Hopf Bifurcation and Dynamic Analysis of an Improved Financial System with Two Delays

Kai, G.; Zhang, W.; Jin, Z.; Wang, C. Z.

doi:https://doi.org/10.1155/2020/3734125

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Abstract Introduction Analysis Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Complexity, Dynamics, Control, and Applications of Nonlinear Systems with Multistability

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Research Article | Open Access

Volume 2020 | Article ID 3734125 | https://doi.org/10.1155/2020/3734125

Hopf Bifurcation and Dynamic Analysis of an Improved Financial System with Two Delays

G. Kai,^1,2,3W. Zhang ,^4,5Z. Jin ,⁶and C. Z. Wang¹

Guest Editor: Viet-Thanh Pham

Received24 Sept 2019

Accepted18 Oct 2019

Published27 Feb 2020

Abstract

The complex chaotic dynamics and multistability of financial system are some important problems in micro- and macroeconomic fields. In this paper, we study the influence of two-delay feedback on the nonlinear dynamics behavior of financial system, considering the linear stability of equilibrium point under the condition of single delay and two delays. The system undergoes Hopf bifurcation near the equilibrium point. The stability and bifurcation directions of Hopf bifurcation are studied by using the normal form method and central manifold theory. The theoretical results are verified by numerical simulation. Furthermore, one feature of the proposed financial chaotic system is that its multistability depends extremely on the memristor initial condition and the system parameters. It is shown that the nonlinear dynamics of financial chaotic system can be significantly changed by changing the values of time delays.

1. Introduction

It is widely recognized that chaos can be obtained in some mathematically simple systems of nonlinear differential equations. With the advent of computers, it is now possible to study the entire parameter space of these systems that result in some desired characteristics of the system. Recently, there has been increasing attention to some unusual examples and application of such systems [1–8]. The financial system is an extremely complex nonlinear dynamical system composed of many elements. The study of the complex nonlinear dynamics behavior of the financial system is an important problem in the fields of micro- and macroeconomy [9]. The uncertain factors bring very important influence to the description of the financial system and make analysis of the financial systems become a very important problem.

Researchers try to explain the core characteristics of economic data: irregular microeconomic fluctuations, instable macroeconomic fluctuations, irregular growth, and syntax changes [10–13]. However, some inappropriate combination of parameters in the financial system may lead to financial markets in trouble or out of control. Therefore, it is necessary to make a systematic and deep study on the internal syntax characteristics of the complicated financial system. The results will reveal the bifurcation phenomena under different parameter combinations, probe into the causes of the complicated nonlinear dynamics phenomena, and predict and control the complicated financial systems [14, 15]. In addition, multistability is a critical property of nonlinear dynamical systems when coexisting attractors can be obtained for the same parameters, but different initial conditions [16–19]. The flexibility in the systems’ performance can be archived without changing parameters.

Wang et al. [20] studied the bifurcation topology and the global complexity of a class of nonlinear financial systems. Ishiyama and Saiki [21] established the macroeconomic growth cycle model and solved the qualitative- and quantitative-related unstable periodic solutions embedded in the chaotic attractor. By using Lyapunov stability theory and Routh–Hurwitz criterion, Zhao et al. [22] studied the global synchronization of the three-dimensional chaotic financial system. Yu et at. [23] used numerical simulation to analyze the Lyapunov exponents and bifurcation diagram of chaotic financial system. Cantore and Levine [24] studied the reparameterized model with evaluation parameters. Gao et al. [25] gave the final bounded estimator set and chaotic synchronization analysis of the financial risks system.

Until now, one of the nonlinear economic and financial dynamic models confirmed by economists comes from a financial system model composed of four subblocks (production, money, security, and labor) [15, 24, 26].where represents the interest rate, represents the investment demand, represents the price index, represents the saving amount, represents the unit investment cost, and represents the elasticity of commodity demand; are all normal numbers.

In real life, with the development of economy, there are more and more factors that restrict the development of economy. Some classical chaotic financial systems can not reflect the laws and changes of economic development well. For example, the factors that affect the change of interest rate are related to the average profit rate besides investment demand and price index, and the average profit rate is proportional to the interest rate. Therefore, we construct the following improved chaotic financial system model:where is the interest rate, is the investment demand, is the price index, is the saving amount, is the unit investment cost, and is the elasticity of commodity demand; are all normal numbers. When the parameters , , and and initial values are at points , system (2a)–(2c) generates chaotic attractors, as shown in Figure 1.

With the development and innovation of financial markets, scholars have found that it will be better to add time delay factor for describing the actual economic markets [27–30]. Chen [31] analyzed the complex nonlinear dynamics, such as periodicity, quasiperiodicity, and chaotic behavior in the delayed feedback of financial systems. Ma and Tu [32] established a class of complex dynamic macroeconomic systems and studied the effect of time delay on savings rate and dynamic financial stability. Holyst and Urbanowicz [33] have shown that the chaotic attractor of the financial model can be stabilized in a periodic track by using Pyragas delayed feedback control. In addition, Ma and Chen [14] added the delayed feedback to the three variables of financial system and gave some results on the existence of Hopf bifurcation and the effect of delayed feedback. Based on political events and other human factors, some scholars have considered the impact of delay and feedback items (see [28, 34]). In practice, financial behaviour is not only affected by a single time delay but often seems to be affected by multiple external shocks. These various external influences embody multiple delays and can be reflected in all variables, i.e., by introducing various delayed feedback items into interest rates of change of interest rate, they will also have a significant impact on system (2a)–(2c). Therefore, we further consider the double-delay system.where and are the two time delays and and are the feedback control intensities.

In this paper, we study the Hopf bifurcation and nonlinear dynamics of an improved financial system with two delays. Firstly, we study the distribution of the roots of the characteristic equations at the equilibrium point. Sufficient conditions for the local stability of the equilibrium point and the existence of Hopf bifurcation are obtained. Secondly, taking two delays as bifurcation parameters and using the canonical form method and the central manifold theorem, we determine the bifurcation direction of the periodic solution and the explicit algorithm for the stability of the bifurcation periodic solution. Under the premise of the existence of local bifurcation, the existence of the bifurcation periodic solution of this system is discussed by using the theory of functional differential equations. Finally, the correctness of the conclusion is verified by numerical simulation.

2. Existence of Hopf Bifurcation in Financial System

In order to study the influence of time delays on nonlinear dynamic system, the three equilibrium points of the system are obtained as follows:

Here, we only analyze the following equilibrium point:

Linear transformations are given as

System (2a)–(2c) becomes the following equations:

The characteristic equation of equations (7a) and (7b) at is

Since the dynamics of differential equations with two delays are very complex, we first discuss the case of , then discuss the case of , with single delay, and finally discuss the case of , .

Case 1. .
The characteristic equation (5) becomesLetand assumeAccording to Routh– Hurwitz criterion, if holds, the equilibrium point of system (7a)–(7c) is locally asymptotically stable.

Case 2. , .
The characteristic equation (5) becomesIf is a solution of equation (12), then the real part and imaginary part are separated and made equal to zero. We can obtainwhere and .
From formulas (13a) and (13b),If all the parameters in system (3a)–(3c) are given, it is easy to calculate the numerical solution of equation (14) by computer. Thus, the following assumptions are given.
Suppose that equation (14) has at least one positive real root.
If is assumed to be true and equation (14) has two positive real roots , we havewhere is a pair of pure virtual root under of equation (12). Let us takeLet be the virtual root of equation (12) near . By differential degeneracy of equation (12) with respect to , we can obtainWhen is substituted into equation (18), we haveAmong them,

Hypothesis 1. :Therefore, there are the following theorems.

Theorem 1. If , , and are assumed to be true, then the equilibrium is locally stable for . When equilibrium is unstable and Hopf bifurcation occurs in system (3a)–(3c) at .

Case 3. , .
The corresponding characteristic equation of system (3a)–(3c) is (8). Now, let delay , and is taken as a parameter. Assuming is the characteristic root under the two delays in the characteristic equation (5), thenFrom (22),Therefore, the equation about can be obtained:Obviously, equation (24) has at most a positive real root of , which is denoted as . Similarly, the following can be obtained:whereThen, there is a point column satisfying equation (26a) and (26b),Correspondingly, and .
When and , equation (8) has a pair of pure virtual root.

Hypothesis 2. In this way, by using the general Hopf bifurcation theorem [35–40] for functional differential equations, the results on the stability and bifurcation of system (3a)–(3c) are obtained.

Theorem 2. Assuming holds and holds, the equilibrium of system (3a)–(3c) is asymptotically stable at . When occurs, Hopf bifurcation occurs in system (3a)–(3c).

3. Period Solution and Stability of Hopf Bifurcation

In this section, we study the relevant properties of Hopf bifurcation in financial system (3a)–(3c) under the condition of delays , .

Using the ideas of Hassard et al. [36], the exact expression of Hopf bifurcation property of system (3a)–(3c) is considered by using central manifold theorem. Here, we consider the Hopf bifurcation of system (4a)–(4c) at the equilibrium point for . The financial system can be converted to the following equation:where

According to systems (3a)–(3c) and (7a)–(7c), it can be seen that

Let

Then,

When the equilibrium point of system (29) passes Hopf bifurcation at , the characteristic equation has a pair of pure virtual root and . According to Ritz representation theorem, there is a matrix function of bounded variation

In fact, we can choose

For , we define

To simplify, equation (29) can be written in the following form:in which

The adjoint operator that defines for is as follows:

In addition, we define a bilinear form

According to the above analysis, and are the eigenvalues of and . Let be the eigenvector corresponding to the eigenvalue of and be the eigenvector corresponding to the eigenvalue of . There are

Through simple calculation, we can getwhere is a constant, making valid.

Therefore, we have

Using the same notation of Ruan et al. [37], we can calculate the center popularity of . Let be the solution of equation (29) when , definingwhere and are the local coordinates of the central epidemic. When ,

Make

Then,

Becausewe can obtain

Based on the above formula (49b), it can be seen that

By using the method of comparison coefficient, we obtain

In order to calculate and , we use

Make

We rewrite (54):

Using (54) and (55), one can obtain

The combination formula (57) is obtained:

Using equations (58a) and (58b) and (60a) and (60b), it is easy to obtain

Make

Combine formulas (58a) and (58b) again to obtain

Substituting equations (62) and (65a)–(65d) into (64a) and (64b), there is

Therefore,where

Letin which

Similarly, we can also have

Letwhere

Furthermore, can be determined by the coefficients and time delays of system (7a)–(7c). Thus, the following values can be calculated using the method in [37]:

Formulas (74a)–(74d) determine the critical point above the central flow . Now, the properties of periodic solutions of system (3a)–(3c) could be obtained. Therefore, we obtain the following theorem.

Theorem 3. For system (3a)–(3c), when ,(1)The symbol determines the orientation of Hopf bifurcation. If , then the Hopf bifurcation is supercritical (subcritical). If , the bifurcation period solution exists.(2)The sign of determines the stability of the bifurcation period solution: if , the period solution is stable (unstable).(3)The symbol determines the period of bifurcation period solution. If , the period solution is increased (decreased).

4. Numerical Results and Analysis

In this section, we will give numerical simulations on the theoretical results of Hopf bifurcation with two delays.

Given the parameters , , , , and , it is verified that at that time , the parameters satisfy the assumption that the equilibrium point of system (3a)–(3c) without time delay is asymptotically stable.(1) If equation (8) has a pair of pure virtual root and , there are two positive roots and . Substituting and into equations (15a) and (15b), we can obtain That is, , and we have the following result for : According to Theorem 1, equilibrium point of system (3a)–(3c) is asymptotically stable for . Hopf bifurcation occurs when , , and a stable period solution is obtained. The time series diagram and phase diagram are shown in Figure 2.(2) Let and consider . Suppose that equation (22) has a pair of pure virtual root and . Then, according to equation (24), it is found that . There is . Substituting into equations (25a) and (25b), we can obtain That is, and According to Theorem 2, we know when , the equilibrium point is asymptotically stable (as shown in Figure 3: and ). When and , Hopf bifurcation occurs and we can have

Therefore, the bifurcation direction is , and the period solution is stable, as shown in Figure 4: and . In addition, the complex dynamics of system (3a)–(3c) could be shown from the bifurcation diagram in Figure 5.

(a)

(b)

(a)

(b)

(a)

(b)

5. Multistability in the Improved Financial System (3a)–(3c) with Two Delays

When and , we will find the multistability in the systems without changing parameters. Given the parameters , , , , and , we have obtained the periodic attractor with initial value in Figure 6(a). However, we also obtain chaos for same parameters’ values but different initial values in Figure 6(b). Therefore, when different initial conditions are taken, the coexisting and different attractors are exhibited. We know multistability is a critical property of nonlinear dynamical systems [41–44]. Since the crisis of the financial system is subject to various factors, the nature of the multi-steady state plays an important role in making correct decisions for government workers.

(a)

(b)

6. Conclusion

Time delay is a very sensitive factor in financial systems with multistability. Financial systems with multiple time delays have richer dynamic characteristics than those with single time delay. Two-delay feedback can effectively control the unstable behavior of financial markets. In this paper, Hopf bifurcation of an improved financial model with two time delays is studied in detail. The existence of the bifurcation period solution of this system is discussed by using the theory of functional differential equations. Complexity of the proposed financial chaotic system is studied from the bifurcation diagram that its multistability depends extremely on the memristor initial condition and the system parameters. In summary, time delay is one of the effective methods to control the stability of the financial market, so it can provide a theoretical reference for relevant departments to regulate economic behavior.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this research work.

Acknowledgments

The authors gratefully acknowledge the support of National Natural Science Foundation of China (nos. 11902163, 11832002, 71763021, and 11427801), the National Social Science Foundation of China (no. BMA180038), Science and Technology Research Project of Universities in Inner Mongolia Autonomous Region (no. NJZY20159), China Postdoctoral Science Foundation funded project (no. 2019M660003XB), and the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHRIHLB).

References

S. I. Batool and H. M. Waseem, “A novel image encryption scheme based on Arnold scrambling and Lucas series,” Multimedia Tools and Applications, vol. 78, no. 19, pp. 27611–27637, 2019.
View at: Publisher Site | Google Scholar
K. M. Ali and M. Khan, “Application based construction and optimization of substitution boxes over 2D mixed chaotic maps,” International Journal of Theoretical Physics, vol. 58, no. 9, pp. 3091–3117, 2019.
View at: Publisher Site | Google Scholar
M. Khan and F. Masood, “A novel chaotic image encryption technique based on multiple discrete dynamical maps,” Multimedia Tools and Applications, vol. 78, pp. 26203–26222, 2019.
View at: Publisher Site | Google Scholar
I. Younas and M. Khan, “A new efficient digital image encryption based on inverse left almost semi group and lorenz chaotic system,” Entropy, vol. 20, no. 12, p. 913, 2018.
View at: Publisher Site | Google Scholar
M. Khan, “A novel image encryption scheme based on multiple chaotic S-boxes,” Nonlinear Dynamics, vol. 82, no. 1-2, pp. 527–533, 2015.
View at: Publisher Site | Google Scholar
Z. Wei, Y. Li, B. Sang et al., “Complex dynamical behaviors in a 3D simple chaotic flow with 3D stable or 3D unstable manifolds of a single equilibrium,” International Journal of Bifurcation and Chaos, vol. 29, no. 07, p. 1950095, 2019.
View at: Publisher Site | Google Scholar
Z. C. Liu, I. Moroz, J. C. Sprott et al., “Hidden hyperchaos and electronic circuit application in a 5D self-exciting homopolar disc dynamo,” Chaos, vol. 27, no. 3, Article ID 033101, 2017.
View at: Publisher Site | Google Scholar
Z. Wei, I. Moroz, J. C. Sprott et al., “Detecting hidden chaotic regions and complex dynamics in the self-exciting homopolar disc dynamo,” International Journal of Bifurcation and Chaos, vol. 27, no. 2, p. 1730008, 2017.
View at: Publisher Site | Google Scholar
R. Wang, Economic Dynamics, Cambridge University Press, Cambridge, England, 2002.
A. C.-L. Chian, “Nonlinear dynamics and chaos in macroeconomics,” International Journal of Theoretical and Applied Finance, vol. 3, no. 3, p. 601, 2000.
View at: Publisher Site | Google Scholar
S. Radelet, J. D. Sachs, R. N. Cooper, and B. P. Bosworth, “The east asian financial crisis: diagnosis, remedies, prospects,” Brookings Papers on Economic Activity, vol. 1998, no. 1, pp. 1–74, 1998.
View at: Publisher Site | Google Scholar
A. Bosworth, “Is there chaos in economic series?” Canadian Journal of Economics, vol. 29, pp. 210–212, 1996.
View at: Google Scholar
Y. Wang and Y. H. Zhai, “Chaos and Hopf bifurcation of a finance system with distributed time delay,” International Journal of Applied Mathematics and Mechanics, vol. 6, no. 20, pp. 1–13, 2010.
View at: Google Scholar
J. H. Ma and Y. Chen, “Study for the bifurcation topological structure and the global complicated character of a kind of non-linear finance system (I),” Applied Mathematics and Mechanics, vol. 22, pp. 1240–1251, 2010.
View at: Google Scholar
C. Ma and X. Wang, “Hopf bifurcation and topological horseshoe of a novel finance chaotic system,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 2, pp. 721–730, 2012.
View at: Publisher Site | Google Scholar
Z. Wei, P. Yu, W. Zhang et al., “Study of hidden attractors, multiple limit cycles from Hopf bifurcation and boundedness of motion in the generalized hyperchaotic Rabinovich system,” Nonlinear Dynamics, vol. 82, no. 1-2, pp. 131–141, 2015.
View at: Publisher Site | Google Scholar
K. Yao, S. Jafari, V.-T. Pham et al., “Antimonotonicity, bifurcation and multistability in the vallis model for el niño,” International Journal of Bifurcation and Chaos, vol. 29, no. 3, p. 1950032, 2019.
View at: Publisher Site | Google Scholar
K. Wei, A. J. M. Khalaf, Z. Wei et al., “Hyperchaos and coexisting attractors in a modified van der Pol-Duffing oscillator,” International Journal of Bifurcation and Chaos, vol. 29, no. 5, p. 1950067, 2019.
View at: Publisher Site | Google Scholar
Z. Pham, V.-T. Pham, T. Kapitaniak et al., “Bifurcation analysis and circuit realization for multiple-delayed Wang-Chen system with hidden chaotic attractors,” Nonlinear Dynamics, vol. 85, no. 3, pp. 1635–1650, 2016.
View at: Publisher Site | Google Scholar
Y. Wang, W. Jiang, and H. Wang, “Delayed feedback control and bifurcation analysis of Rossler chaotic system,” Nonlinear Dynamics, vol. 61, no. 4, pp. 707–715, 2010.
View at: Publisher Site | Google Scholar
K. Ishiyama and Y. Saiki, “Unstable periodic orbits and chaotic economic growth,” Chaos, Solitons & Fractals, vol. 26, no. 8, pp. 33–42, 2005.
View at: Publisher Site | Google Scholar
X. S. Zhao, Z. B. Li, and S. Li, “Synchronization of a chaotic finance system,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 6031–6039, 2011.
View at: Publisher Site | Google Scholar
H. J. Yu, G. L. Cai, and Y. X. Li, “Dynamic analysis and control of a new hyperchaotic finance system,” Nonlinear Dynamics, vol. 67, no. 23, pp. 2171–2182, 2012.
View at: Publisher Site | Google Scholar
C. Cantore and P. Levine, “Getting normalization right: dealing with ‘dimensional constants’ in macroeconomics,” Journal of Economic Dynamics and Control, vol. 36, no. 2, pp. 1931–1949, 2012.
View at: Publisher Site | Google Scholar
W. Gao, L. Yan, M. Saeedi et al., “Ultimate bound estimation set and chaos synchronization for a financial risk system,” Mathematics and Computers in Simulation, vol. 154, pp. 19–33, 2018.
View at: Publisher Site | Google Scholar
W. Saberi Nik, “Integrability analysis of chaotic and hyperchaotic finance systems,” Nonlinear Dynamics, vol. 94, no. 1, pp. 443–459, 2018.
View at: Publisher Site | Google Scholar
L. Zhang, G. Cai, and X. Fang, “Stability for a novel time-delay financial hyperchaotic system by adaptive periodically intermittent linear control,” Journal of Applied Analysis and Computation, vol. 7, no. 1, pp. 79–91, 2017.
View at: Google Scholar
W. Zhang, J. Cao, A. Alsaedi et al., “Synchronization of time delayed fractional order chaotic financial system,” Discrete Dynamics in Nature and Society, vol. 2017, no. 29–32, pp. 1–5, 2017.
View at: Publisher Site | Google Scholar
Y. Alsaadi and J. Cao, “Bifurcation analysis and chaos switchover phenomenon in a nonlinear financial system with delay feedback,” International Journal of Bifurcation and Chaos, vol. 25, no. 12, pp. 2671–2691, 2015.
View at: Publisher Site | Google Scholar
Z. Wei, B. Zhu, J. Yang, M. Perc, and M. Slavinec, “Bifurcation analysis of two disc dynamos with viscous friction and multiple time delays,” Applied Mathematics and Computation, vol. 347, pp. 265–281, 2019.
View at: Publisher Site | Google Scholar
W. C. Chen, “Dynamics and control of a financial system with time-delayed feedbacks,” Chaos, Solitons & Fractals, vol. 37, no. 5, pp. 1198–1207, 2008.
View at: Publisher Site | Google Scholar
J. Ma and H. Tu, “Analysis of the stability and Hopf bifurcation of money supply delay in complex macroeconomic models,” Nonlinear Dynamics, vol. 76, no. 1, pp. 497–508, 2014.
View at: Publisher Site | Google Scholar
J. A. Holyst and K. Urbanowicz, “Chaos control in economical model by time-delayed feedback method,” Physica A, vol. 287, no. 3-4, pp. 587–598, 2000.
View at: Publisher Site | Google Scholar
Y. Ding, W. Jiang, and H. Wang, “Hopf-pitchfork bifurcation and periodic phenomena in nonlinear financial system with delay,” Chaos, Solitons & Fractals, vol. 45, no. 8, pp. 1048–1057, 2012.
View at: Publisher Site | Google Scholar
J. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 1977.
B. Hassard, D. Kazarino, and Y. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, England, 1981.
S. Ruan and J. Wei, “On the zero of some transcendential functions with applications to stability of delay differential equations with two delays,” Dynamics of Continuous, Discrete and Impulsive Systems, vol. 10, pp. 863–874, 2003.
View at: Google Scholar
Q. Shi, J. Shi, and Y. Song, “Hopf bifurcation in a reaction-diffusion equation with distributed delay and Dirichlet boundary condition,” Journal of Differential Equations, vol. 263, no. 10, pp. 6537–6575, 2017.
View at: Publisher Site | Google Scholar
Z. Wang, X. Wang, Y. Li et al., “Stability and Hopf bifurcation of fractional-order complex-valued single neuron model with time delay,” International Journal of Bifurcation and Chaos, vol. 27, no. 13, p. 1750209, 2017.
View at: Publisher Site | Google Scholar
G. Huang, W. Zhang, Z. C. Wei et al., “Hopf bifurcation, positively invariant set and physical realization of a new four-dimensional hyperchaotic financial system,” Mathematical Problems in Engineering, vol. 2017, Article ID 2490580, 2017.
View at: Publisher Site | Google Scholar
J. P. Singh and B. K. Roy, “The simplest 4-D chaotic system with line of equilibria, chaotic 2-torus and 3-torus behaviour,” Nonlinear Dynamics, vol. 89, no. 3, pp. 1845–1862, 2017.
View at: Publisher Site | Google Scholar
J. P. Singh and B. K. Roy, “Hidden attractors in a new complex generalised Lorenz hyperchaotic system, its synchronisation using adaptive contraction theory, circuit validation and application,” Nonlinear Dynamics, vol. 92, no. 2, pp. 373–394, 2018.
View at: Publisher Site | Google Scholar
J. P. Singh, K. Lochan, N. V. Kuznetsov et al., “Coexistence of single- and multi-scroll chaotic orbits in a single-link flexible joint robot manipulator with stable spiral and index-4 spiral repellor types of equilibria,” Nonlinear Dynamics, vol. 90, no. 2, pp. 1277–1299, 2017.
View at: Publisher Site | Google Scholar
C. B. Roy, W. J. C. Thio, J. C. Sprott et al., “Constructing Infinitely many attractors in a programmable chaotic circuit,” IEEE Access, vol. 6, pp. 29003–29012, 2018.
View at: Publisher Site | Google Scholar

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