Classification and evolution of bifurcation curves for a one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity

Yu-Hao Liang; Shin-Hwa Wang

doi:10.3934/dcds.2020071

Previous Article
Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. I. Asymptotic dynamics in fixed unbounded domain
DCDS Home
This Issue
Next Article
Unlikely intersections over finite fields: Polynomial orbits in small subgroups

February 2020, 40(2): 1075-1105. doi: 10.3934/dcds.2020071

Classification and evolution of bifurcation curves for a one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity

Yu-Hao Liang ^1,, and Shin-Hwa Wang ^2,

1.	Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan
2.	Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan

^* Corresponding author: Yu-Hao Liang

Received May 2019 Revised August 2019 Published November 2019

Fund Project: This work is partially supported by the Ministry of Science and Technology of the Republic of China under grant No. MOST 105-2115-M-007-002-MY3 and No. MOST 107-2115-M-390-006-MY2.

We study the classification and evolution of bifurcation curves of positive solutions of the one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity given by

$ \left \{ \begin{array} [c]{l}u^{\prime \prime}(x)+\lambda(-\varepsilon u^{3}+u^{2}+u+1) = 0,\;0<x<1,\\ u(0) = 0,\ u^{\prime}(1) = -c<0, \end{array} \right. $

where

$ 1/10\leq \varepsilon \leq1/5 $

. It is interesting to find that the evolution of bifurcation curves is not completely identical with that for the one-dimensional perturbed Gelfand equations, even though it is the same for these two problems with zero Dirichlet boundary conditions. In fact, we prove that there exist a positive number

$ \varepsilon^{\ast}\,(\approx0.178) $

and three nonnegative numbers

$ c_{0}(\varepsilon)<c_{1}(\varepsilon)<c_{2}(\varepsilon) $

defined on

$ [1/10,1/5] $

with

$ c_{0} = 0 $

$ 1/10<\varepsilon \leq \varepsilon^{\ast} $

and

$ c_{0}>0 $

$ \varepsilon^{\ast}<\varepsilon \leq1/5 $

, such that, on the

$ (\lambda,\Vert u\Vert_{\infty}) $

-plane, (ⅰ) when

$ 0<c\leq c_{0}(\varepsilon) $

and

$ c\geq c_{2}(\varepsilon) $

, the bifurcation curve is strictly increasing; (ⅱ) when

$ c_{0}(\varepsilon)<c<c_{1}(\varepsilon) $

, the bifurcation curve is

$ S $

-shaped; (ⅲ) when

$ c_{1}(\varepsilon)\leq c<c_{2}(\varepsilon) $

, the bifurcation curve is

$ \subset $

-shaped.

Keywords: Bifurcation, positive solution, cubic nonlinearity, S-shaped bifurcation curve, $ \subset $-shaped bifurcation curve, time map.

Mathematics Subject Classification: 34B18, 74G35.

Citation: Yu-Hao Liang, Shin-Hwa Wang. Classification and evolution of bifurcation curves for a one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 1075-1105. doi: 10.3934/dcds.2020071

References:

