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Discrete & Continuous Dynamical Systems - S

November 2019 , Volume 12 , Issue 7

Issue of DCDS-S dedicated to the 70th birthday of Norman Dancer

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Preface: Special issue of DCDS-S dedicated to the 70th birthday of Norman Dancer
Zalman Balanov, Wiesław Krawcewicz and Jianshe Yu
2019, 12(7): ⅰ-ⅱ doi: 10.3934/dcdss.20197i +[Abstract](679) +[HTML](86) +[PDF](1369.58KB)
Abstract:
The work of Norman Dancer
Yihong Du
2019, 12(7): 1807-1833 doi: 10.3934/dcdss.2019119 +[Abstract](1316) +[HTML](473) +[PDF](563.04KB)
Abstract:

In this article, a sample of Norman Dancer's published works are briefly described, to give the reader a taste of his deep and important research on nonlinear functional analysis, nonlinear ODE and PDE problems, and dynamical systems. The sample covers a variety of topics where Norman Dancer has made remarkable contributions.

The author takes this opportunity to express his deep admiration of the work of Professor Norman Dancer, and to thank him for the kind help to the development of the author's career, which has been greatly influenced by him and his work. 

On a degree associated with the Gross-Pitaevskii system with a large parameter
E. Norman Dancer
2019, 12(7): 1835-1839 doi: 10.3934/dcdss.2019120 +[Abstract](1176) +[HTML](314) +[PDF](252.8KB)
Abstract:

In a number of cases we calculate the sum of the degrees of the small positive solutions of the Gross-Pitaevskii system when the interaction is strong.

Subharmonic solutions for a class of Lagrangian systems
Anouar Bahrouni, Marek Izydorek and Joanna Janczewska
2019, 12(7): 1841-1850 doi: 10.3934/dcdss.2019121 +[Abstract](1182) +[HTML](539) +[PDF](370.68KB)
Abstract:

We prove that second order Hamiltonian systems \begin{document}$ -\ddot{u} = V_{u}(t,u) $\end{document} with a potential \begin{document}$ V\colon \mathbb{R} \times \mathbb{R} ^N\to \mathbb{R} $\end{document} of class \begin{document}$ C^1 $\end{document}, periodic in time and superquadratic at infinity with respect to the space variable have subharmonic solutions. Our intention is to generalise a result on subharmonics for Hamiltonian systems with a potential satisfying the global Ambrosetti-Rabinowitz condition from [14]. Indeed, we weaken the latter condition in a neighbourhood of \begin{document}$ 0\in \mathbb{R} ^N $\end{document}. We will also discuss when subharmonics pass to a nontrivial homoclinic orbit.

On good deformations of $ A_m $-singularities
Zalman Balanov and Yakov Krasnov
2019, 12(7): 1851-1866 doi: 10.3934/dcdss.2019122 +[Abstract](1034) +[HTML](374) +[PDF](422.7KB)
Abstract:

The standard way to study a compound singularity is to decompose it into the simpler ones using either blow up techniques or appropriate deformations. Among deformations, one distinguishes between miniversal deformations (related to deformations of a basis of the local algebra of singularity) and good deformations (one-parameter deformations with simple singularities coalescing into a multiple one). In concrete settings, explicit construction of a good deformation is an art rather than a science. In this paper, we discuss some cases important from the application viewpoint when explicit good deformations can be constructed and effectively used. Our applications include: (a) an \begin{document}$ n $\end{document}-dimensional Euler-Jacobi formula with simple and double roots, and (b) a simple approach to the known classification of phase portraits of planar differential systems around linearly non-zero equilibrium.

The Morse property for functions of Kirchhoff-Routh path type
Thomas Bartsch, Anna Maria Micheletti and Angela Pistoia
2019, 12(7): 1867-1877 doi: 10.3934/dcdss.2019123 +[Abstract](1138) +[HTML](343) +[PDF](433.5KB)
Abstract:

For a bounded domain \begin{document}$ \Omega\subset\mathbb{R}^n $\end{document} let \begin{document}$ H_\Omega:\Omega\times\Omega\to\mathbb{R} $\end{document} be the regular part of the Dirichlet Green function for the Laplace operator. Given a fixed arbitrary \begin{document}$ {\mathcal C}^2 $\end{document} function \begin{document}$ f:{\mathcal D}\to\mathbb{R} $\end{document}, defined on an open subset \begin{document}$ {\mathcal D}\subset\mathbb{R}^{nN} $\end{document}, and fixed coefficients \begin{document}$ \lambda_1,\dots,\lambda_N\in\mathbb{R}\setminus\{0\} $\end{document} we consider the function \begin{document}$ f_\Omega:{\mathcal D}\cap\Omega^N\to\mathbb{R} $\end{document} defined as

