This study examines the two most attractive characteristics, memory and chaos, in simulations of financial systems. A fractional-order financial system is proposed in this study. It is a generalization of a dynamic financial model recently reported in the literature. The fractional-order financial system displays many interesting dynamic behaviors, such as fixed points, periodic motions, and chaotic motions. It has been found that chaos exists in fractional-order financial systems with orders less than 3. In this study, the lowest order at which this system yielded chaos was 2.35. Period doubling and intermittency routes to chaos in the fractional-order financial system were found.

Nonlinear dynamics and chaos in a fractional-order financial system

In this paper, a delayed fractional order financial system is proposed and the complex dynamical behaviors of such a system are discussed by numerical simulations. A great variety of interesting dynamical behaviors of such a system including single-periodic, multiple-periodic, and chaotic motions are displayed. In particular, the effect of time delay on the chaotic behavior is investigated, it is found that an approximate time delay can enhance or suppress the emergence of chaos. Meanwhile, corresponding to different values of delay, the lowest orders for chaos to exist in the delayed fractional order financial systems are determined, respectively.

Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay

The complex dynamical behavior of a finance system is investigated in this paper The Ruelle–Takens route to chaos and strange nonchaotic attractors (SNA) are found through numerical simulations Then the system with time-delayed feedback is considered and the stability and Hopf bifurcation of the controlled system are investigated This research has important theoretical and practical meanings

Chaos and Hopf bifurcation of a finance system

Control of an uncertain fractional order economic system via adaptive sliding mode

Modeling and analysis of financial systems have been interesting topics among researchers. The more precisely we know dynamic of systems, the better we can deal with them. This way, in this paper, we investigate the effect of market confidence on a financial system from the perspective of fractional calculus. Market confidence, which is a significant concern in economic systems, is considered, and its effects are comprehensively investigated. The system has been studied through numerical simulations and analyses, such as the Lyapunov exponents, bifurcation diagrams, and phase portrait. It is shown that the system enters chaos through experiencing a cascade of period doublings, and the existence of chaos is verified. Finally, an analog circuit of the chaotic system is designed and implemented to prove its feasibility in real-world applications. Also, through the circuit implementation, the effects of different factors on the behavior of the systems are investigated.

The effect of market confidence on a financial system from the perspective of fractional calculus: Numerical investigation and circuit realization

Based on the work discussed on the former study, this article first starts from the mathematical model of a kind of complicated financial system, and analyses all possible things that the model shows in the operation of our country's macro-financial system: balance, stable periodic, fractal, Hopf-bifurcation, the relationship between parameters and Hopf-bifurcation, and chaotic motion etc. By the changes of parameters of all economic meanings, the conditions on which the complicated behaviors occur in such a financial system, and the influence of the adjustment of the macro-economic policies and adjustment of some parameter on the whole financial system behavior have been analyzed. This study will deepen people's understanding of the lever function of all kinds of financial policies.

https://www.amm.shu.edu.cn/CN/article/downloadArticleFile.do?attachType=PDF&id=11666

Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system (I)

The Lyapunov exponent is important quantitative index for describing chaotic attractors. Since Wolf put up the trajectory algorithm to Lyapunov exponent in 1985, how to calculate the Lyapunov exponent with accuracy has become a very important question. Based on the theoretical algorithm of Zuo Binwu, the matric algorithm of Lyapunov exponent is given, and the results with the results of Wolf's algorithm are compared. The calculating results validate that the matric algorithm has sufficient accuracy, and the relationship between the character of attractor and the value of Lyapunov exponent is studied in this paper. The corresponding conclusions are given in this paper.

The matric algorithm of lyapunov exponent for the experimental data obtained in dynamic analysis

The prediction methods for nonlinear dynamic systems which are decided by chaotic time series are mainly studied as well as structures of nonlinear self-related chaotic models and their dimensions. By combining neural networks and wavelet theories, the structures of wavelet transform neural networks were studied and also a wavelet neural networks learning method was given. Based on wavelet networks, a new method for parameter identification was suggested, which can be used selectively to extract different scales of frequency and time in time series in order to realize prediction of tendencies or details of original time series. Through pre-treatment and comparison of results before and after the treatment, several useful conclusions are reached: High accurate identification can be guaranteed by applying wavelet networks to identify parameters of self-related chaotic models and more valid prediction of the chaotic time series including noise can be achieved accordingly.

Study on prediction methods for dynamic systems of nonlinear chaotic time series

The paper not only studies the noise reduction methods of chaotic time series with noise and its reconstruction techniques, but also discusses prediction techniques of chaotic time series and its applications based on chaotic data noise reduction. In the paper, we first decompose the phase space of chaotic time series to range space and null noise space. Secondly we restructure original chaotic time series in range space. Lastly on the basis of the above, we establish order of the nonlinear model and make use of the nonlinear model to predict some research. The result indicates that the nonlinear model has very strong ability of approximation function, and Chaos predict method has certain tutorial significance to the practical problems.

Prediction techniques of chaotic time series and its applications at low noise level

Certain deterministic nonlinear systems may show chaotic behavior.We consider the motion of qualitative information and the practicalities of extractinga part from chaotic experimental data.Our approach based on a theorem of Takensdraws on the ideas from the generalized theory of information known as singular sys-tem analysis.We illustrate this technique by numerical data from the chaotic regionof the chaotic experimental data.The method of the singular-value decompositionis used to calculate the eigenvalues of embedding space matrix.The correspondingconcrete algorithm to calculate eigenvectors and to obtain the basis of embeddingvector space is proposed in this paper.The projection on the orthogonal basis gener-ated by eigenvectors of timeseries data and concrete paradigm are also provided here.Meanwhile the state space reconstruction technology of different kinds of chaotic dataobtained from dynamical system has also been discussed in detail.

马军海

Papers

Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system (I)

The matric algorithm of lyapunov exponent for the experimental data obtained in dynamic analysis

Study on prediction methods for dynamic systems of nonlinear chaotic time series

Prediction techniques of chaotic time series and its applications at low noise level

The state space reconstruction technology of different kinds of chaotic data obtained from dynamical system