[1]	T. Boddington, P. Gray and C. Robinson, Thermal explosion and the disappearance of criticality at small activation energies: Exact results for the slab, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 368 (1979), 441-461. doi: 10.1098/rspa.1979.0140. Google Scholar
[2]	K. J. Brown, M. M. A. Ibrahim and R. Shivaji, S-shaped bifurcation curves, Nonlinear Anal., 5 (1981), 475-486. doi: 10.1016/0362-546X(81)90096-1. Google Scholar
[3]	M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. doi: 10.1007/BF00282325. Google Scholar
[4]	Y. Du and Y. Lou, Proof of a conjecture for the perturbed Gelfand equation from combustion theory, J. Differential Equations, 173 (2001), 213-230. doi: 10.1006/jdeq.2000.3932. Google Scholar
[5]	J. Goddard II, Q. Morris, R. Shivaji and B. Son, Bifurcation curves for singular and nonsingular problems with nonlinear boundary conditions, Electron. J. Differential Equations, 2018 (2018), Paper No. 26, 12 pp. Google Scholar
[6]	J. Goddard II, R. Shivaji and E. K. Lee, A double S-shaped bifurcation curve for a reaction-diffusion model with nonlinear boundary conditions, Bound. Value Probl., 2010 (2010), Art. ID 357542, 23 pp. doi: 10.1155/2010/357542. Google Scholar
[7]	P. V. Gordon, E. Ko and R. Shivaji, Multiplicity and uniqueness of positive solutions for elliptic equations with nonlinear boundary conditions arising in a theory of thermal explosion, Nonlinear Anal.: Real World Appl., 15 (2014), 51-57. doi: 10.1016/j.nonrwa.2013.05.005. Google Scholar
[8]	S.-Y. Huang and S.-H. Wang, On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion, Discret. Contin. Dyn. Syst., 35 (2015), 4839-4858. doi: 10.3934/dcds.2015.35.4839. Google Scholar
[9]	S.-Y. Huang and S.-H. Wang, Proof of a conjecture for the one-dimensional perturbed Gelfand problem from combustion theory, Arch. Ration. Mech. Anal., 222 (2016), 769-825. doi: 10.1007/s00205-016-1011-1. Google Scholar
[10]	K.-C. Hung and S.-H. Wang, A theorem on S-shaped bifurcation curve for a positone problem with convex-concave nonlinearity and its applications to the perturbed Gelfand problem, J. Differential Equations, 251 (2011), 223-237. doi: 10.1016/j.jde.2011.03.017. Google Scholar
[11]	K.-C. Hung and S.-H. Wang, Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications, Trans. Amer. Math. Soc., 365 (2013), 1933-1956. doi: 10.1090/S0002-9947-2012-05670-4. Google Scholar
[12]	K.-C. Hung, S.-H. Wang and C.-H. Yu, Existence of a double $S$-shaped bifurcation curve with six positive solutions for a combustion problem, J. Math. Anal. Appl., 392 (2012), 40-54. doi: 10.1016/j.jmaa.2012.02.036. Google Scholar
[13]	P. Korman and Y. Li, On the exactness of an S-shaped bifurcation curve, Proc. Amer. Math. Soc., 127 (1999), 1011-1020. doi: 10.1090/S0002-9939-99-04928-X. Google Scholar
[14]	P. Korman, Y. Li and T. Ouyang, A simplified proof of a conjecture for the perturbed Gelfand equation from combustion theory, J. Differential Equations, 263 (2017), 2874-2885. doi: 10.1016/j.jde.2017.04.016. Google Scholar
[15]	T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1–13. doi: 10.1512/iumj.1971.20.20001. Google Scholar
[16]	Y.-H. Liang, S.-H. Wang, Detailed proofs of some results in the article: Classification and evolution of bifurcation curves for a one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity, Available from: http://www.math.nthu.edu.tw/\symbol126shwang/SupplMatCubic.pdf. Google Scholar
[17]	Y.-H. Liang and S.-H. Wang, Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions, J. Differential Equations, 260 (2016), 8358-8387. doi: 10.1016/j.jde.2016.02.021. Google Scholar
[18]	Y.-H. Liang and S.-H. Wang, Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions Ⅱ, Electron. J. Differential Equations, 2017 (2017), Paper No. 61, 12 pp. Google Scholar
[19]	M. Mimura and K. Sakamoto, Multi-dimensional transition layers for an exothermic reaction-diffusion system in long cylindrical domains, J. Math. Sci. Univ. Tokyo, 3 (1996), 109-179. Google Scholar
[20]	J. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531. doi: 10.1006/jfan.1999.3483. Google Scholar
[21]	J. Shi, Multi-parameter bifurcation and applications, Topological Methods, Variational Methods and Their Applications (Taiyuan, 2002), World Sci. Publ., River Edge, NJ, 2003, 211–221. Google Scholar
[22]	R. Shivaji, Remarks on an S-shaped bifurcation curve, J. Math. Anal. Appl., 111 (1985), 374-387. doi: 10.1016/0022-247X(85)90223-9. Google Scholar
[23]	C.-C. Tsai, S.-H. Wang and S.-Y. Huang, Classification and evolution of bifurcation curves for a one-dimensional Neumann-Robin problem and its applications, Electron. J. Qual. Theory Differ. Equ., 2018 (2018), Paper No. 85, 30 pp. Google Scholar
[24]	C.-C. Tzeng, K.-C. Hung and S.-H. Wang, Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity, J. Differential Equations, 252 (2012), 6250-6274. doi: 10.1016/j.jde.2012.02.020. Google Scholar
[25]	S.-H. Wang, On S-shaped bifurcation curves, Nonlinear Anal., 22 (1994), 1475-1485. doi: 10.1016/0362-546X(94)90183-X. Google Scholar
[26]	X. Zhang and M. Feng, Double bifurcation diagrams and four positive solutions of nonlinear boundary value problems via time maps, Commun. Pure Appl. Anal., 17 (2018), 2149-2171. doi: 10.3934/cpaa.2018103. Google Scholar

show all references

References:

