Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2009 (2009), Article ID 785068, 28 pages
http://dx.doi.org/10.1155/2009/785068
Research Article

Existence of Solutions for a Nonlinear Algebraic System

1School of Science, Tianjin University of Commerce, Tianjin 300134, China
2School of Science, Qingdao Technological University, Fushun Road 11, Qingdao 266033, China

Received 30 April 2009; Accepted 15 August 2009

Academic Editor: Binggen Zhang

Copyright © 2009 Guang Zhang and Liang Bai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

As well known, the existence and nonexistence of solutions for nonlinear algebraic systems are very important since they can provide the necessary information on limiting behaviors of many dynamic systems, such as the discrete reaction-diffusion equations, the coupled map lattices, the compartmental systems, the strongly damped lattice systems, the complex dynamical networks, the discrete-time recurrent neural networks, and the discrete Turing models. In this paper, both the existence of nonzero solution pairs and the nonexistence of nontrivial or nonzero solutions for a nonlinear algebraic system will be considered by using the critical point theory and Lusternik-Schnirelmann category theory. The process of proofs on the obtained results is simple, the conditions of theorems are also easy to be verified, however, some of them improve the known ones even if the system is reduced to the precial cases, in particular, others of them are still new.

1. Introduction

In this paper, the nonlinear algebraic system,

will be considered, where is a parameter,

are column vectors with is a continuous function defined on and for and , and is a positive integer. Also is an square matrix that there exists a positive diagonal matrix such that is nonnegative definite. The letter will denote transposition.

For a given , a column vector is said to be a solution of (1.1) corresponding to it if substitution of and into (1.1) renders it an identity. The vector is said to be positive if for , negative if for , and nonzero if for . Positive, negative, and (strongly) nonzero vector are denoted by and respectively. If there exists such that , it will be called nontrivial solution of (1.1). In this case, it is denoted by .

First note that is always a trivial solution of (1.1). Also note that if is a solution of (1.1), then is a solution of (1.1) as well. Therefore, we always consider solution pairs, .

Nonlinear systems of the form (1.1) arise in many applications such as the discrete models of steady-state equations of reaction–diffusion equations (see [16]), the discrete analogue of the periodic boundary value problems (see [711]), the steady-state equations of coupled map lattices (see [1225]), the discrete periodic boundary value problems (see [2629]), the steady-state equations of compartmental system (see [3034]), the steady-state models on complex dynamical network ([3539]), the steady-state systems of discrete Turing instability models (see [5, 33, 4068]). Thus, the existence and the nonexistence of solutions on (1.1) are very important.

In fact, the special cases of (1.1) have been extensively studied by a number of authors, see [2629, 3638] and the listed references therein. However, our results improve and extend the known ones even if (1.1) is reduced to their cases, in particular, some of them are new.

In this paper, the existence of solution pairs for the nonlinear algebraic system (1.1) will be considered by using the critical point theory and Lusternik-Schnirelmann category theory [69] or [70]. The nonexistence of nontrivial and nonzero solutions of (1.1) will also be established. The present paper is organized as follows. In Section 2, problems in various areas are transformed into system (1.1). In Section 3, we discuss turing instability. Then the nonexistence of solutions for (1.1) will be studied in Section 4, here, all results are new. Furthermore, in Section 5, the existence of solution pairs for (1.1) will be considered, some known results will be extended and improved, in particular, the method of proofs is different from previous ones. Some applications will be presented in Section 6.

2. Problems Expressed by (1.1)

A lot of problems in various areas can be expressed by (1.1). In this section, we will pick some typical examples.

2.1. Periodic Boundary Value Problems

As well known, steady-state equations of many important models in application, such as the nonlinear reaction—diffusion equations [6], the generalized reaction Duffing model [4], and the Fisher equation [1], can be expressed by the following equation:

Thus, existence of solutions for second-order differential equation with periodic boundary value condition has been extensively studied by a number of authors (see [811]).

By using finite differences [7], discrete analogue of the periodic boundary value problem

can be written by

where with , with , for all , , and

In view of (2.3), we can obtain (1.1) with the square matrix having the form

The conditions and lead to , for . At the same time, the matrix has a zero eigenvalue and all other eigenvalues are positive, see [7].

Let

the conditions

imply that the matrix is symmetric. As an simiple example, we consider

then there exists the positive diagonal matrix such that

which is nonnegative definite.

Pattern dynamics in coupled map lattices (CMLs) have been extensively studied (see [1219, 22]). It has been found that CMLs exhibit a variety of space-time patterns such as kink-antikinks, traveling waves, space-time periodic structures, space-time intermittence and spatiotemporal chaos. It is believed that CMLs possess the potential to explain phenomena associated with turbulence and other spatiotemporal systems.