We prove that \begin{document}$ f_\Omega $\end{document} is a Morse function for most domains \begin{document}$ \Omega $\end{document} of class \begin{document}$ {\mathcal C}^{m+2,\alpha} $\end{document}, any \begin{document}$ m\ge0 $\end{document}, \begin{document}$ 0<\alpha<1 $\end{document}. This applies in particular to the Robin function \begin{document}$ h:\Omega\to\mathbb{R} $\end{document}, \begin{document}$ h(x) = H_\Omega(x,x) $\end{document}, and to the Kirchhoff-Routh path function where \begin{document}$ \Omega\subset\mathbb{R}^2 $\end{document}, \begin{document}$ {\mathcal D} = \{x\in\mathbb{R}^{2N}: x_j\ne x_k \; \text{for }\; j\ne k \} $\end{document}, and

Dihedral molecular configurations interacting by Lennard-Jones and Coulomb forces
Irina Berezovik, Wieslaw Krawcewicz and Qingwen Hu
2019, 12(7): 1879-1903 doi: 10.3934/dcdss.2019124 +[Abstract](1007) +[HTML](322) +[PDF](300.33KB)
Abstract:

In this paper, we investigate nonlinear periodic vibrations of a group of particles with a planar dihedral configuration governed by the Lennard-Jones and Coulomb forces. Using the gradient equivariant degree, we provide a full topological classification of the periodic solutions with both temporal and spatial symmetries. In the process, we provide general formulae for the spectrum of the linearized system of equations describing the above configuration, which allows us to obtain the critical frequencies of the particles' motions. The obtained frequencies represent the set of all critical periods for small amplitude periodic solutions emerging from a given stationary symmetric orbit of solutions.

Global bifurcations and a priori bounds of positive solutions for coupled nonlinear Schrödinger Systems
Guowei Dai, Rushun Tian and Zhitao Zhang
2019, 12(7): 1905-1927 doi: 10.3934/dcdss.2019125 +[Abstract](1214) +[HTML](594) +[PDF](530.21KB)
Abstract:

In this paper, we consider the following coupled elliptic system

Under symmetric assumptions \begin{document}$ \lambda_1 = \lambda_2, \mu_1 = \mu_2 $\end{document}, we determine the number of \begin{document}$ \gamma $\end{document}-bifurcations for each \begin{document}$ \beta\in(-1, +\infty) $\end{document}, and study the behavior of global \begin{document}$ \gamma $\end{document}-bifurcation branches in \begin{document}$ [-1, 0]\times H_r^1\left( \mathbb{R} ^N\right)\times H_r^1\left( \mathbb{R} ^N\right) $\end{document}. Moreover, several results for \begin{document}$ \gamma = 0 $\end{document}, such as priori bounds, are of independent interests, which are improvements of corresponding theorems in [6] and [35].

Least energy solutions for fractional Kirchhoff type equations involving critical growth
Yinbin Deng and Wentao Huang
2019, 12(7): 1929-1954 doi: 10.3934/dcdss.2019126 +[Abstract](1361) +[HTML](552) +[PDF](526.06KB)
Abstract:

We study the following fractional Kirchhoff type equation:

where \begin{document}$ a, \ b>0 $\end{document} are constants, \begin{document}$ 2^*_s = \frac{6}{3-2s} $\end{document} with \begin{document}$ s\in(0, 1) $\end{document} is the critical Sobolev exponent in \begin{document}$ \mathbb{R} ^3 $\end{document}, \begin{document}$ V $\end{document} is a potential function on \begin{document}$ \mathbb{R} ^3 $\end{document}. Under some more general assumptions on \begin{document}$ f $\end{document} and \begin{document}$ V $\end{document}, we prove that the given problem admits a least energy solution by using a constrained minimization on Nehari-Pohozaev manifold and monotone method.