[1]	T. Boddington, P. Gray and C. Robinson, Thermal explosion and the disappearance of criticality at small activation energies: Exact results for the slab, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 368 (1979), 441-461. doi: 10.1098/rspa.1979.0140. Google Scholar
[2]	K. J. Brown, M. M. A. Ibrahim and R. Shivaji, S-shaped bifurcation curves, Nonlinear Anal., 5 (1981), 475-486. doi: 10.1016/0362-546X(81)90096-1. Google Scholar
[3]	M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. doi: 10.1007/BF00282325. Google Scholar
[4]	Y. Du and Y. Lou, Proof of a conjecture for the perturbed Gelfand equation from combustion theory, J. Differential Equations, 173 (2001), 213-230. doi: 10.1006/jdeq.2000.3932. Google Scholar
[5]	J. Goddard II, Q. Morris, R. Shivaji and B. Son, Bifurcation curves for singular and nonsingular problems with nonlinear boundary conditions, Electron. J. Differential Equations, 2018 (2018), Paper No. 26, 12 pp. Google Scholar
[6]	J. Goddard II, R. Shivaji and E. K. Lee, A double S-shaped bifurcation curve for a reaction-diffusion model with nonlinear boundary conditions, Bound. Value Probl., 2010 (2010), Art. ID 357542, 23 pp. doi: 10.1155/2010/357542. Google Scholar
[7]	P. V. Gordon, E. Ko and R. Shivaji, Multiplicity and uniqueness of positive solutions for elliptic equations with nonlinear boundary conditions arising in a theory of thermal explosion, Nonlinear Anal.: Real World Appl., 15 (2014), 51-57. doi: 10.1016/j.nonrwa.2013.05.005. Google Scholar
[8]	S.-Y. Huang and S.-H. Wang, On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion, Discret. Contin. Dyn. Syst., 35 (2015), 4839-4858. doi: 10.3934/dcds.2015.35.4839. Google Scholar
[9]	S.-Y. Huang and S.-H. Wang, Proof of a conjecture for the one-dimensional perturbed Gelfand problem from combustion theory, Arch. Ration. Mech. Anal., 222 (2016), 769-825. doi: 10.1007/s00205-016-1011-1. Google Scholar
[10]	K.-C. Hung and S.-H. Wang, A theorem on S-shaped bifurcation curve for a positone problem with convex-concave nonlinearity and its applications to the perturbed Gelfand problem, J. Differential Equations, 251 (2011), 223-237. doi: 10.1016/j.jde.2011.03.017. Google Scholar
[11]	K.-C. Hung and S.-H. Wang, Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications, Trans. Amer. Math. Soc., 365 (2013), 1933-1956. doi: 10.1090/S0002-9947-2012-05670-4. Google Scholar
[12]	K.-C. Hung, S.-H. Wang and C.-H. Yu, Existence of a double $S$-shaped bifurcation curve with six positive solutions for a combustion problem, J. Math. Anal. Appl., 392 (2012), 40-54. doi: 10.1016/j.jmaa.2012.02.036. Google Scholar
[13]	P. Korman and Y. Li, On the exactness of an S-shaped bifurcation curve, Proc. Amer. Math. Soc., 127 (1999), 1011-1020. doi: 10.1090/S0002-9939-99-04928-X. Google Scholar
[14]	P. Korman, Y. Li and T. Ouyang, A simplified proof of a conjecture for the perturbed Gelfand equation from combustion theory, J. Differential Equations, 263 (2017), 2874-2885. doi: 10.1016/j.jde.2017.04.016. Google Scholar
[15]	T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1–13. doi: 10.1512/iumj.1971.20.20001. Google Scholar
[16]	Y.-H. Liang, S.-H. Wang, Detailed proofs of some results in the article: Classification and evolution of bifurcation curves for a one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity, Available from: http://www.math.nthu.edu.tw/\symbol126shwang/SupplMatCubic.pdf. Google Scholar
[17]	Y.-H. Liang and S.-H. Wang, Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions, J. Differential Equations, 260 (2016), 8358-8387. doi: 10.1016/j.jde.2016.02.021. Google Scholar
[18]	Y.-H. Liang and S.-H. Wang, Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions Ⅱ, Electron. J. Differential Equations, 2017 (2017), Paper No. 61, 12 pp. Google Scholar
[19]	M. Mimura and K. Sakamoto, Multi-dimensional transition layers for an exothermic reaction-diffusion system in long cylindrical domains, J. Math. Sci. Univ. Tokyo, 3 (1996), 109-179. Google Scholar
[20]	J. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531. doi: 10.1006/jfan.1999.3483. Google Scholar
[21]	J. Shi, Multi-parameter bifurcation and applications, Topological Methods, Variational Methods and Their Applications (Taiyuan, 2002), World Sci. Publ., River Edge, NJ, 2003, 211–221. Google Scholar
[22]	R. Shivaji, Remarks on an S-shaped bifurcation curve, J. Math. Anal. Appl., 111 (1985), 374-387. doi: 10.1016/0022-247X(85)90223-9. Google Scholar
[23]	C.-C. Tsai, S.-H. Wang and S.-Y. Huang, Classification and evolution of bifurcation curves for a one-dimensional Neumann-Robin problem and its applications, Electron. J. Qual. Theory Differ. Equ., 2018 (2018), Paper No. 85, 30 pp. Google Scholar
[24]	C.-C. Tzeng, K.-C. Hung and S.-H. Wang, Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity, J. Differential Equations, 252 (2012), 6250-6274. doi: 10.1016/j.jde.2012.02.020. Google Scholar
[25]	S.-H. Wang, On S-shaped bifurcation curves, Nonlinear Anal., 22 (1994), 1475-1485. doi: 10.1016/0362-546X(94)90183-X. Google Scholar
[26]	X. Zhang and M. Feng, Double bifurcation diagrams and four positive solutions of nonlinear boundary value problems via time maps, Commun. Pure Appl. Anal., 17 (2018), 2149-2171. doi: 10.3934/cpaa.2018103. Google Scholar