Consider the following coupled map lattice:

where denotes the time and denotes the spatial coordinate, is positive and treated as a parameter. This is a discrete analogue of the well-known Nagumo equation of the form

where is a positive constant. The continuous Nagumo equation (2.11) is used as a model for the spread of genetic traits [2] and for the propagation of nerve pulses in a nerve axon, neglecting recovery [3, 5].

Solution of (2.10) is said to be stationary wave solution if for all and . In view of (2.10), we have

Now, we consider the existence of -periodic solution of (2.12). Clearly, this is equal to the existence of solutions for (1.1), nonnegative definite matrix,

Recently, Zhou et al. [29] consider the discrete time second order dynamical systems

where and for any . Our results are also valid for the problem (2.14). In this case, the corresponding results also improve and extend the main theorem in [29].

Cai et al. [27] considered existence and multiplicity of periodic solution for the fourth-order difference equation

by using linking theorem, where the function is defined on with for a given positive integer and for all .

However, (2.15) is equal to (1.1), where

which is nonnegative definite.

Clearly, and of (2.15) can also be replaced by and of (2.14), respectively. In this case, our all results are new.

2.2. Compartmental System

Dynamic models of many processes in the biological and physical sciences give systems of ordinary differential equations called compartmental systems (see [3034] and references therein). For example, Jacquez and Simon [32] considered the following system:

where represents the mass of compartment . , is the transfer or rate coefficient from compartment to compartment , is the transfer coefficient from compartment to environment, represents the flows into the compartment from outside the system, or inflow. The entries of the matrix have three properties:

A system of the form of (2.17) for which satisfies (2.18) is called a compartmental system.

Considering the compartmental system that the transfer coefficient from compartment to compartment is equal to that from compartment to compartment for all the compartments in the system, and the inflow is determined by the mass of compartment , say, . Then steady-state equation of this kind compartmental system is

When is an odd function, we can obtain (1.1) with the square matrix having the form

By using (2.18), we can get that the matrix is nonnegative definite. To our knowledge, few results on the system (2.19) are found in literature.

2.3. Strongly Damped Lattice System

Recently, Li and Zhou [21] considered the following second-order lattice dynamic system:

where , , , are given, is a vector with the components and can be ordered as the following form of -dimensional vector in :

where , , . is a nonnegative definite matrix on with eigenvalues , and is the simple and minimal eigenvalue with corresponding eigenvector . Also , denote the th component of , , respectively. An example of is , at this time, (2.21) can be regarded as the discrete analogue of the initial-boundary value problem of the following continuous strongly damped wave equation:

which arises in wave phenomena in various areas in mathematical physics (see [20, 21, 2325] and references therein).

Steady-state equation of (2.21) is

where

When is an odd function, (2.24) can be expressed by (1.1). Thus, the existence of (2.24) is important, however, to our knowledge few results are seen in literature.

2.4. Complex Dynamical Network

Recently, complex dynamical network have been considered by Li et al. in [35]. Suppose that a complex network consists of identical linearly and diffusively coupled nodes, with each node being an -dimensional dynamical system. The state equations of this dynamical network are given by

where are the state variables of node , the constant represents the coupling strength between node and node , is a matrix linking coupled variables, and if some pairs , , with , then it means two coupled nodes are linked through their th and th state variables, respectively. The coupling matrix represents the coupling configuration of the network, which is assumed as a random network described by the E-R model or a scale-free network described by the B-A model. If there is a connection between node and node , then ; otherwise, . If the degree of node is defined to be the number of its outreaching connections, then

Let the diagonal elements be ,

In [35] the authors assumed there exists a generous stationary state for network (2.26) which is defined as

They suppose that is positive semidefinite and apply the pinning control strategy on a small fraction of the nodes to achieves the stabilization control of the goal (2.28). The network (2.26) can be rewritten by the system

where is the state vector, and the denotes the -dimensional functional value vector of , and

is nonnegative definite in [35].

When , steady-state equation of (2.29) can be expressed by (1.1). We think that the assumption (2.28) is not fact for a complex dynamical network. Thus, it is necessary to consider the existence of the other solutions. On the other hand, for any we also obtain a nonlinear algebraic system from (2.26), which is the special case of (1.1).

2.5. Discrete Neural Networks

Recently, Zhou et al. [39] considered the following discrete-time recurrent neural networks, which is thought to describe the dynamical characteristics of transiently chaotic neural network:

where is the internal state of neuron , is the output of neuron , is the input bias of neuron , is the self-recurrent bias of neuron , represents the damping factor of nerve membrane, and is the connection weight from neuron to neuron which is written as

In order to obtain the results on asymptotically stability, authors in [39], assume that the input-output function is , inverse function of exists, , there exists a matrix with for such that , and is positive definite, this implies that the matrix is nonnegative definite.

The steady-state equation of (2.31) is

where

When is an odd function, (2.33) can be expressed by (1.1).

