Chaotic Financial System Research Articles

Dynamics Analysis and Adaptive Synchronization of a Class of Fractional-Order Chaotic Financial Systems

It is of practical significance to realize a stable and controllable financial system by using chaotic synchronization theory. In this paper, the dynamics and synchronization are studied for a class of fractional-order chaotic financial systems. First, the stability and dynamics of the fractional-order chaotic financial system are analyzed by using the phase trajectory diagram, time series diagram, bifurcation diagram, and Lyapunov exponential diagram. Meanwhile, we obtain the range of each parameter that puts the system in a periodic state, and we also reveal the relationship of the derivative order and the chaotic behaviors. Then, the adaptive control strategy is designed to achieve synchronization of the chaotic financial system. Finally, the theoretical results and control method are verified by numerical simulations.

Predefined-time stabilization of Lorenz system with applications for stabilizing and synchronizing chaotic finance systems

This article investigates the predefined-time stabilization (PtS) of the canonical Lorenz system at first, and then applies the derived results into the chaotic finance systems (CFSs) so as to realize the stabilization and synchronization, respectively. Compared with the traditional finite-/fixed-time stability analysis, the upper-bound of convergence time (UbCT) in this investigation can be set beforehand in need, which is an explicit constant regardless of initial values, system dimension, and controlling parameters. Moreover, the designed control schemes are non-chattering, which do not contain the conventional discontinuous signum and absolute value functions anymore. Via adopting the second Lyapunov method, the sufficient conditions are obtained successively for guaranteeing the realization of PtS for Lorenz system, CFS, as well as the predefined-time synchronization between two CFSs. The numerical experiments are finally arranged to manifest the correctness and effectiveness of the theoretical fruits, in which some comparison and perturbation analysis are made.

Robust synchronization of four-dimensional chaotic finance systems with unknown parametric uncertainties

ABSTRACT The inherent randomness in economic factors causes complex and irregular behaviour that affects financial system stability and economic growth. Such chaotic behaviour can make it difficult to synchronize financial systems. The chaotic finance system synchronization precision maintains financial stability and economic growth. In this paper, the controller design procedure assumes that the financial system is exposed to unknown bounded exogenous disturbances and model uncertainties. This research proposes a novel direct adaptive control strategy that achieves robust synchronization of two identical four-dimensional finance chaotic (FDFC) systems. The proposed controller establishes a faster, smoother synchronization error vector convergence to zero. The controller design procedure does not eliminate the closed-loop's nonlinear terms and is independent of the financial system parameters. These controller's attributes accomplish the closed-loop robust performance. Further, this controller uses real-time estimates of unknown model uncertainties and bounds to compensate for unknown exogenous disturbances. Computer simulation results and proofs of theoretical analysis based on the Lyapunov stability theory confirm that the proposed control technique compels the error vector trajectories to the origin in a short transient time with less active oscillations for all signals. The paper includes comparative computer simulations for verifying the theoretical findings.

A novel financial system with one stable and two unstable equilibrium points: Dynamics, coexisting attractors, complexity analysis and synchronization using integral sliding mode control

In the world of economics, financial risk is a significant issue that concerns countries and all economic agents. It refers to the potential for experiencing losses due to unpredictable fluctuations in financial or investment activities caused by endogenous factors. Chaos theory provides an explanation for the complexity observed in economic time series, as it supports an endogenous view. This article proposed an enhanced chaos-based financial risk system with one stable and two unstable equilibrium points. The existence of stable and unstable equilibrium points shows that the proposed system has both hidden and self-excited chaotic attractors. By analyzing its dynamic behavior, the study unveils the transition from stability to chaos in the improved financial risk system, while also describing how the complex dynamics of the system affect changes in financial market risk. In addition, this study explores the fundamental dynamic behavior of the financial risk system, including stability analysis, Lyapunov exponent, bifurcation analysis, coexisting attractors, 0–1 test, algorithm and spectral entropy (SE). Finally, with the help of integral sliding mode control (ISMC), we discover the outcomes for the asymptotic synchronization of two chaotic financial systems that are constructed as master and slave systems. Simulations are provided to accompany and support the findings from this research.