Figure 1. Three different types of exactly $ S $-shaped bifurcation curves $ S_{c} $ with $ \lambda_{0}>0 $ and $ \left \Vert u_{\lambda_{0}}\right \Vert _{\infty}>0 $. (ⅰ) Type 1: $ \lambda_{0}<\lambda_{\ast}<\lambda^{\ast} $. (ⅱ) Type 2: $ \lambda_{0} = \lambda_{\ast}<\lambda^{\ast} $. (ⅲ) Type 3: $ \lambda_{\ast}<\lambda_{0}<\lambda^{\ast} $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. Exactly $ \subset $-shaped bifurcation curve $ S_{c} $ with $ \lambda _{0}>0 $ and $ \left \Vert u_{\lambda_{0}}\right \Vert _{\infty}>0 $

Figure Options

Download full-size image

Download as PowerPoint slide

17,Fig. 4])">Figure 3. Numerical simulations of bifurcation curves $ \bar{S} $ of (5) and $ \bar{S}_{c} $ of (6) with $ \bar{f}_{a}(u) = \exp \left( \frac{au}{a+u}\right) , $ $ a = 5 $ and with varying $ c>0 $ on the $ (\lambda,\left \Vert u\right \Vert _{\infty}) $-plane of the bi-logarithm coordinates. Here $ c_{1,2}^{-}<c_{1,2}\,(\approx 0.488)<c_{1,2}^{+}<c_{1}\,(\approx1.365)<c_{1}^{+}<c_{2}\,(\approx 7.718)<c_{2}^{+}<c_{3}\,(\approx47.711)<c_{3}^{+} $ (adopted from [17,Fig. 4])

Figure Options

Download full-size image

Download as PowerPoint slide

Fig. 3)">Figure 4. Numerical simulations of bifurcation curves $ S $ of (2) and $ S_{c} $ of (1) with $ \varepsilon = 1/10 $ and varying $ c>0 $ on the $ (\lambda,\left \Vert u\right \Vert _{\infty}) $-plane. Here $ c_{1,2}^{-}<c_{1,2}\,(\approx0.384)<c_{1,2}^{+} <c_{1}\,(\approx1.117)<c_{1}^{+}<c_{2}\,(\approx39.438)<c_{2}^{+} $ and $ \beta_{1/10}\approx10.992 $ (cf. Fig. 3)