On the other hand, Wang and Cheng [3638] considered the existence of steady-state solutions for the discrete neural networks

with the periodic boundary value conditions:

Our results are also valid for their problems and improve their theorems. In particular, for the more general system of the form

our results are also valid, where and are similar with (2.14).

3. Turing Instability

In 1952, Turing [68] suggested that, under certain conditions, chemicals can react and diffusion in such a way as to produce steady-state heterogeneous spatial patterns of chemical of morphogenic concentration. His idea is a simple but profound one. In view of Turing's theory framework, many Turing's patterns have been obtained by the observations, the numerical simulations, the animal coat patterns, the wavelength of the electrochemical system, the vegetation in many semiarid regions, the skeletal pattern formation of chick limb, and so forth, see [33, 4043, 4547, 49, 5357, 6267].

However, the mathematical theory of Turing's patterns is not clear, see the recent papers [44, 5052, 5961]. In fact, when the diffusion term is added, the steady-state solutions are different with the primary model. Some new solutions will be increased. On the other hand, all numerical simulations will use the discrete analogue of the corresponding reaction diffusion equations or systems. Thus, it is necessary to consider the existence of solutions for the discrete steady-state equations. In general, such equation can be expressed by a partial difference equation of the form

with the periodic boundary value conditions

However, we will give the other explanation in the later.

Because the existence of solutions of (3.1)-(3.2) is equal to (1.1), see Zhang et al. [71] or Zhang and Feng [72]. Clearly, and of (3.1) can also be replaced by and of (2.14), respectively.

4. Nonexistence

Usually, the existence of solutions is important. In fact, the nonexistence is also important because it can give some useful information for the existence of solutions. Thus, in this section, we will firstly give the nonexistence results of nontrivial solutions and nonzero solutions on the system (1.1). The obtained results are new.

When the matrix is nonnegative definite, we know that its eigenvalues are nonnegative and denote

and the corresponding orthonormal eigenvectors are .

First of all, we let be a nontrivial solution of (1.1). Multiplying (1.1) by on the left we get

which implies that

In view of the reference [73], we know that

Thus, we have

which implies that the following nonexistence result is fact.

Theorem 4.1. If there exists such that or then the system (1.1) has no nontrivial solutions.

Now, we assume that and denote

Clearly, the condition implies that the infimum

exists and

The condition implies that the supremum

exists and

Thus, Theorem 4.1 implies that the following results hold.

Corollary 4.2. Assume that . Then the condition implies that the system (1.1) has no nontrivial solutions when there exists such that holds. The condition implies that for any , the system (1.1) has no nontrivial solutions when holds.

When and satisfy , we easily obtain the following results.

Corollary 4.3. Assume that and , then condition (4.13) or (4.14) implies that the system (1.1) has no nontrivial solutions.

Similarly, when and satisfy the conditions: (H1) and , (H2) and , (H3) and , (H4) and , we have the following result.

Corollary 4.4. Assume that . If condition (H1) or (H2) holds, then (4.13) implies that the system (1.1) has no nontrivial solutions. If the condition (H3) or (H4) holds, then (4.14) implies that the system (1.1) has no nontrivial solutions.

Now, we again assume that the system (1.1) has a nonzero solution , then we have

Thus, we have the following result.

Theorem 4.5. Assume that there exist and such that and Then the system (1.1) has no nonzero solutions.

In view of Theorem 4.5, we can also obtain some corollaries, here, we only give a clear result.

Corollary 4.6. The conditions or for and imply that the system (1.1) has no positive-negative solutions.

Corollary 4.6 can be immediately obtained by (4.16).

5. Existence

In view of Section 4, we find that the existence of solutions may become fact when the function crosses the eigenvectors spaces. This motivates us to use Lusternik-Schnirelmann category theory and leads to new methods compared with previous ones, see [2629, 3638].

For a given symmetric matrix , the index of the corresponding quadratic form on , , is the largest dimension of a subspace such that for all , . The following result is specialized to our cases, see the references [69] or [70].

Lemma 5.1. If with is a even function on of the form , where is a quadratic form of index , and such that for large (where ) and that as , then has at least pairs, , nonzero critical points.

For using Lemma 5.1, we reformulate our problem as a critical point problem. For any and , we let

At this time, we can define the functions by

where . Since

we see that a column vector is a critical point of the functional corresponding to if and only if is a solution of (1.1) corresponding to .

Let

such that

where

In this case, we have

where .

By using the condition (5.6), we easily get that

For the functional , we ask that for large , which implies that for and large . In this case, if there exists and such that

then the matrix has exactly negative eigenvalues. Lemma 5.1 implies that the following result holds.

Theorem 5.2. If there exist and such that and that further suppose that there is such that for and . Then the system (1.1) has at least nonzero solution pairs.

By using Theorem 5.2, we clearly obtain the following results.