Branching and bifurcation
Patrick M. Fitzpatrick and Jacobo Pejsachowicz
2019, 12(7): 1955-1975 doi: 10.3934/dcdss.2019127 +[Abstract](1323) +[HTML](499) +[PDF](459.95KB)
Abstract:

By relating the set of branch points \begin{document}$ \mathcal{B} (f) $\end{document} of a Fredholm mapping \begin{document}$ f $\end{document} to linearized bifurcation, we show, among other things, that under mild local assumptions at a single point, the set \begin{document}$ \mathcal B(f) $\end{document} is sufficiently large to separate the domain of the mapping. In the variational case, we will also provide estimates from below for the number of connected components of the complement of \begin{document}$ \mathcal B(f). $\end{document}

Multiple solutions for a critical quasilinear equation with Hardy potential
Fengshuang Gao and Yuxia Guo
2019, 12(7): 1977-2003 doi: 10.3934/dcdss.2019128 +[Abstract](1377) +[HTML](564) +[PDF](592.06KB)
Abstract:

In this paper, we investigate the following quasilinear equation involving a Hardy potential:

where \begin{document}$ 2^ * = \frac{2N}{N-2} $\end{document} is the Sobolev critical exponent for the embedding of \begin{document}$ H_0^1(\Omega) $\end{document} into \begin{document}$ L^p(\Omega) $\end{document}, \begin{document}$ a>0 $\end{document} is a constant and \begin{document}$ \Omega\subset \mathbb{R}^N $\end{document} is an open bounded domain which contains the origin. We will prove that under some suitable assumptions on \begin{document}$ a_{ij} $\end{document}, when \begin{document}$ N\geq 7 $\end{document} and \begin{document}$ \mu\in[0,\mu^*) $\end{document} for some constant \begin{document}$ \mu^* $\end{document}, problem (P) admits an unbounded sequence of solutions. To achieve this goal, we perform the subcritical approximation and the regularization perturbation.

Global symmetry-breaking bifurcations of critical orbits of invariant functionals
Anna Goƚȩbiewska, Norimichi Hirano and Sƚawomir Rybicki
2019, 12(7): 2005-2017 doi: 10.3934/dcdss.2019129 +[Abstract](1251) +[HTML](482) +[PDF](430.91KB)
Abstract:

In this article we present a method of study of a global symmetry-breaking bifurcation of critical orbits of invariant functionals. As a topological tool we use the degree for equivariant gradient maps. We underline that many known results on bifurcations of non-radial solutions of elliptic PDE's from the families of radial ones are consequences of our theory.

Selective Pyragas control of Hamiltonian systems
Edward Hooton, Pavel Kravetc, Dmitrii Rachinskii and Qingwen Hu
2019, 12(7): 2019-2034 doi: 10.3934/dcdss.2019130 +[Abstract](1342) +[HTML](545) +[PDF](486.58KB)
Abstract:

We consider a Newton system which has a branch (surface) of neutrally stable periodic orbits. We discuss sufficient conditions which allow arbitrarily small delayed Pyragas control to make one selected cycle asymptotically stable. In the case of small amplitude periodic solutions we give conditions in terms of the asymptotic expansion of the right hand side, while in the case of larger cycles we frame the conditions in terms of the Floquet modes of the target orbit as a solution of the uncontrolled system.

Existence of positive ground state solutions for Choquard equation with variable exponent growth
Gui-Dong Li and Chun-Lei Tang
2019, 12(7): 2035-2050 doi: 10.3934/dcdss.2019131 +[Abstract](1517) +[HTML](521) +[PDF](481.63KB)
Abstract:

In this paper, we investigate the following Choquard equation

where \begin{document}$ N\geq 3 $\end{document}, \begin{document}$ \alpha\in (0,N) $\end{document} and \begin{document}$ I_\alpha $\end{document} is the Riesz potential. If

where \begin{document}$ 1< p <\frac{N+\alpha}{N-2} $\end{document} and \begin{document}$ \Omega\subset\mathbb{R}^N $\end{document} is a bounded set with nonempty, we obtain the existence of positive ground state solutions by using the Nehari manifold.

