Discrete Dynamics in Nature and Society

Volume 2010, Article ID 405639, 12 pages

http://dx.doi.org/10.1155/2010/405639

## Neimark-Sacker Bifurcation in a Discrete-Time Financial System

^{1}Nonlinear Dynamics and Chaos Group, School of Management, Tianjin University, Tianjin 300072, China^{2}Center for Applied Mathematics, School of Economics and Management, Shandong University of Science and Technology, Qingdao 266510, China

Received 14 May 2010; Revised 16 July 2010; Accepted 28 August 2010

Academic Editor: Akio Matsumoto

Copyright © 2010 Baogui Xin et al. 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

A discrete-time financial system is proposed by using forward Euler scheme. Based on explicit Neimark-Sacker bifurcation (also called Hopf bifurcation for map) criterion, normal form method and center manifold theory, the system's existence, stability and direction of Neimark-Sacker bifurcation are studied. Numerical simulations are employed to validate the main results of this work. Some comparison of bifurcation between the discrete-time financial system and its continuous-time system is given.

#### 1. Introduction

Economic dynamics have recently become more prominent in mainstream economics [1]. The real financial and economic systems show a lot of complex dynamical phenomena, such as, business cycle, financial crisis, irregular growth, and bullwhip effect. Many nonlinear dynamical models of economics and finance [2–9] present various complex dynamical behaviors such as, chaos, fractals, and bifurcation.

Bifurcation refers to a class of phenomena in dynamic systems such that the dynamic properties of the system cause a sudden “qualitative” or topological change when the parameter values (the bifurcation parameters) cross a boundary. Bifurcation boundaries, for example, Hopf bifurcations [10–13], have been discovered in many macroeconomic systems [14]. Hopf bifurcations occur at points where the system has a nonhyperbolic equilibrium with a pair of purely imaginary eigenvalues, but without zero eigenvalues. For a financial or economic system, there can be disequilibrium thresholds where society decides it cannot afford the increasing cost of misallocated resources as disequilibrium increases. Such a threshold then forces a restructuring of the market system. This concept of restructuring to maintain the survival of the system is known as bifurcation theory. A bifurcation in a financial or economic system is a point (or threshold) where the system is restructured to operate at a more acceptable or stable level of disequilibrium. Bifurcations do not usually lead to equilibrium conditions, only to a stable or comfortable disequilibrium condition under which the system can continue to survive [15].

Huang and Li [16] proposed a nonlinear financial model as follows: where denotes the interest rate, denotes the investment demand, denotes the price index, is the saving amount, is the cost per investment, is the demand elasticity of commercial markets, and all three constants .

Chen [1] and Ma et al. [11–13] studied some complex dynamics in system (1.1), such as, a steady state, a periodic oscillation, a quasiperiodic motion and a chaotic motion. In this paper, we will apply the forward Euler scheme to system (1.1) in order to obtain an autonomous discrete-time financial system as follows: where is the step size.

An arduous task in the study of nonlinear dynamical systems like system (1.2) is to identify different types of complex nonlinear behaviors and to present how the behavior evolves as a system parameter varies [17, 18]. Thereinto, bifurcation is a very important nonlinear behavior which can indicate a qualitative change of system properties as a system parameter changes. Neimark-Sacker bifurcations give rise to closed invariant curves which present more interesting complex behaviors. The criterion of Hopf bifurcation in continuous-time system can be stated in terms of eigenvalues or the coefficients of characteristic polynomial [19, 20]. The later method, called Schur-Cohn stability criterion, which is more convenient and efficient for detecting the existence of Hopf bifurcation in high-order and multiparameters systems was also demonstrated in discrete dynamical systems [21–23].

The remainder of this paper is organized as follows. In Section 2, we present some preliminaries. In Section 3, we prove stabilities of the fixed points in system (1.2). In Section 4, we analyze the existence of Neimark-Sacker bifurcation in system (1.2) by means of Wen’s Neimark-Sacker bifurcation criterion. In Section 5, we study the stability and direction of Neimark-Sacker bifurcation in system (1.2) by utilizing Kuznetsov’s normal form method and center manifold theory. In Section 6, we illustrate the Neimark-Sacker bifurcation in system (1.2). In Section 7, we give some comparison of bifurcation between the continuous-time system (1.1) and the discrete-time system (1.2). Finally conclusions in Section 8 close the paper.

#### 2. Preliminaries

Lemma 2.1 (see [24]). *Let from to be . Assume is a period- point. Let be the eigenvalues of .*(i)*If all the eigenvalues of have , then the periodic orbit is attracting. *(ii)*If one eigenvalue of has , then the periodic orbit is unstable. *(iii)*If all the eigenvalues of have , then the periodic orbit is repelling.*

Next, we will study the stability of a nonlinear discrete dynamical system which can be described as follows:.

Theorem 2.2. *Let be a fixed point of system (2.1), that is, and is the Jacobian matrix at the point ; then the necessary condition for asymptotically stability of the point is that *(i)* for all *(ii)* for all **where denotes the trace of A and the determinant of A.*

*Proof. *Assume that the point is asymptotically stable and let be the eigenvalues of the Jacobian matrix A at the point . Then it follows from Lemma 2.1 that all the eigenvalues satisfy
Thus
The theorem is proved.

Theorem 2.3. *Let be a fixed point of system (2.1), that is, and is the Jacobian matrix at the point ; then the necessary condition for repellent of the point is that *(i)* for all *(ii)* for all **where denotes the trace of A and the determinant of A.*

*Proof. *Assume that the point is repelling and let be the eigenvalues of the Jacobian matrix A at the point . Then it follows from Lemma 2.1 that all the eigenvalues satisfy
Thus
The theorem is proved.

Lemma 2.4 (An explicit criterion of Neimark-Sacker bifurcation [22]). *For an th-order discrete-time dynamical system like system (1.2), assume first that at the fixed point , its characteristic polynomial of Jacobian matrix takes the following form:
**
where , is the bifurcation parameter, and is the control parameter or the other to be determined. Consider the sequence of determinants where
**
If the following conditions hold,*(H1)*eigenvalue assignment when is even (or odd, resp.),*(H2)*transversality condition *(H3)*nonresonance condition or resonance condition where and **then a Neimark-Sacker bifurcation occurs at .*

#### 3. Stability of the Fixed Points

The fixed points of system (1.2) satisfy the following equations: By the analysis of roots for (3.1), one obtains the following proposition.

Proposition 3.1. *() If , system (1.2) has only one fixed point **() If , system (1.2) has three fixed points:
**
The Jacobian matrix of system (1.2) evaluated at the fixed point is given by
*

Following from Theorem 2.2, it is easy to obtain the following propositions.

Proposition 3.2. *When , the fixed point is not asymptotically stable if
*

Proposition 3.3. *When , *(1)*the fixed point is not asymptotically stable if
*(2)*the fixed points are not asymptotically stable if **
That is, if one of Propositions 3.2 and 3.3 holds, it is possible that bifurcation occurs in system (1.2).**The Jacobian matrix of the system (1.2) evaluated at the fixed point is given by
**
Its eigenvalues can be written as
*

Following from Theorem 2.2, it is easy to obtain the following propositions.

Proposition 3.4. *When *(1)*the fixed point is asymptotically stable if *(2)*the fixed point is unstable if *(3)*a bifurcation occurs at the fixed point if *

#### 4. Existence of Neimark-Sacker Bifurcation

The main task of this paper is to determine the value of bifurcation parameter when the system (1.2) has only one fixed point with .

The characteristic polynomial of the Jacobian matrix (3.7) is where According to Lemma 2.4, for , we can get the following equalities and inequalities: According to (4.3), the critical value of Neimark-Sacker bifurcation of system (1.2) can be obtained as Thus, it follows (3.8) that the eigenvalues modules satisfy the condition (H1) in Lemma 2.4, that is, Neimark-Sacker bifurcation occurs at the fixed point .

#### 5. Direction and Stability of the Neimark-Sacker Bifurcations

In this section, we will use Kuznetsov’s normal form method and center manifold theory [25] to investigate the direction and stability of the Neimark-Sacker bifurcations in system (1.2).

Since the fixed point is not the origin , the needs to be transformed to the origin by the change of variables This transforms system (1.2) into the following equivalent system: This system can be written as where is the Jacobin matrix of system (5.2) evaluated at the origin as follows. And the multilinear functions and are defined, respectively, by For the system (5.2), The eigenvalues of the matrix are

Let be a complex eigenvector of the matrix corresponding to given by (5.7), and satisfy

Let be a complex eigenvector of the transposed matrix corresponding to given by (5.7), and satisfy Then we can obtain For the eigenvector , to normalize , let where We have , where means the standard scalar product in

So the coefficients of the normal of the system (5.2) can be computed by the formulas as follows: The direction coefficient of bifurcation of a closed invariant curve can be obtained by following formula Thus we can obtain the theorem as follows.

Proposition 5.1. *The direction and stability of Neimark-Sacker bifurcation of system (1.2) can be determined by the sign of . If , then the Neimark-Sacker bifurcation of system (1.2) at is supercritical (subcritical), and the unique closed invariant curve bifurcating from is asymptotically stable (unstable).*

#### 6. Numerical Simulations

In this section, we will give an example to illustrate above analytic results.

Let with an initial state ; we can obtain the critical saving amount . By substituting this into (5.14), we have . It follows from Proposition 5.1 that system (1.2) undergoes a supercritical Neimark-Sacker bifurcation at . When we give a small perturbation , a sufficiently small positive real number, that is, , system (1.2) has a stable closed invariant curve around the equilibrium (quasiperiodic solution), as shown in Figures 1 and 2.

#### 7. Comparison

For system (1.1) at the fixed point with , Ma and Chen [11] gave the critical value of Hopf bifurcation as follows. By simple calculation, we can get the following conclusions.

Proposition 7.1. *Hopf bifurcations of continuous-time system (1.1) and discrete-time system (1.2) occur simultaneously at when .*

Proposition 7.2. *The continuous-time system (1.1) undergoes Hopf bifurcation earlier than the discrete-time system (1.2) when *(I)* and **or*(II)* and .*

Proposition 7.3. *The discrete-time system (1.2) undergoes Hopf bifurcation earlier than the continuous-time system (1.1) when *(I)* and **or*(II)* and .*

#### 8. Conclusion

In this paper, we introduce a discrete-time financial system obtained by Euler method. The existence of Neimark-Sacker bifurcation is studied by means of Wen’s Neimark-Sacker bifurcation criterion. The stability and direction of Neimark-Sacker bifurcation are proved by utilizing Kuznetsov’s normal form method and center manifold theory. Numerical simulations are used to illustrate the above main results. We give Some comparison of bifurcation between the discrete-time financial system and its continuous-time system.

#### Acknowledgments

The authors are very grateful to the referees for their valuable suggestions, which help to improve the paper significantly. This work is supported partly by the China Postdoctoral Science Foundation (Grant no. 20100470783 ) and the Specialized Research Fund for the Doctoral Program of Higher Education from Ministry of Education of China (Grant no. 20090032110031).

#### References

- W. Chen, “Dynamics and control of a financial system with time-delayed feedbacks,”
*Chaos, Solitons & Fractals*, vol. 37, pp. 1198–1207, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Xin, J. Ma, and Q. Gao, “The complexity of an investment competition dynamical model with imperfect information in a security market,”
*Chaos, Solitons & Fractals*, vol. 42, no. 4, pp. 2425–2438, 2009. View at Publisher · View at Google Scholar - W. Ji, “Chaos and control of game model based on heterogeneous expectations in electric power triopoly,”
*Discrete Dynamics in Nature and Society*, vol. 2009, Article ID 469564, 8 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Tramontana, L. Gardini, R. Dieci, and F. Westerhoff, “The emergence of bull and bear dynamics in a nonlinear model of interacting markets,”
*Discrete Dynamics in Nature and Society*, vol. 2009, Article ID 310471, 30 pages, 2009. View at Google Scholar · View at Zentralblatt MATH - A. C.-L. Chian, E. L. Rempel, and C. Rogers, “Complex economic dynamics: chaotic saddle, crisis and intermittency,”
*Chaos, Solitons & Fractals*, vol. 29, no. 5, pp. 1194–1218, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. Chiarella, P. Flaschel, and G. Wells, “The dynamics of Keynesian monetary growth,”
*Macroeconomic Dynamics*, vol. 7, pp. 473–475, 2003. View at Google Scholar - C. Chiarella, X. He, H. Hung, and P. Zhu, “An analysis of the cobweb model with boundedly rational heterogeneous producers,”
*Journal of Economic Behavior & Organization*, vol. 61, pp. 750–768, 2006. View at Publisher · View at Google Scholar - X.-Z. He and F. H. Westerhoff, “Commodity markets, price limiters and speculative price dynamics,”
*Journal of Economic Dynamics & Control*, vol. 29, no. 9, pp. 1577–1596, 2005. View at Publisher · View at Google Scholar - M. Zheng,
*Dynamics of asset pricing models with heterogeneous beliefs*, Ph.D. thesis, Peiking University, Peiking, China, 2007. - J. Benhabib and K. Nishimura, “The Hopf bifurcation and the existence and stability of closed orbits in multisector models of optimal economic growth,”
*Journal of Economic Theory*, vol. 21, no. 3, pp. 421–444, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. H. Ma and Y. S. Chen, “Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system. I,”
*Applied Mathematics and Mechanics*, vol. 22, no. 11, pp. 1240–1251, 2001. View at Publisher · View at Google Scholar - J. H. Ma and Y. S. Chen, “Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system. II,”
*Applied Mathematics and Mechanics*, vol. 22, no. 12, pp. 1375–1382, 2001. View at Publisher · View at Google Scholar - Q. Gao and J. Ma, “Chaos and Hopf bifurcation of a finance system,”
*Nonlinear Dynamics*, vol. 58, no. 1-2, pp. 209–216, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W. Barnett and Y. He, “Bifurcations in macroeconomic models,” in
*Economic Growth and Macroeconomic Dynamics: Recent Developments in Economic Theory*, S. Dowrick, P. Rohan, and S. Turnovsky, Eds., pp. 95–112, Cambridge University Press, Cambridge, UK, 2004. View at Google Scholar - D. Nawrocki, “Entropy, bifurcation and dynamic market disequilibrium,”
*Financial Review*, vol. 19, pp. 266–284, 1984. View at Publisher · View at Google Scholar - D. Huang and H. Li,
*Theory and Method of the Nonlinear Economics*, Sichuan University Press, Chengdu, China, 1993. - W. Wu, Z. Chen, and Z. Yuan, “The evolution of a novel four-dimensional autonomous system: among 3-torus, limit cycle, 2-torus, chaos and hyperchaos,”
*Chaos Solitons & Fractals*, vol. 39, pp. 2340–2356, 2009. View at Publisher · View at Google Scholar - W. Wu and Z. Chen, “Hopf bifurcation and intermittent transition to hyperchaos in a novel strong four-dimensional hyperchaotic system,”
*Nonlinear Dynamics*, vol. 60, no. 4, pp. 615–630, 2010. View at Publisher · View at Google Scholar - Q. Lu,
*Bifurcation and Singularity*, Shanghai Scientific and Technological Education, Shanghai, China, 1995. - L. Duan, Q. Lu, and Q. Wang, “Two-parameter bifurcation analysis of firing activities in the Chay neuronal model,”
*Neurocomputing*, vol. 72, pp. 341–351, 2008. View at Publisher · View at Google Scholar - G. Wen, D. Xu, and X. Han, “On creation of Hopf bifurcations in discrete-time nonlinear systems,”
*Chaos*, vol. 12, no. 2, pp. 350–355, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. Wen, “Criterion to identify Hopf bifurcations in maps of arbitrary dimension,”
*Physical Review E*, vol. 72, no. 2, Article ID 026201, 4 pages, 2005. View at Publisher · View at Google Scholar - E. Li, G. Li, G. Wen, and H. Wang, “Hopf bifurcation of the third-order Hénon system based on an explicit criterion,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 70, no. 9, pp. 3227–3235, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. C. Robinson,
*An Introduction to Dynamical Systems: Continuous and Discrete*, Pearson Prentice Hall, Upper Saddle River, NJ, USA, 2004. - Y. A. Kuznetsov,
*Elements of Applied Bifurcation Theory*, vol. 112 of*Applied Mathematical Sciences*, Springer, New York, NY, USA, 2nd edition, 1998.