Figure Options

Download full-size image

Download as PowerPoint slide

Fig. 4)">Figure 5. Numerical simulation of the bifurcation surface $\Gamma \equiv \left \{ (\lambda,c,\left \Vert u_{\lambda,c}\right \Vert _{\infty}):\lambda,c>0\text{ and }u_{\lambda,c}\text{ is a positive solution of (1) )}\right \} $ with $ \varepsilon = 1/10 $ in the $ (\lambda,c,\left \Vert u\right \Vert _{\infty}) $-space. (cf. Fig. 4)

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6. Numerical simulations of bifurcation curves $ S $ of (2) and $ S_{c} $ of (1) with $ \varepsilon = 1/5 $ and varying $ c>0 $ on the $ (\lambda,\left \Vert u\right \Vert _{\infty}) $-plane. Here $ c_{0}^{-}<c_{0}\,(\approx0.121)<c_{1,2}^{-} <c_{1,2}\,(\approx0.609)<c_{1,2}^{+}<c_{1}\,(\approx1.120)<c_{1}^{+} <c_{2}\,(\approx19.052)<c_{2}^{+} $ and $ \beta_{1/5}\approx5.977 $

Figure Options

Download full-size image

Download as PowerPoint slide

Fig. 6)">Figure 7. Numerical simulation of the bifurcation surface $\Gamma \equiv \left \{ (\lambda,c,\left \Vert u_{\lambda,c}\right \Vert _{\infty}):\lambda,c>0\text{ and }u_{\lambda,c}\text{ is a positive solution of (1)}\right \} $ with $ \varepsilon = 1/5 $ in the $ (\lambda,c,\left \Vert u\right \Vert _{\infty}) $-space. (cf. Fig. 6)

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8. Numerical simulations of bifurcation curves $ S $ and $ S_{c} $ with $ \varepsilon = 12/5 $ and varying $ c>0 $ on the $ (\lambda,\left \Vert u\right \Vert _{\infty}) $-plane. Here $ \beta_{12/5}\approx1.120 $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9. Numerical simulations of bifurcation curves $ S $ of (2) and $ S_{c} $ of (1) with $ \varepsilon = 7/4 $ and varying $ c>0 $ on the $ (\lambda,\left \Vert u\right \Vert _{\infty}) $-plane. Here $ c_{0}^{-}<c_{0}\,(\approx1.432)<c_{0}^{+} <c_{2}\,(\approx2.382)<c_{2}^{+} $ and $ \beta_{7/4}\approx1.327 $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 10. (a): The conjectured classification of bifurcation curves $ S_{c} = S_{\varepsilon,c} $ of (1) on the first quadrant of $ (\varepsilon,c) $-plane. (b)–(i): Distinct shapes of bifurcation curves $ S_{c} $ on the $ (\lambda,\left \Vert u\right \Vert _{\infty}) $-plane in the different partitions of the first quadrant of $ (\varepsilon,c) $-plane

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 11. Numerical simulations of bifurcation curves $ \hat{S} $ of (52) and $ \hat{S}_{c} $ of (51) with $ \varepsilon = 1/10 $ and varying $ c>0 $ on the $ (\lambda,\left \Vert u\right \Vert _{\infty}) $-plane. Here $ c_{0,1} ^{-}<c_{0,1}(\approx0.750)<c_{0,1}^{+}<c_{1}\,(\approx4.009)<c_{1}^{+} <c_{2}\,(\approx28.174)<c_{2,3}^{-}<c_{2,3}\,(\approx28.570)<c_{2,3}^{+} <c_{3}\,(\approx28.727)<c_{3}^{+}\, $ and $ \hat{\beta}_{1/10}\approx9.014 $