Solutions of nonlinear periodic Dirac equations with periodic potentials
Xiaoyan Lin and Xianhua Tang
2019, 12(7): 2051-2061 doi: 10.3934/dcdss.2019132 +[Abstract](1378) +[HTML](457) +[PDF](403.85KB)
Abstract:

This paper is concerned with the nonlinear Dirac equation \begin{document}$ -i\sum_{k = 1}^{3}\alpha_{k}\partial_{k}u + [V(x)+a]\beta u + \omega u = f(x, u) $\end{document} in \begin{document}$ \mathbb{R}^3 $\end{document}, where \begin{document}$ V(x) $\end{document} and \begin{document}$ f(x, u) $\end{document} are periodic in \begin{document}$ x $\end{document}, \begin{document}$ f(x, u) $\end{document} is asymptotically linear and superlinear as \begin{document}$ |u|\rightarrow \infty $\end{document}. Under weaker assumptions on \begin{document}$ f $\end{document}, we obtain the existence of one nontrivial solution for the above equation.

A Leslie-Gower predator-prey model with a free boundary
Yunfeng Liu, Zhiming Guo, Mohammad El Smaily and Lin Wang
2019, 12(7): 2063-2084 doi: 10.3934/dcdss.2019133 +[Abstract](1402) +[HTML](540) +[PDF](388.99KB)
Abstract:

In this paper, we consider a Leslie-Gower predator-prey model in one-dimensional environment. We study the asymptotic behavior of two species evolving in a domain with a free boundary. Sufficient conditions for spreading success and spreading failure are obtained. We also derive sharp criteria for spreading and vanishing of the two species. Finally, when spreading is successful, we show that the spreading speed is between the minimal speed of traveling wavefront solutions for the predator-prey model on the whole real line (without a free boundary) and an elliptic problem that follows from the original model.

Periodic and subharmonic solutions for a 2$n$th-order $\phi_c$-Laplacian difference equation containing both advances and retardations
Peng Mei, Zhan Zhou and Genghong Lin
2019, 12(7): 2085-2095 doi: 10.3934/dcdss.2019134 +[Abstract](1401) +[HTML](469) +[PDF](381.32KB)
Abstract:

We consider a 2\begin{document}$n$\end{document}th-order nonlinear difference equation containing both many advances and retardations with \begin{document}$\phi_c$\end{document}-Laplacian. Using the critical point theory, we obtain some new and concrete criteria for the existence and multiplicity of periodic and subharmonic solutions in the more general case of the nonlinearity.

Traveling waves in Fermi-Pasta-Ulam chains with nonlocal interaction
Alexander Pankov
2019, 12(7): 2097-2113 doi: 10.3934/dcdss.2019135 +[Abstract](1338) +[HTML](522) +[PDF](406.92KB)
Abstract:

The paper is devoted to traveling waves in FPU type particle chains assuming that each particle interacts with several neighbors on both sides. Making use of variational techniques, we prove that under natural assumptions there exist monotone traveling waves with periodic velocity profile (periodic waves) as well as waves with localized velocity profile (solitary waves). In fact, we obtain periodic waves by means of a suitable version of the Mountain Pass Theorem. Then we get solitary waves in the long wave length limit.

Ground states of nonlinear Schrödinger equations with fractional Laplacians
Zupei Shen, Zhiqing Han and Qinqin Zhang
2019, 12(7): 2115-2125 doi: 10.3934/dcdss.2019136 +[Abstract](1312) +[HTML](525) +[PDF](387.3KB)
Abstract:

Inspired by Schaftingen [15], we develop a symmetric variational principle for the field equation involving a fractional Laplacians

As an application, we prove the existence of symmetric ground states in the fractional Sobolev space \begin{document}$ H^\alpha (\mathbb{R}^N) $\end{document}. These results improve some known ones in the literature. An example is also given to illustrate our results.

On periodic solutions in the Whitney's inverted pendulum problem
Roman Srzednicki
2019, 12(7): 2127-2141 doi: 10.3934/dcdss.2019137 +[Abstract](1228) +[HTML](464) +[PDF](381.34KB)
Abstract:

In the book "What is Mathematics?" Richard Courant and Herbert Robbins presented a solution of a Whitney's problem of an inverted pendulum on a railway carriage moving on a straight line. Since the appearance of the book in 1941 the solution was contested by several distinguished mathematicians. The first formal proof based on the idea of Courant and Robbins was published by Ivan Polekhin in 2014. Polekhin also proved a theorem on the existence of a periodic solution of the problem provided the movement of the carriage on the line is periodic. In the present paper we slightly improve the Polekhin's theorem by lowering the regularity class of the motion and we prove a theorem on the existence of a periodic solution if the carriage moves periodically on the plane.