Corollary 5.3. For any , if there exists such that and there is such that for , then the system (1.1) has at least nonzero solution pairs.

Corollary 5.4. If there exists such that further suppose that there is such that for and . Then the system (1.1) has at least nonzero solution pairs, particularly, the system (1.1) has at least one positive-negative solution pair.

Corollary 5.5. For any and , the conditions and imply that the system (1.1) has at least nonzero solution pairs.

Now, we consider the functional . Similarly, we have the following result by using Lemma 5.1.

Theorem 5.6. If there exist and such that and that further suppose that there is such that Then the system (1.1) has at least nonzero solution pairs.

Corollary 5.7. Assume that the condition holds for any , and there exist and such that the condition (5.13) is valid, then the system (1.1) has at least nonzero solution pairs.

Corollary 5.8. For any and , the conditions and imply that the system (1.1) has at least nonzero solution pairs.

Consider the algebraic equations

which is the special case of (1.1). Obviously, for any all conditions of Theorem 5.6 hold. Thus, (5.14) has at least one nonzero solution pair. In fact, it has the exact nonzero solution pair . Thus, the conditions of Theorem 5.6 are sharp for (5.14).

6. Applications

Clearly, the theorems and corollaries established earlier are useful to solve the problems listed in Section 2. Some simple illustrative examples and remarks will be listed in this section.

6.1. Periodic Solutions

Guo and Yu [28] considered the existence of -periodic solution for

The main result they derived is described as follows.

Assume satisfies the following condition:

(i), and there exists positive integer such that for any , ;(ii)for any , , and ;(iii)there exists , such that for any , then for any positive integer , (6.1) has at least three -periodic solutions.

Now, we consider the existence of -periodic solution of the following nonlinear second-order difference equation:

where is a given positive integer, where is a parameter, and for , and there exists a positive integer such that for any , , . A solution of (6.3) is called to be nonzero if its every term is not zero.

The above problem is equal to the system (1.1) where the matrix is defined by (2.13) with and has the eigenvalues

Note that

then we have

when is even and

when is odd.

Again let

By using Theorems 5.2 and 5.6, we can obtain the following facts.

Theorem 6.1. For any , if there exist and such that further suppose that there is such that for and . Then the problem (6.3) has at least nonzero -periodic solutions.

Theorem 6.2. When is a even positive integer and there exist and such that (6.10) holds, further suppose that there is such that then the system (6.3) has at least nonzero -periodic solutions; If is an odd positive integer and there exist and such that (6.10) and (6.11) hold, then the system (6.3) has at least nonzero -periodic solutions.

In view of Theorem 6.1, we can immediately obtain the following result.

Corollary 6.3. For any or hold, then the problem (6.3) has at least nonzero -periodic solutions.

Remark 6.4. Corollary 6.3 improves the main result in [28].

When for any , we have

Corollary 6.3 implies that for any positive integer and positive number the equation

has at least nonzero -periodic solutions. For example, let and , we can consider the existence of solutions for nonlinear system:

which has at least nonzero solution pairs in view of Corollary 6.3. However, Theorem in [28] can only obtain two nonzero solutions because they conclude a zero solution. In fact, we can solve its the nonzero numerical solutions:

However, we find that the number of nonzero solutions of (6.16) is more than . In fact, (6.16) has also nontrivial solutions

Thus, we have the following open problem.

Open Problem  1
Obtain better existence results for the system (1.1) when .
When for any , we have
For any Theorem 6.1 implies that the periodic boundary value problem of the form has at least nonzero solutions.
On the other hand, Zhou et al. [29] consider the discrete time second-order dynamical systems:
where and for any . Our results are also valid for the problem (6.22). In this case, the corresponding results also improve the main theorem in [29]. In fact, we let be an unit matrix, then the problem (6.22) is equal to the system (1.1) where which has the eigenvalues

6.2. Steady-State Solutions on Discrete Neural Networks

In [36], Wang and Cheng considered the existence of steady-state solutions for the discrete neural network

with the periodic boundary value conditions

In fact, the steady-state equation can be written by

which can be rewritten by the system (1.1), where , the coefficient matrix is defined by(2.13)

In this case, in Theorems 6.1 and 6.2   and are, respectively, replaced by and , then, they are valid for the problem (6.27). At this time, Theorem 6.1 improves the corresponding result in [36], but Theorem 6.2 is new.

On the other hand, for the more general system of the form

our results are also valid, where and are similar with (2.14) and the method is similar with Section 6.2. Thus, the main results in [36, 38] are also extended and improved.

We also find that Wang and Cheng [37] considered the existence of the steady-state solutions for the discrete neural network:

with the periodic boundary value condition (6.26). Similarly, the problem (6.30)–(6.26) be rewritten by the system (1.1), where , is defined by (2.13), and

By using similar method, we can obtain the improved and extended results. Clearly, we can also consider the existence of steady-state solutions for the general system of the form

where is a -vector for each .

6.3. Periodic Solutions for Fourth-Order Difference Equation

When , the problem (2.15) exists -periodic solutions if and only if the nonlinear system

has nontrivial solutions. The matrix

has the eigenvalues and . When , the problem (2.15) exists -periodic solutions if and only if the nonlinear system

has nontrivial solutions. The matrix

has the eigenvalues .

When , the problem (2.15) exists -periodic solutions if and only if the nonlinear system

has nontrivial solutions. The matrix

has the eigenvalues and .

When , (2.15) exists -periodic solutions if and only if the nonlinear system (1.1) has nontrivial solutions, where the matrix is defined in Subsection 2.1.

We can obtain eigenvalues of from the fourth-order linear difference equation

with the periodic boundary value conditions:

Let

In view of (6.39), we have

which implies that

From (6.40), we see that

thus, we have

for and . In view of (6.43), we have obtained

Note that

then

By using Theorems 5.2 and 5.6, we can clearly obtain similar results as Theorems 6.1 and 6.2 and improve and extend the results in [27]. They will be omitted.

6.4. On Partial Difference Equation

In Turing pattern analysis, the positive steady-state solutions are usually needed. The authors in [41] think that persistent puzzle in the field of biological electron transfer is the conserved iron-sulfur cluster motif in both high potential iron-sulfur protein (HiPIP) and ferredoxin (Fd) active sites. However, the voltage in cell can be negative and there exists the negative threshold, see [45]. Thus, the negative steady-state should be also considered for Turing pattern analysis. Our results will likely find important implications in other real evolutionary processes.

By the discussion in Section 2, we have known that the models of many applied problems can be expressed by the partial difference equation of the form

with periodic boundary value conditions

In fact, we can also give the other explanation.

Indeed, let us consider neuron units placed on a torus. Let denote the state value of the th neuron unit during the time periodic . Assume that each neuron unit is random activated by its four neighbors so that the change of state values between two consecutive time periods is given by

with the periodic boundary value conditions

where is the connection weight, stands for the bias mechanism inherent in the th neuron unit.

In order to utilize the neural network modeled by the aforementioned evolutionary system, it is of interest to predict the existence of steady-state solution such that for and . This then leads us to finding solutions of the steady system of (6.49)-(6.50) or more generally (1.1).

We can similarly obtain the eigenvalues of corresponding (6.49)–(6.50):

for , thus, the existence and nonexistence of solutions for (6.49)–(6.50) can also be established. They will omitted.

In the present paper, we ask that the coefficient matrix and the nonlinear term of the system (1.1), respectively, satisfy the symmetry and the odd symmetry. Clearly, a number of application problems are not valid. They will be considered in the further paper.

Acknowledgment

This work is supported by the special Fundation of Tianjin Education Committee (2006ZH91) and Recearch Fundation of Tianjin Education Committee (20050404).

References

  1. H. I. Abdel-Gawad, “A model for a finite memory transport in the Fisher equation,” Applied Mathematical Modelling, vol. 32, no. 9, pp. 1883–1893, 2008. View at Publisher · View at Google Scholar · View at Scopus
  2. D. G. Aronson and H. F. Weiberger, “Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation,” in Partial Differential Equations and Related Topics, vol. 446 of Lecture Notes in Mathematics, pp. 5–49, Springer, Berlin, Germany, 1975. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. R. F. Hugh, “Impulse and physiological states in models of nerve membrance,” Biophysical Journal, vol. 1, pp. 445–466, 1961. View at Google Scholar
  4. D.-J. Huang and H.-Q. Zhang, “The extended first kind elliptic sub-equation method and its application to the generalized reaction Duffing model,” Physics Letters A, vol. 344, no. 2-4, pp. 229–237, 2005. View at Publisher · View at Google Scholar · View at Scopus
  5. J. Nagumo, S. Ariomot, and S. Yoshizawa, “An active pulse transmission line simulating nerve axon,” Proceedings of the IRE, vol. 50, pp. 2061–2070, 1962. View at Google Scholar
  6. J. I. Ramos, “Exponential methods for one-dimensional reaction-diffusion equations,” Applied Mathematics and Computation, vol. 170, no. 1, pp. 380–398, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer, New York, NY, USA, 1997.
  8. I. Rachunkova, “Existence of two positive solutions of a singular nonlinear periodic boundary value problem,” Journal of Computational and Applied Mathematics, vol. 113, no. 1-2, pp. 27–34, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  9. S. Sedziwy, “Nonlinear periodic boundary value problem for a second order ordinary differential equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 32, no. 7, pp. 881–890, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. Q. Yao, “Positive solutions of nonlinear second-order periodic boundary value problems,” Applied Mathematics Letters, vol. 20, no. 5, pp. 583–590, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Z. Zhang and J. Wang, “On existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equations,” Journal of Mathematical Analysis and Applications, vol. 281, no. 1, pp. 99–107, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. M. Abel and M. Spicci, “Nonlinear localized periodic solutions in a coupled map lattice,” Physica D, vol. 119, no. 1-2, pp. 22–33, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. S. Aubry, “Breathers in nonlinear lattices: existence, linear stability and quantization,” Physica D, vol. 103, no. 1–4, pp. 201–250, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  14. S. Aubry and G. Abramovici, “Chaotic trajectories in the standard map. The concept of anti-integrability,” Physica D, vol. 43, no. 2-3, pp. 199–219, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. S.-N. Chow and W. X. Shen, “Dynamics in a discrete Nagumo equation: spatial topological chaos,” SIAM Journal on Applied Mathematics, vol. 55, no. 6, pp. 1764–1781, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. R. Coutinho and B. Fernandez, “Extended symbolic dynamics in bistable CML: existence and stability of fronts,” Physica D, vol. 108, no. 1-2, pp. 60–80, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. J. P. Crutchfield and K. Kaneko, “Phenomenology of spatio-temporal chaos,” in Directions in Chaos, Vol. 1, vol. 3 of World Scientific Series Directions in Condensed Maltier Physics, pp. 272–353, World Scientific, Singapore, 1987. View at Google Scholar · View at MathSciNet
  18. K. Kaneko, “Pattern dynamics in spatiotemporal chaos. Pattern selection, diffusion of defect and pattern competition intermittency,” Physica D, vol. 34, no. 1-2, pp. 1–41, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. J. P. Keener, “Propagation and its failure in coupled systems of discrete excitable cells,” SIAM Journal on Applied Mathematics, vol. 47, no. 3, pp. 556–572, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. M. Levi, F. C. Hoppensteadt, and W. L. Miranker, “Dynamics of the Josephson junction,” Quarterly of Applied Mathematics, vol. 36, no. 2, pp. 167–198, 1978. View at Google Scholar · View at MathSciNet
  21. H. Li and S. Zhou, “Structure of the global attractor for a second order strongly damped lattice system,” Journal of Mathematical Analysis and Applications, vol. 330, no. 2, pp. 1426–1446, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. R. S. MacKay and S. Aubry, “Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators,” Nonlinearity, vol. 7, no. 6, pp. 1623–1643, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. M. Qian and G. Wang, “Structure of the global attractor for discretized sine-Gordon equation,” International Journal of Mathematics, Game Theory, and Algebra, vol. 13, no. 6, pp. 443–454, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. G. Wang and S. Zhu, “Dimension of the global attractor for the discretized damped sine-Gordon equation,” Applied Mathematics and Computation, vol. 117, no. 2-3, pp. 257–265, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. S. Zhou, “Attractor and dimension for discretization of a damped wave equation with periodic nonlinearity,” Topological Methods in Nonlinear Analysis, vol. 15, no. 2, pp. 267–281, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. F. M. Atici and A. Cabada, “Existence and uniqueness results for discrete second-order periodic boundary value problems,” Computers & Mathematics with Applications, vol. 45, no. 6-9, pp. 1417–1427, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. X. Cai, J. Yu, and Z. Guo, “Existence of periodic solutions for fourth-order difference equations,” Computers & Mathematics with Applications, vol. 50, no. 1-2, pp. 49–55, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. Z. M. Guo and J. S. Yu, “Existence of periodic solutions and subharmonic solutions on second order superlinear difference equations,” Science in China. Series A, vol. 33, no. 3, pp. 226–235, 2003. View at Google Scholar
  29. Z. Zhou, J. Yu, and Z. Guo, “Periodic solutions of higher-dimensional discrete systems,” Proceedings of the Royal Society of Edinburgh. Section A, vol. 134, no. 5, pp. 1013–1022, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. R. Bru, R. Cants, and B. Ricarte, “Modelling nitrogen dynamics in citrus trees,” Mathematical and Computer Modelling, vol. 38, no. 10, pp. 975–987, 2003. View at Publisher · View at Google Scholar · View at Scopus
  31. W. M. Haddad, T. Hayakawa, and J. M. Bailey, “Adaptive control for nonlinear compartmental dynamical systems with applications to clinical pharmacology,” Systems & Control Letters, vol. 55, no. 1, pp. 62–70, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. J. A. Jacquez and C. P. Simon, “Qualitative theory of compartmental systems with lags,” Mathematical Biosciences, vol. 180, pp. 329–362, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. J. R. Schramski, D. K. Gattie, B. C. Patten, S. R. Borrett, B. D. Fath, and S. J. Whipple, “Indirect effects and distributed control in ecosystems: distributed control in the environ networks of a seven-compartment model of nitrogen flow in the Neuse River Estuary, USA-Time series analysis,” Ecological Modelling, vol. 206, no. 1-2, pp. 18–30, 2007. View at Publisher · View at Google Scholar · View at Scopus
  34. S. Walcher, J. Altschuh, K.-W. Schramm, and S. Mayer, “Estimates in deterministic fate modelling of environmental chemicals,” Environmental Modelling and Software, vol. 18, no. 10, pp. 929–936, 2003. View at Publisher · View at Google Scholar · View at Scopus
  35. X. Li, X. Wang, and G. Chen, “Pinning a complex dynamical network to its equilibrium,” IEEE Transactions on Circuits and Systems I, vol. 51, no. 10, pp. 2074–2087, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  36. G.-Q. Wang and S. S. Cheng, “Multiple steady state solutions of a circular neural network,” Chaos, Solitons & Fractals, vol. 37, no. 4, pp. 1228–1232, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. G.-Q. Wang and S. S. Cheng, “Fixed point theorems arising from seeking steady states of neural networks,” Applied Mathematical Modelling, vol. 33, no. 1, pp. 499–506, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. G.-Q. Wang and S. S. Cheng, “Notes on periodic solutions of discrete steady state systems,” Portugaliae Mathematica, vol. 64, no. 1, pp. 3–10, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. Z. Zhou, J. Wang, Z. Jing, and R. Wang, “Complex dynamical behaviors in discrete-time recurrent neural networks with asymmetric connection matrix,” International Journal of Bifurcation and Chaos, vol. 16, no. 8, pp. 2221–2233, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  40. C. de Joussineau, J. Soulé, M. Martin, C. Anguille, P. Montcourrier, and D. Alexandre, “Delta-promoted filopodia mediate long-range lateral inhibition in Drosophila,” Nature, vol. 426, no. 6966, pp. 555–559, 2003. View at Publisher · View at Google Scholar · View at Scopus
  41. A. Dey, F. E. Jenney Jr., M. W. W. Adams et al., “Solvent tuning of electrochemical potentials in the active sites of HiPIP versus ferredoxin,” Science, vol. 318, no. 5855, pp. 1464–1468, 2007. View at Publisher · View at Google Scholar · View at Scopus
  42. R. Eftimie, G. de Vries, and M. A. Lewis, “Complex spatial group patterns result from different animal communication mechanisms,” Proceedings of the National Academy of Sciences of the United States of America, vol. 104, no. 17, pp. 6974–6979, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  43. S. K. Eswaramoorthy, J. M. Howe, and G. Muralidharan, “In situ determination of the nanoscale chemistry and behavior of solid-liquid systems,” Science, vol. 318, no. 5855, pp. 1437–1440, 2007. View at Publisher · View at Google Scholar · View at Scopus
  44. C. P. Grant, “Spinodal decomposition for the Cahn-Hilliard equation,” Communications in Partial Differential Equations, vol. 18, no. 3-4, pp. 453–490, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  45. Q. Hu, Physiology, Anhui Science and Technology Press, Hefei, China, 1986.
  46. T.-X. Jiang, H.-S. Jung, R. B. Widelitz, and C.-M. Chuong, “Self-organization of periodic patterns by dissociated feather mesenchymal cells and the regulation of size, number and spacing of primordia,” Development, vol. 126, no. 22, pp. 4997–5009, 1999. View at Google Scholar · View at Scopus
  47. M. Leonetti, E. Dubois-Violette, and F. Homble, “Pattern formation of stationary transcellular ionic currents in Fucus,” Proceedings of the National Academy of Sciences of the United States of America, vol. 101, no. 28, pp. 10243–10248, 2004. View at Publisher · View at Google Scholar · View at Scopus
  48. Y.-J. Li, J. Oslonovitch, N. Mazouz, F. Plenge, K. Krischer, and G. Ertl, “Turing-type patterns on electrode surfaces,” Science, vol. 291, no. 5512, pp. 2395–2398, 2001. View at Publisher · View at Google Scholar · View at Scopus
  49. K. Kytta, K. Kaski, and R. A. Barrio, “Complex turing patterns in non-linearly coupled systems,” Physica A, vol. 385, no. 1, pp. 105–114, 2007. View at Publisher · View at Google Scholar · View at Scopus
  50. S. Maier-Paape, E. Sander, and T. Wanner, “Spinodal decomposition: a survey of recent results,” in International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), B. Fiedler, K. Groger, and J. Sprekels, Eds., pp. 1288–1299, World Scientific, River Edge, NJ, USA, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  51. S. Maier-Paape and T. Wanner, “Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions—part I: probability and wavelength estimate,” Communications in Mathematical Physics, vol. 195, no. 2, pp. 435–464, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  52. S. Maier-Paape and T. Wanner, “Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: nonlinear dynamics,” Archive for Rational Mechanics and Analysis, vol. 151, no. 3, pp. 187–219, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  53. J. D. Murray, “Parameter space for Turing instability in reaction diffusion mechanisms: a comparison of models,” Journal of Theoretical Biology, vol. 98, no. 1, pp. 143–163, 1982. View at Publisher · View at Google Scholar · View at MathSciNet
  54. J. D. Murray, Mathematical Biology. I: An Introduction, vol. 17 of Interdisciplinary Applied Mathematics, Springer, New York, NY, USA, 3rd edition, 2002. View at MathSciNet
  55. K. M. Page, P. K. Maini, and N. A. M. Monk, “Complex pattern formation in reaction-diffusion systems with spatially varying parameters,” Physica D, vol. 202, no. 1-2, pp. 95–115, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  56. K. J. Painter, P. K. Maini, and H. G. Othmer, “Stripe formation in juvenile Pomacanthus explained by a generalized Turing mechanism with chemotaxis,” Proceedings of the National Academy of Sciences of the United States of America, vol. 96, no. 10, pp. 5549–5554, 1999. View at Publisher · View at Google Scholar · View at Scopus
  57. J. E. Pearson, “Complex patterns in a simple system,” Science, vol. 261, no. 5118, pp. 189–192, 1993. View at Google Scholar · View at Scopus
  58. U. K. Rößler, A. N. Bogdanov, and C. Pfleiderer, “Spontaneous skyrmion ground states in magnetic metals,” Nature, vol. 442, no. 7104, pp. 797–801, 2006. View at Publisher · View at Google Scholar · View at Scopus
  59. E. Sander and T. Wanner, “Monte Carlo simulations for spinodal decomposition,” Journal of Statistical Physics, vol. 95, no. 5-6, pp. 925–948, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  60. E. Sander and T. Wanner, “Unexpectedly linear behavior for the Cahn-Hilliard equation,” SIAM Journal on Applied Mathematics, vol. 60, no. 6, pp. 2182–2202, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  61. E. Sander and T. Wanner, “Pattern formation in a nonlinear model for animal coats,” Journal of Differential Equations, vol. 191, no. 1, pp. 143–174, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  62. I. Salazar-Ciudad, J. Jernvall, and S. A. Newman, “Mechanisms of pattern formation in development and evolution,” Development, vol. 130, no. 10, pp. 2027–2037, 2003. View at Publisher · View at Google Scholar · View at Scopus
  63. G. Sempo, S. Depickere, and C. Detrain, “Spatial organization in a dimorphic ant: caste specificity of clustering patterns and area marking,” Behavioral Ecology, vol. 17, no. 4, pp. 642–650, 2006. View at Publisher · View at Google Scholar · View at Scopus
  64. E. Szathmy, “Evolution: developmental circuits rewired,” Nature, vol. 411, no. 6834, pp. 143–145, 2001. View at Publisher · View at Google Scholar · View at Scopus
  65. R. V. Solé and J. Bascompte, “Chaotic Turing structures,” Physics Letters A, vol. 179, no. 4-5, pp. 325–331, 1993. View at Publisher · View at Google Scholar · View at Scopus
  66. R. V. Solé, J. Bascompte, and J. Valls, “Stability and complexity of spatially extended two-species competition,” Journal of Theoretical Biology, vol. 159, no. 4, pp. 469–480, 1992. View at Publisher · View at Google Scholar · View at Scopus
  67. L. Trinkle-Mulcahy and A. I. Lamond, “Toward a high-resolution view of nuclear dynamics,” Science, vol. 318, no. 5855, pp. 1402–1407, 2007. View at Publisher · View at Google Scholar · View at Scopus
  68. A. M. Turing, “The chemical basis of morphogenesis,” Philosophical Transactions of the Royal Society, vol. 237, pp. 37–72, 1952. View at Google Scholar
  69. Y. S. Choi, I. Koltracht, and P. J. McKenna, “On eigen-structure of a nonlinear map in Rn,” Linear Algebra and Its Applications, vol. 399, pp. 141–155, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  70. D. C. Clark, “A variant of the Lusternik-Schnirelman theory,” Indiana University Mathematics Journal, vol. 22, no. 1, pp. 65–74, 1972. View at Publisher · View at Google Scholar · View at MathSciNet
  71. G. Zhang, H. T. Yuan, and L. Bai, “Existence of solution pairs for a nonlinear algebraic aystem,” in preparation.
  72. G. Zhang and W. Feng, “On the number of positive solutions of a nonlinear algebraic system,” Linear Algebra and Its Applications, vol. 422, no. 2-3, pp. 404–421, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  73. P. Lancaster and M. Tismenetsky, The Theory of Matrices, Computer Science and Applied Mathematics, Academic Press, Orlando, Fla, USA, 2nd edition, 1985. View at MathSciNet