k_6$ $10^7$ $\rm {M^{-2}s^{-1}}$
$B$ 0.10 M $k_3$ $10^7$ $\rm {M^{-1}s^{-1}}$ $k_8$ $15$ $\rm {M^{-2}s^{-1}}$
$C$ (Exp. Ⅰ, Ⅲ) 6.3 mM $k_{-3}$ $10^3$ $\rm {M^{-1}s^{-1}}$ $k_9$ $2$ $\rm {M^{-1}s^{-1}}$
$F$ (Exp. Ⅲ) 0.13 mM $k_4$ $8.0\times 10^4$ $\rm {M^{-2}s^{-1}}$ $k_{-9}$ $2.0\times 10^8$ $\rm {M^{-2}s^{-1}}$
$F$ (Exp. Ⅱ) 0.65 mM $k_{-4}$ $8.9\times 10^3$ $\rm {M^{-1}s^{-1}}$ $k_{10}$ $2.0 \times 10^3$ $\rm s^{-1}$
$H$ 1.29 M $k_5$ $1.7\times 10^4$ $\rm {M^{-1}s^{-1}}$ $k_{11}$ $0.8$ $\rm {M^{-1}s^{-1}}$
Table 4.  Relation between signs of $q'_2$, $q_{10}$, $q_{20}$, and bifurcation type
$q'_2$ $q_{10}$ $q_{20}$type
+++T1
++0T5
++-T3
+0+T6
+00T3
+0-T3
+-+T3
+-0T3
+--T3
0++T7
0+0T0
0+-T0
00+T9
000T0
00-T0
0-+T8
0-0T0
0--T0
-++T4
-+0T0
-+-T0
-0+T4
-00T0
-0-T0
--+T2
--0T0
---T0
Table Options

Download as excel

$q'_2$ $q_{10}$ $q_{20}$type
+++T1
++0T5
++-T3
+0+T6
+00T3
+0-T3
+-+T3
+-0T3
+--T3
0++T7
0+0T0
0+-T0
00+T9
000T0
00-T0
0-+T8
0-0T0
0--T0
-++T4
-+0T0
-+-T0
-0+T4
-00T0
-0-T0
--+T2
--0T0
---T0
[1]

Ben Niu, Weihua Jiang. Dynamics of a limit cycle oscillator with extended delay feedback. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1439-1458. doi: 10.3934/dcdsb.2013.18.1439
[2]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805
[3]

Bourama Toni. Upper bounds for limit cycle bifurcation from an isochronous period annulus via a birational linearization. Conference Publications, 2005, 2005 (Special) : 846-853. doi: 10.3934/proc.2005.2005.846
[4]

Qiongwei Huang, Jiashi Tang. Bifurcation of a limit cycle in the ac-driven complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 129-141. doi: 10.3934/dcdsb.2010.14.129
[5]

Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997
[6]

Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045
[7]

Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152
[8]

Zhiqin Qiao, Deming Zhu, Qiuying Lu. Bifurcation of a heterodimensional cycle with weak inclination flip. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 1009-1025. doi: 10.3934/dcdsb.2012.17.1009
[9]

Valery A. Gaiko. The geometry of limit cycle bifurcations in polynomial dynamical systems. Conference Publications, 2011, 2011 (Special) : 447-456. doi: 10.3934/proc.2011.2011.447
[10]

Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098
[11]

Fernando Antoneli, Ana Paula S. Dias, Rui Paiva. Coupled cell networks: Hopf bifurcation and interior symmetry. Conference Publications, 2011, 2011 (Special) : 71-78. doi: 10.3934/proc.2011.2011.71
[12]

R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure & Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147
[13]

Fatihcan M. Atay. Delayed feedback control near Hopf bifurcation. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 197-205. doi: 10.3934/dcdss.2008.1.197
[14]

Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247
[15]

Orit Lavi, Doron Ginsberg, Yoram Louzoun. Regulation of modular Cyclin and CDK feedback loops by an E2F transcription oscillator in the mammalian cell cycle. Mathematical Biosciences & Engineering, 2011, 8 (2) : 445-461. doi: 10.3934/mbe.2011.8.445
[16]

Jihua Yang, Liqin Zhao. Limit cycle bifurcations for piecewise smooth integrable differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2417-2425. doi: 10.3934/dcdsb.2017123
[17]

Fangfang Jiang, Junping Shi, Qing-guo Wang, Jitao Sun. On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047
[18]

Stijn Luca, Freddy Dumortier, Magdalena Caubergh, Robert Roussarie. Detecting alien limit cycles near a Hamiltonian 2-saddle cycle. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1081-1108. doi: 10.3934/dcds.2009.25.1081
[19]

Sze-Bi Hsu, Junping Shi. Relaxation oscillation profile of limit cycle in predator-prey system. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 893-911. doi: 10.3934/dcdsb.2009.11.893
[20]

Magdalena Caubergh, Freddy Dumortier, Robert Roussarie. Alien limit cycles in rigid unfoldings of a Hamiltonian 2-saddle cycle. Communications on Pure & Applied Analysis, 2007, 6 (1) : 1-21. doi: 10.3934/cpaa.2007.6.1

2016 Impact Factor: 0.994

Tools

Metrics

  • PDF downloads (18)
  • HTML views (305)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2018 American Institute of Mathematical Sciences