Positive solutions of doubly coupled multicomponent nonlinear Schrödinger systems
Jiabao Su, Rushun Tian and Zhi-Qiang Wang
2019, 12(7): 2143-2161 doi: 10.3934/dcdss.2019138 +[Abstract](1386) +[HTML](471) +[PDF](507.51KB)
Abstract:

In this paper, we study the following doubly coupled multicomponent system

where \begin{document}$ \Omega\subset \mathbb{R} ^N $\end{document} and \begin{document}$ N = 2,3 $\end{document}; \begin{document}$ \lambda_j, \gamma_{jk} = \gamma_{kj}, \mu_j, \beta_{jk} = \beta_{kj} $\end{document} are constants, \begin{document}$ j, k = 1, 2, ..., n $\end{document}, \begin{document}$ n\geq 2 $\end{document}. We prove some existence and nonexistence results for positive solutions of this system. If the system is fully symmetric, i.e. \begin{document}$ \lambda_j\equiv\lambda, \gamma_{jk}\equiv\gamma, \mu_j\equiv\mu, \beta_{jk}\equiv\beta $\end{document}, we study the multiplicity and bifurcation phenomena of positive solution.

Ground state homoclinic solutions for a second-order Hamiltonian system
Xiaoping Wang
2019, 12(7): 2163-2175 doi: 10.3934/dcdss.2019139 +[Abstract](1405) +[HTML](486) +[PDF](447.37KB)
Abstract:

Consider the second-order Hamiltonian system

where \begin{document}$ t\in {\mathbb{R}}, u\in {\mathbb{R}}^{N} $\end{document}, \begin{document}$ L: \mathbb{R}\rightarrow {\mathbb{R}}^{N\times N} $\end{document} and \begin{document}$ W: {\mathbb{R}}\times {\mathbb{R}}^{N}\rightarrow {\mathbb{R}} $\end{document}. We mainly study the case when both \begin{document}$ L $\end{document} and \begin{document}$ W $\end{document} are periodic in \begin{document}$ t $\end{document} and \begin{document}$ 0 $\end{document} belongs to a spectral gap of \begin{document}$ \sigma\left(-\frac{d^2}{dt^2} +L\right) $\end{document}. We prove that the above system possesses a ground state homoclinic solution under assumptions which are weaker than the ones known in the literature.

The mean and noise of FPT modulated by promoter architecture in gene networks
Kunwen Wen, Lifang Huang, Qiuying Li, Qi Wang and Jianshe Yu
2019, 12(7): 2177-2194 doi: 10.3934/dcdss.2019140 +[Abstract](1279) +[HTML](505) +[PDF](475.08KB)
Abstract:

Increasing experimental evidences suggest that cell phenotypic variation often depends on the accumulation of some special proteins. Recently, a lot of studies have shown that the complexity of promoter architecture plays a major role in regulating transcription and controlling expression dynamics and further phenotype. One unanswered question is why the organism chooses such a complex promoter architecture and how the promoter architecture affects the timing of proteins amount up to a given threshold. To address this issue, we study the effect of promoter architecture on the first-passage time (FPT) by formulating a multi-state gene model, that may reflect the complexity of promoter architecture. We derive analytical formulae for FPT moments in each case of irreversible promoter and reversible promoter regulation, which is the first time to give these analytical results in the existing literature. We show that the mean and noise of FPT increase with the state number of promoter architecture if the mean residence time at \begin{document}$ off$\end{document} states is not fixed. Inversely, if the mean residence time at \begin{document}$ off$\end{document} states is fixed, then complex promoter architecture will not vary the mean of FPT but will tend to decrease the noise of FPT. Our results show that, in the same inactive promoter states, the noise of FPT with promoters in irreversible case is always less than that in reversible case. In conclusion, our results reveal the effect of the promoter architecture on FPT and enhance understanding of the regulation mechanism of gene expression.

Bifurcation analysis of a stage-structured predator-prey model with prey refuge
Qing Zhu, Huaqin Peng, Xiaoxiao Zheng and Huafeng Xiao
2019, 12(7): 2195-2209 doi: 10.3934/dcdss.2019141 +[Abstract](1423) +[HTML](478) +[PDF](367.13KB)
Abstract:

A stage-structured predator-prey model with prey refuge is considered. Using the geometric stability switch criteria, we establish stability of the positive equilibrium. Stability and direction of periodic solutions arising from Hopf bifurcations are obtained by using the normal form theory and center manifold argument. Numerical simulations confirm the above theoretical results.

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