Modeling and Analysis of the Fuzzy-Fractional Chaotic Financial System Using the Extended He–Mohand Algorithm in a Fuzzy-Caputo Sense

This paper contains modeling of a fuzzy-fractional financial chaotic model based on triangular fuzzy numbers (TFNs) to predict the idea that long-term dependency and uncertainty both have an impact on the financial market. For solution purposes, the He–Mohand algorithm is proposed where homotopy perturbation is hybrid with Mohand transform in a fuzzy-Caputo sense. In analysis, solutions and corresponding errors at upper and lower bounds are estimated. The obtained numerical results are displayed in tables to show the reliability and efficiency of the proposed methodology. Upper bound errors range from 10 − 6 to 10 − 12 and lower bound errors from 10 − 6 to 10 − 11 . For graphical analysis, system profiles are illustrated as two-dimensional and three-dimensional plots at diverse values of fractional parameters and time to comprehend the physical behavior of the proposed fuzzy-fractional model. These plots demonstrate that the interest rate, price index, and investment demand decrease with the increase of the value of the r -cut at the lower bound. At the upper bound, this behavior is totally opposite. The chaotic behavior of the system at smaller values of saving rate, elasticity of demands, and per-investment cost is greater in contrast to their larger value. Analysis reveals that the proposed methodology (He–Mohand algorithm) provides a new way of understanding the complicated structure of financial systems and provides new insights into the dynamics of financial markets. This algorithm has potential applications in risk management, portfolio optimization, and trading strategies.

Realizing stochastic fixed-time synchronization between two nonlinear chaotic finance systems

The issue of stochastic fixed-time synchronization (SFxS) between two nonlinear chaotic finance systems (CFSs) in noisy environment is addressed by this investigation. Two novel strategies, including the adaptive and fixed-time control, are integrated organically, which steers the CFSs to realize synchronization within a fixed-time, and meanwhile the controlling gains can be updated by the designed adaptive law. Particularly, the designed adaptive fixed-time controlling (AFC) scheme is differentiable, so that the chattering phenomenon can be alleviated and even eliminated effectively. Furthermore, by the stochastic Lyapunov stability analysis, the sufficient criterion is derived for realizing the SFxS of CFSs, and the upper-bound expectation of settling-time (UET) is estimated as well. At last, the numerical experiments are arranged to verify the correctness and feasibility of analytical results, the influence of noisy perturbation on the convergence rate is uncovered, and the interesting similar Dragon–King events are observed.

Bifurcation & stochastic analysis of chaotic finance system through entanglement function

In this novel chaotic financial system, by varying the set-aside bifurcation parameter, the existence of the Hopf bifurcation is investigated. The results are used to discuss equilibrium stability and bifurcations in the modern chaotic system. Added to that, there is an element of chaos in mathematical systems with only one positive Lyapunov exponent. Furthermore, there are conditions in which bifurcation is required to achieve chaotic entanglement. The entanglement functions are designed to form an artificially chaotic system such that two or more stable linear subsystems evolve chaotically, no matter how far apart they are. The Gaussian noise is introduced to normalize or bridge the empirical distribution function of the random variables system and check the convergence. Beyond that, the stochastic model emphasizes the relationships between dissipation, phase space flow, and stability. The maximum dissipation of the system increased as the stochastic financial stability increased.

Applications of Modified Bessel Polynomials to Solve a Nonlinear Chaotic Fractional-Order System in the Financial Market: Domain-Splitting Collocation Techniques

We propose two accurate and efficient spectral collocation techniques based on a (novel) domain-splitting strategy to handle a nonlinear fractional system consisting of three ODEs arising in financial modeling and with chaotic behavior. One of the major numerical difficulties in designing traditional spectral methods is in the handling of model problems on a long computational domain, which usually yields to loss of accuracy. One remedy is to split the underlying domain and apply the spectral method locally in each subdomain rather than on the global domain of interest. To treat the chaotic financial system numerically, we use the generalized version of modified Bessel polynomials (GMBPs) in the collocation matrix approaches along with the domain-splitting strategy. Whereas the first matrix collocation scheme is directly applied to the financial model problem, the second one is a combination of the quasilinearization method and the direct first numerical matrix method. In the former approach, we arrive at nonlinear algebraic matrix equations while the resulting systems are linear in the latter method and can be solved more efficiently. A convergence theorem related to GMBPs is proved and an upper bound for the error is derived. Several simulation outcomes are provided to show the utility and applicability of the presented matrix collocation procedures.

Optimal type-3 fuzzy control and analysis of complicated financial systems

The financial systems have complicated dynamics and are perturbed by various uncertainties and disturbances. Chaos theory provides a practical approach to analyzing financial systems. The chaotic systems have unpredictable random characteristics that help to analyze the financial systems better. Recently, type-3 (T3) fuzzy logic systems (FLSs) have been developed for high-uncertain systems. T3-FLSs provide a reliable tool to cope with high-noisy environments. In T3-FLSs, the upper/lower bounds of uncertainties are fuzzy values. This property results in a strong tool to model more levels of uncertainties. Control, modeling, and forecasting accuracy in financial systems are so important. Then, better systems with higher accuracy are required. In this paper, a new T3-FLS based controller is introduced for chaotic financial systems. By solving a Riccati equation, sufficient conditions are concluded for optimality and robustness. T3-FLSs are learned to minimize the error and stabilize the whole system. A new optimal learning rules are extracted for T3-FLSs. Various benchmark chaotic model of financial systems are considered for examining the efficacy of the introduced approach, and the excellent response and superiority of the suggested approach is verified. Also, a comparison with other methods demonstrates the better efficiency of the suggested scheme.

A modified Salp Swarm Algorithm for parameter estimation of fractional-order chaotic systems

For the parameter estimation problem in research related to the fractional-order chaotic systems (FOCSs), a modified optimization algorithm based on Salp Swarm Algorithm (SSA) was developed in this paper. The proposed algorithm introduced several improvements on SSA: adding a grouping step, introducing “betrayal” behavior, and improving the update method of the followers. We applied multiple classical optimization algorithms to conduct the parameter estimation experiments on the fractional-order Lorenz chaotic system (Lorenz-FOCS) and the fractional-order Financial chaotic system (Financial-FOCS). In addition, we explored the impact of searching space on parameters estimation through experiments. The experimental results confirmed the feasibility of the modified Salp Swarm Algorithm (MSSA). The MSSA performed better than the SSA and other classical optimization algorithms in terms of the estimation accuracy and convergence rate.

An Intelligence Computational Approach for the Fractional 4D Chaotic Financial Model

The main purpose of the study is to present a numerical approach to investigate the numerical performances of the fractional 4-D chaotic financial system using a stochastic procedure. The stochastic procedures mainly depend on the combination of the artificial neural network (ANNs) along with the Levenberg-Marquardt Backpropagation (LMB) i.e., ANNs-LMB technique. The fractional-order term is defined in the Caputo sense and three cases are solved using the proposed technique for different values of the fractional order α. The values of the fractional order derivatives to solve the fractional 4-D chaotic financial system are used between 0 and 1. The data proportion is applied as 73%, 15%, and 12% for training, testing, and certification to solve the chaotic fractional system. The acquired results are verified through the comparison of the reference solution, which indicates the proposed technique is efficient and robust. The 4-D chaotic model is numerically solved by using the ANNs-LMB technique to reduce the mean square error (MSE). To authenticate the exactness, and consistency of the technique, the obtained performances are plotted in the figures of correlation measures, error histograms, and regressions. From these figures, it can be witnessed that the provided technique is effective for solving such models to give some new insight into the physical behavior of the model.

Predefined-time control of chaotic finance/economic system based on event-triggered mechanism

<abstract><p>Aiming at the problem that the convergence time of the chaotic finance/economic system cannot be set independently and the continuous macro-control is required, this paper investigates the predefined-time control of the chaotic finance/economic system based on event-triggered mechanism. The predefined-time control approach ensures the chaotic finance system quickly converge to the stable state within a pre-determined time. Moreover, in order to avoid continuous macro-control, an event-trigger mechanism is added into the above predefined-time control approach, which guarantees that the control input is updated only when some predefined event occurs. Rigorous theoretical derivation is presented and concrete simulation experiments are carried out to validate the feasibility and applicability of the proposed control strategy.</p></abstract>

Dynamical Analysis and Adaptive Finite-Time Sliding Mode Control Approach of the Financial Fractional-Order Chaotic System

In this work, we studied the complex behaviors of the fractional-order financial chaotic system, consisting of a simple, relatively chaotic system with two quadratic nonlinearities (QN) and a sextic nonlinearity (SN). We completed and enriched the results presented in the study of Subartini et al. (2021). As a result of this, our study focused more on the fractional order and adaptive finite-time sliding mode control in the financial risk chaotic system. The dynamical behaviors of the financial chaotic system (FCS) with two QN and an SN were analyzed, and the stability was investigated via the Cardano method. The stability analysis showed that the real part of all the roots was negative, which confirmed the stability of the new system under the typical parameters. By using the MATLAB simulation, these properties were characterized, including the phase portraits, 0-1 test, Poincaré map, bifurcation diagram, and Lyapunov exponent. The analysis showed that the financial risk chaotic system of fractional order was able to exhibit chaotic behavior and periodical behavior. In spite of external perturbations and uncertainty, an adaptive finite-time sliding mode control strategy was devised to guide the states of the financial chaotic system to the origin in a finite amount of time. MATLAB phase plots were employed in this study to illustrate all the main results.

Fixed-Time Output-Constrained Synchronization of Unknown Chaotic Financial Systems Using Neural Learning

This article addresses the challenging problem of fixed-time output-constrained synchronization for master–slave chaotic financial systems with unknown parameters and perturbations. A fixed-time neural adaptive control approach is originally proposed with the aid of the barrier Lyapunov function (BLF) and neural network (NN) identification. The BLF is introduced to preserve the synchronization errors always within the predefined output constraints. The NN is adopted to identify the compound unknown item in the synchronization error system. Unlike the conventional NN identification, the concept of indirect NN identification is employed, and only a single adaptive learning parameter is required to be adjusted online. According to the stability argument, the proposed controller can ensure that all error variables in the closed-loop system regulate to the minor residual sets around zero in fixed time. Finally, simulations and comparisons are conducted to verify the efficiency and benefits of the proposed control strategy. It can be concluded from the simulation results that the proposed fixed-time neural adaptive controller is capable of achieving better synchronization performance than the compared linear feedback controller.

Fractional-Order Financial System and Fixed-Time Synchronization

This study is concerned with the dynamic investigation and fixed-time synchronization of a fractional-order financial system with the Caputo derivative. The rich dynamic behaviors of the fractional-order financial system with variations of fractional orders and parameters are discussed analytically and numerically. Through using phase portraits, bifurcation diagrams, maximum Lyapunov exponent diagrams, 0–1 testing and time series, it is found that chaos exists in the proposed fractional-order financial system. Additionally, a complexity analysis is carried out utilizing approximation entropy SE and C0 complexity to detect whether chaos exists. Furthermore, a synchronization controller and an adaptive parameter update law are designed to synchronize two fractional-order chaotic financial systems and identify the unknown parameters in fixed time simultaneously. The estimate of the setting time of synchronization depends on the parameters of the designed controller and adaptive parameter update law, rather than on the initial conditions. Numerical simulations show the effectiveness of the theoretical results obtained.

Dynamics analysis, adaptive control, synchronization and anti-synchronization of a novel modified chaotic financial system

Dynamics analysis, adaptive control, synchronization and anti-synchronization of a novel modified chaotic financial system

Adaptive fixed-time synchronization of Lorenz systems with application in chaotic finance systems

This paper concerns the problem of fixed-time synchronization of master–slave Lorenz systems. The adaptive control and fixed-time control strategies are successfully integrated so that not only the Lorenz systems can be synchronized within a fixed-time, but also the related controlling gains are not necessary to select beforehand. Distinguished from the conventional fixed-time control schemes, the proposed controller do not contain the signum function anymore, thereby the chattering behavior is avoided owing to its smoothness. The synchronizing condition is deduced according to the theory of fixed-time stability, and the upper bound of settling time is also estimated, which is irrelevant to the initial states of Lorenz systems. Apart from the correctness and effectiveness of theoretical analysis is validated by simulating Lorenz systems, the spirit of adaptive fixed-time control strategy is applied into synchronizing two finance systems finally.

Dynamic analysis and adaptive modified projective synchronization for systems with Atangana-Baleanu-Caputo derivative: A financial model with nonconstant demand elasticity

The inherent instability of the financial system itself may lead to chaos and unpredictable economic disorder. It is meaningful to study the stability and control theory of financial chaotic system. Based on the definition of Atangana-Baleanu-Caputo fractional derivative, this work extends the integer-order financial chaotic system with nonconstant demand elasticity to a fractional-order system, and analyzes its nonlinear dynamic properties. The dynamic behavior of the system is discussed by phase diagram, bifurcation diagram, the Largest Lyapunov Exponent (LLE) and 0–1 test method. An adaptive controller is designed to realize the modified projection synchronization of the system. The results can help to improve the understanding of the complex financial system, and provide theoretical support for the formulation of financial intervention strategies.

Impulsive Stabilization on Hyper-Chaotic Financial System under Neumann Boundary

This paper proposes a novel technique to obtain sufficient conditions for the existence and stabilization of positive solutions for a kind of hyper-chaotic financial model. Since some important economic indexes are heavily related to region, the authors consider a nonlinear chaotic financial system with diffusion, which leads to some mathematical difficulties in dealing with the infinite-dimension characteristic. In order to overcome these difficulties, novel analysis techniques have to be proposed on the basis of Laplacian semigroup and impulsive control. Sufficient conditions are provided for existence and global exponential stabilization of positive solution for the system. It is interesting to discover that the impulse strength can be larger than 1 in the newly obtained stability criterion. Numerical simulations show the effectiveness of theoretical analysis.

Investigation of chaos behavior and integral sliding mode control on financial risk model

<abstract> <p>This paper reports the finding of a new financial chaotic system. A new control law for completely synchronizing the new financial chaotic system with itself has been established using adaptive integral sliding mode control. We also find that the new financial chaotic system has fascinating traits including symmetry, equilibrium points, multistability, Lyapunov exponents and bifurcation diagrams. We illustrate all the main results of this research work using MATLAB phase plots. The Lyapunov characteristic exponents and analysis using bifurcation diagrams have resulted in a new financial chaos system showing chaos phenomena in the intervals of parameters 0 < <italic>a</italic> < 15, and parameters 0 < <italic>b</italic> < 0.25. The results of this study can be used to predict if there is chaos in financial risk. Chaotic systems have many applications in engineering like cryptosystems and secure communication systems.</p> </abstract>