Figure Options

Download full-size image

Download as PowerPoint slide

[1]	Tzung-shin Yeh. S-shaped and broken s-shaped bifurcation curves for a multiparameter diffusive logistic problem with holling type-Ⅲ functional response. Communications on Pure & Applied Analysis, 2017, 16 (2) : 645-670. doi: 10.3934/cpaa.2017032
[2]	Sabri Bensid, Jesús Ildefonso Díaz. Stability results for discontinuous nonlinear elliptic and parabolic problems with a S-shaped bifurcation branch of stationary solutions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1757-1778. doi: 10.3934/dcdsb.2017105
[3]	Shao-Yuan Huang, Shin-Hwa Wang. On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 4839-4858. doi: 10.3934/dcds.2015.35.4839
[4]	Chih-Yuan Chen, Shin-Hwa Wang, Kuo-Chih Hung. S-shaped bifurcation curves for a combustion problem with general arrhenius reaction-rate laws. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2589-2608. doi: 10.3934/cpaa.2014.13.2589
[5]	Brian Ryals, Robert J. Sacker. Bifurcation in the almost periodic $ 2 $D Ricker map. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021089
[6]	Mohan Mallick, R. Shivaji, Byungjae Son, S. Sundar. Bifurcation and multiplicity results for a class of $n\times n$ $p$-Laplacian system. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1295-1304. doi: 10.3934/cpaa.2018062
[7]	Valeria Banica, Luis Vega. Singularity formation for the 1-D cubic NLS and the Schrödinger map on $\mathbb S^2$. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1317-1329. doi: 10.3934/cpaa.2018064
[8]	Sugata Gangopadhyay, Goutam Paul, Nishant Sinha, Pantelimon Stǎnicǎ. Generalized nonlinearity of $ S$-boxes. Advances in Mathematics of Communications, 2018, 12 (1) : 115-122. doi: 10.3934/amc.2018007
[9]	Piotr Bizoń, Dominika Hunik-Kostyra, Dmitry Pelinovsky. Stationary states of the cubic conformal flow on $ \mathbb{S}^3 $. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 1-32. doi: 10.3934/dcds.2020001
[10]	Andrus Giraldo, Neil G. R. Broderick, Bernd Krauskopf. Chaotic switching in driven-dissipative Bose-Hubbard dimers: When a flip bifurcation meets a T-point in $ \mathbb{R}^4 $. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021217
[11]	Xue Dong He, Roy Kouwenberg, Xun Yu Zhou. Inverse S-shaped probability weighting and its impact on investment. Mathematical Control & Related Fields, 2018, 8 (3&4) : 679-706. doi: 10.3934/mcrf.2018029
[12]	Omer Gursoy, Kamal Adli Mehr, Nail Akar. Steady-state and first passage time distributions for waiting times in the $ MAP/M/s+G $ queueing model with generally distributed patience times. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021078
[13]	Xianjun Wang, Huaguang Gu, Bo Lu. Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron. Electronic Research Archive, 2021, 29 (5) : 2987-3015. doi: 10.3934/era.2021023
[14]	Juan Dávila, Manuel Del Pino, Catalina Pesce, Juncheng Wei. Blow-up for the 3-dimensional axially symmetric harmonic map flow into $ S^2 $. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 6913-6943. doi: 10.3934/dcds.2019237
[15]	Pak Tung Ho. Prescribing $ Q $-curvature on $ S^n $ in the presence of symmetry. Communications on Pure & Applied Analysis, 2020, 19 (2) : 715-722. doi: 10.3934/cpaa.2020033
[16]	Tetsuya Ishiwata, Takeshi Ohtsuka. Evolution of a spiral-shaped polygonal curve by the crystalline curvature flow with a pinned tip. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5261-5295. doi: 10.3934/dcdsb.2019058
[17]	Guoyuan Chen, Yong Liu, Juncheng Wei. Nondegeneracy of harmonic maps from $ {{\mathbb{R}}^{2}} $ to $ {{\mathbb{S}}^{2}} $. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3215-3233. doi: 10.3934/dcds.2019228
[18]	Chengxin Du, Changchun Liu. Time periodic solution to a two-species chemotaxis-Stokes system with $ p $-Laplacian diffusion. Communications on Pure & Applied Analysis, 2021, 20 (12) : 4321-4345. doi: 10.3934/cpaa.2021162
[19]	Jaime Angulo Pava, César A. Hernández Melo. On stability properties of the Cubic-Quintic Schródinger equation with $\delta$-point interaction. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2093-2116. doi: 10.3934/cpaa.2019094
[20]	Sugata Gangopadhyay, Constanza Riera, Pantelimon Stănică. Gowers $ U_2 $ norm as a measure of nonlinearity for Boolean functions and their generalizations. Advances in Mathematics of Communications, 2021, 15 (2) : 241-256. doi: 10.3934/amc.2020056

2020 Impact Factor: 1.392

Tools

Metrics

Tools

Article outline

Figures and Tables

2020 Impact Factor: 1.392

Tools

Figures and Tables

2020 Impact Factor: 1.392

American Institute of Mathematical Sciences

Classification and evolution of bifurcation curves for a one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Classification and evolution of bifurcation curves for a one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors