WCNA'2004

Dynamical systems: bifurcations and applications

session survey

(USA, Orlando, July, 2004)

F.Botelho

The University of Memphis

Department of Mathematical Sciences

373 Dunn Hall, Memphis, Tennessee 38152-3420, USA

mbotelho@memphis.edu

V.A.Gaiko

Department of Mathematics

Belarusian State University of Informatics and Radioelectronics

L.Beda Str. 6-4, Minsk 220040, Belarus

vlrgk@yahoo.com

1. Introduction

In this paper, we review the talks presented at the Fourth World Congress of Nonlinear Analysts (WCNA'2004) in the session entitled "Dynamical Systems: Bifurcations and Applications". This session bridged theoretical and applied areas of mathematics.

The six invited talks covered topics from low dimensional dynamics and planar polynomial systems to neural network models and difference equations in Banach spaces.

1.          J.Murdock, F.Botelho (University of Memphis, Memphis, USA): A Map with Invariant Cantor Set of Positive Measure.

2.          J.Giné (Universitat de Lleida, Lleida, Spain), H.Giacomini (Université de Tours, Tours, France): Generalized Nonlinear Superposition Principles of Planar Polynomial Vector Fields.

3.          S.Lynch, Z.Bandar (Manchester Metropolitan University, Manchester, UK): Bistable Neuromodules.

4.          F.Botelho, J.Jamison (University of Memphis, Memphis, USA): Chaoticity Generated by a Learning Model.

5.          V.Gaiko (Belarusian State University of Informatics and Radioelectronics, Minsk, Belarus): Rotation Parameters and Limit Cycles.

6.          A.Reinfelds (University of Latvia, Riga, Latvia): Equivalence of Time-Depending Difference Systems in a Banach Space.

2. From low to higher dimensional dynamical systems

Interval maps provide important models to a wide variety of real phenomena [1]. Many examples exist of one-dimensional systems that are topologically conjugate to the shift operator in two symbols and therefore are chaotic [2, 3]. Most of these invariant Cantor sets are of Lebesgue measure zero. There is a correlation between the degree of smoothness of a map and the thickness of the invariant Cantor sets that map has, [3].

It is described in [4] the construction of a positive measure Cantor set and a piecewise monotone interval map with that particular Cantor set invariant. A positive measure Cantor set is the intersection of a nested sequence of closed intervals. This follows the well-known middle thirds construction. The selected middle intervals now decrease in size at an exponentially fast rate. This way, the Cantor set obtained has positive measure.

At each stage of the construction it is defined a piecewise linear map that permutes the selected intervals in a convenient manner, [4]. The resulting map emerges as the uniform limit of a sequence of these piecewise linear maps and therefore is piecewise monotone. As a consequence, this map leaves invariant the Cantor set constructed and is differentiable everywhere except at the endpoints of the selected middle intervals. In addition, it can be modified at those endpoints via a cubic polynomial in order to obtain everywhere differentiability.

This new map is not continuously differentiable. It turns out that a continuously differentiable map that leaves invariant the Cantor set defined in [4] cannot exist. The rate that the selected intervals contract restricts smoothness. Techniques controlling the speed that the selected intervals contract and the differentiability properties of the resulting map are being investigated.

Linear and piecewise linear systems are well understood and easily implemented. Questions concerning the dynamical equivalence between a given system and its linearization are of great interest from both theoretical and applied point of view, [5].

The Hartman-Grobman theorem for a diffeomorphism or a Lipschitz vector field on some Euclidean space assures the existence of a local conjugacy between the given system and its linear associate provided that a hyperbolicity condition holds. Contraction fixed point theorems imply the existence of conjugating maps [6]. The extension of Grobman-Hartman theorem to Banach spaces required new techniques. The conjugating homeomorphisms are solutions of functional integral equations, [7].

Difference equations appear often as discrete models to real phenomena or as discretizations of continuous ones. Conjugating theorems for systems of difference equations were proved in [7, 8]. These results reduce the study of complicated systems to much simpler ones, [7, 8].

3. Planar polynomial systems

The main problem of qualitative theory of the planar polynomial dynamical systems is Hilbert's sixteenth problem on the maximum number and relative position of limit cycles. There are three local bifurcations of limit cycles: 1) Andronov-Hopf bifurcation (from a singular point of center or focus type); 2) separatrix cycle bifurcation (from a homoclinic or heteroclinic orbit); 3) multiple limit cycle bifurcation (from a multiple limit cycle of even or odd multiplicity). All these bifurcations depend on the rotation properties of the vector fields generated by the dynamical systems under consideration. We construct canonical systems with field-rotation parameters, connect all local bifurcations of limit cycles by means of the Wintner-Perko termination principle and develop a new global approach to the solution of Hilbert's sixteenth problem. In the quadratic case, for example, we construct a canonical system with four field-rotation parameters and suggest a new geometric proof of the theorem stating that the maximum number of limit cycles surrounding a focus is equal to three, supporting our conjecture that a quadratic system can have at most four limit cycles and only in (3:1) distribution on the whole phase plane [9].

Developing Erugin's two-isoclines method [9], we apply our approach to the global qualitative analysis of cubic systems [10]. In particular, we construct a canonical cubic system of Kukles type and carry out the global analysis of its special case corresponding to a generalized Liénard equation. We prove that the foci of such a Liénard system can be at most of second order and that this particular system can have at least three limit cycles on the whole phase plane. Moreover, unlike all previous works on the Kukles-type systems, we study global bifurcations of limit and separatrix cycles, using arbitrary (including as large as possible) field-rotation parameters of our canonical system. As a result, we have obtained a classification of all possible types of separatrix cycles for this generalized Liénard system and also all possible distributions of its limit cycles, conjecturing that this system has at most three limit cycles [10].

Our geometric methods can be successfully applied to more general polynomial systems. In particular, we study a classical Liénard system with a polynomial of degree 2k + 1. We prove that limit cycle bifurcations depend on only odd parameters of this system which are field-rotation parameters; then, using these parameters, we obtain exactly k limit cycles and, applying some geometric reasons, suggest a prove of the well-known conjecture [11] stating that k is the maximal number of limit cycles for the given system.

It is possible also to study the existence problem of first integrals for the corresponding polynomial equations. For example, we can find the first integrals in the Painlevé form which are expressed in unknown particular solutions of the corresponding equation [12, 13]. For certain systems, some of the particular solutions remain arbitrary and the others are explicitly determined or functionally related to the arbitrary particular solutions. In this way, we can obtain a nonlinear superposition principle that generalizes the classical nonlinear superposition principle of the Lie theory: in general, the first integral contains some arbitrary solutions of the system, but also quadratures of these solutions and an explicit dependence on the independent variable [12, 13].

A new approach to this existence problem is developed in [14]. In particular, here it is detected (in an algorithmic way) the systems which admit a first integral of the Painlevé form and, specially, the cases when some of the particular solutions remain arbitrary. In these cases, an expression of the first integral, what is called a generalized nonlinear superposition principle, can be obtained. Then, in some cases, it can be concluded that a given polynomial system has a first integral of the Painlevé form with the proposed number of factors, but the algorithm does not determine all arbitrary particular solutions of the corresponding equation: a first integral of the polynomial system can be constructed from a finite number of particular solutions and this number can be determined. When all such particular solutions are determined, they are algebraic functions. For these cases, it can be introduced an algorithm which represents an alternative method for determining such type of solutions [14]. Finally, it can be proved that for the polynomial dynamical system admitting the first integral of the Painlevé form, where all particular solutions of the corresponding equation are determined, all these particular solutions are algebraic functions [14].

4. Applications

Nonlinear mathematical models can provide, for instance, the rich dynamical spectrum encountered in biological systems. Consider first artificial neural networks (ANNs) which are mechanical devices composed by interconnected processing units to mimic specific brain functions [15]. Examples of such functions:

-        associative memory (the ability to associate features of possibly different types such as a name and an appearance to a picture);

-        pattern recognition (the ability to recover missing information in an input such as the recovery of a whole picture from an incomplete one);

-        learning (the ability to adapt to its surroundings).

The activity of an ANN is modeled by either a discrete or a continuous system of equations depending on a variety of pre-assigned parameters. Nonlinear dynamics, bifurcation theory, and operator theory have been successfully applied to analyze long-term evolution of ANNs. Over a small parameter range, we can observe global convergence, bistability and hysteresis phenomena, as well as chaoticity and turbulence, see [16]. The two essential ingredients to generate complex dynamical behavior are nonlinearity and feedback. These processes are inherently present in neuronal dynamics. Bistable phenomena has been proved to exist in a wide range of physical forms not only as human brain dynamics but also in economics, astrochemical cloud models, and psychology [17, 18]. Parameters such as weights, biases and gradients of transfer functions can be altered when an artificial neural network is learning.

There are several mathematical models for learning. Here we mention a discrete learning algorithm introduced by Oja [19, 20]. This algorithm can be implemented as an artificial neural network and it provides an updating scheme for the set of connecting weights. The main goal is to determine the network's connecting weights for a given initial assignment. The algorithm iterative rule is nonlinear and it depends on the input correlation matrix. If convergence occurs, this rule determines a vector of connecting weights given an initial one. Since the unit eigenvector associated with the largest eigenvalue is stable we say that this model behaves as a maximal eigenfilter or principal component analyzer.

A generalization of Oja's model and its stability behavior are investigated in [21]. The underlying space is a Hilbert space and the input correlation is a self-adjoint, nonnegative, compact operator. The spectral theorem of compact operators implies the decomposition of the space into a direct sum of finite dimensional eigenspaces, except possibly for the kernel of the correlation operator.

A reduction method converts the problem into the evolution of real valued maps. This technique was illustrated in the particular case of statistically independent input vectors. The underlying space is decomposed into the direct sum of a kernel and an eigenspace of dimension one. The analysis is therefore reduced to the dynamics of unimodal-type maps.

We observe that chaotic behavior can only take place on finitely many eigenspaces. In this situation, the algorithm does not converge and a natural set of connecting weights cannot be selected. Nevertheless, as the eigenvalues decrease toward zero, convergence occurs in larger and larger regions allowing a natural set of connecting weights to emerge.

References

1.               J.Murray. Mathematical Biology. New York: Springer, 1993.

2.               R.Devaney. An Introduction to Chaotic Dynamical Systems. New York: Addison-Wesley, 1992.

3.               R. C. Robinson. An Introduction to Dynamical Systems. New Jersey: Prentice Hall, 2004.

4.               J. Murdock, F.Botelho. A map with invariant Cantor set of positive measure. To appear in the WCNA-2004 Proceedings.

5.               D.Arrowsmith, C.Place. An Introduction to Dynamical Systems. Cambridge: Cambridge University Press, 1990.

6.               A.Reinfelds. Dynamical equivalence of dynamical systems. Univ. Iagel. Acta Math., 36, 149-155, 1998.

7.               A. Reinfelds. Grobman-Hartman's theorem for time-dependent difference equations. Latv. Univ. Zināt. Rakti, 605, 9-13, 1997.

8.               A.Reinfelds. The reduction principle for discrete dynamical, semidynamical systems in metric spaces. Z.Angew. Math. Phys. 45, 933-955, 1994.

9.               V.A.Gaiko. Global Bifurcation Theory, Hilbert's Sixteenth Problem. Boston: Kluwer Academic Publishers, 2003.

10.           V.A.Gaiko, W.T. van Horssen. Global bifurcations ofљ limit, separatrix cycles in a generalized Liénard system. To appear in Nonlin. Analysis 2004, doi:10.1016/j.na.2004.07.010.

11.           S.Smale. Mathematical problems for the next century. Math. Intelligencer, 20, 7-15, 1998.

12.           I.A.García, J.Giné. Generalized cofactors, nonlinear superposition principles. Appl. Math. Lett., 16, 1137-1141, 2003.

13.           I.A.García, J.Giné. Non-algebraic invariant curves for polynomial planar vector fields. Discrete Contin. Dynam. Syst., 10, 755-768, 2004.

14.           J.Giné. Generalized nonlinear superposition principle of planar polynomial vector fields. To appear in the WCNA-2004 Proceedings.

15.           P. Wilde. Neural Network Models. New York: Springer, 1997.

16.           E. Izhikevich. Multiple cusp bifurcations. Neural Networks, 11, 495-508, 1998.

17.           S.Lynch. Dynamical Systems with Applications Using Maple. Basel, Birkhäuser, 2000.

18.           S.Lynch, Z.Bandar. Bistable neuromodules. To appear in the WCNA-2004 Proceedings.

19.           E.Oja. A simplified neuron model as a principal component analyzer. J. Math. Biology, 15, 267-273, 1982.

20.           E.Oja. Principal components, minor components, linear neural networks. Neural Networks 5, 927-936, 1992.

21.           F. Botelho, J. E. Jamison. Chaoticity generated by a learning model. J. Math. Analysis Appl., 286, 618-635, 2004.

 

 

Fernanda Botelho, Dr., The University of Memphis. Research interests: dynamical systems and complex dynamics, ergodic theory and point set topology.

Valery A. Gaiko, Dr., Department of Mathematics, Belarusian State University of Informatics and Radioelectronics. Research interests: qualitative theory of differential equations, dynamical systems and bifurcation theory, nonlinear analysis and applications.