Narrow Results

This study investigates the chaotic behavior and stability of a nonlinear fractional-order financial system utilizing the Caputo fractional derivative. Initially, we formulate the nonlinear fractional-order financial model and establish the problem framework. Next, we prove the existence and uniqueness of solutions by applying the Banach and Schauder fixed-point theorems to the proposed system. Additionally, we analyze the Ulam–Hyers stability and discuss other significant findings related to the system’s stability. To simulate the proposed model, we develop numerical schemes based on fractional calculus, employing Lagrange polynomial interpolation. Finally, we present the numerical simulations to validate the theoretical results, highlighting the significant impact of fractional-order derivatives on the system’s behavior.

The aim of this study is to give a deep investigation into the dynamics of the simplified modified Camassa–Holm equation (CHe) for shallow water waves. Taking advantage of the semi-inverse method, we develop the variational principle, based on which the Hamiltonian of the system is extracted. By means of the Galilean transformation, the governing equation is transformed into a planar dynamical system. Then, the bifurcation analysis is presented via employing the theory of the planar dynamical system. Correspondingly, the quasi-periodic and chaotic behaviors of the system are also discussed by introducing two different kinds of perturbed terms. Finally, the variational method is based on the variational principle and Ritz method, and the Kudryashov method is used to construct the diverse solitary wave solutions, which include the bright solitary, dark solitary, kink solitary and the bright–dark solitary wave solutions. The graphic depictions of the obtained diverse solitary wave solutions are presented to elucidate the physical properties. The findings of this research enable us to gain a deeper understanding of the nonlinear dynamic characteristics of the considered equation.

Analysis of a dynamic system helps scientists understand its properties and utilize it properly in different applications. This study analyzes the effects of various external excitements on a recently proposed mathematical neuron model derived from the original Fitzhugh–Nagumo model. Different bifurcation analyses on this system are conducted to detect chaotic behaviors that are common and of great importance in biological systems, considering the effects of different types of external excitements. Lyapunov exponents (LEs) confirm the existence of chaotic patterns. Furthermore, a bifurcation diagram that looks into the changes in the system dynamics caused by the simultaneous application of the external stimulants is represented. Neurons are bound to play a role in a network in which synchrony is an analytical quality. Therefore, the potential of a network of this model in showing synchronization is examined using the master stability function (MSF) technique. Ultimately, it is concluded that this neural model can produce chaotic behaviors and synchronous networks.

Systems of complex partial differential equations, which include the famous nonlinear Schrödinger, complex Ginzburg–Landau and Nagumo equations, as examples, are important from a practical point of view. These equations appear in many important fields of physics. The goal of this paper is to concentrate on this class of complex partial differential equations and study the fixed points and their stability analytically, the chaotic behavior and chaos control of their unstable periodic solutions. The presence of chaotic behavior in this class is verified by the existence of positive maximal Lyapunov exponent.The problem of chaos control is treated by applying the method of Pyragas. Some conditions on the parameters of the systems are obtained analytically under which the fixed points are stable (or unstable).

A dynamical system falling in between the elementary binary cellular automata and the coupled map lattices is presented. It is composed by a one-dimensional cellular automaton in which the values within each cell are continuous instead of discrete. However, in this case, the coupling between cells is made through a logistic map instead of making the fuzzification of the disjunctive normal form describing the corresponding Boolean rule. The system resembles the CA rule 90 evolution, since the future value of a cell depends only on a combination of the values of the nearest neighbors cells. The basic dynamical and stability properties of the system are analyzed. The system displays different types from attractors (fixed points, cycles and chaotic attractors), depending on the growth rate parameter used for the logistic map coupling. If the cell values are binary, i.e., only values 0 and 1 are allowed within each cell, the dynamical evolution of the rule 90 automaton is recovered.

In this paper, we present a pseudo-random bit generator (PRBG) based on two lag time series of the logistic map using positive and negative values in the bifurcation parameter. In order to hidden the map used to build the pseudo-random series we have used a delay in the generation of time series. These new series when they are mapped x_n against x_n+1 present a cloud of points unrelated to the logistic map. Finally, the pseudo-random sequences have been tested with the suite of NIST giving satisfactory results for use in stream ciphers.

In this work, we developed a hybrid scheme for Gilson–Pickering equation by extracting advantageous features of collocation method and B-splines. Collocation method involves satisfying a differential equation to some tolerance at finite number of points and has low computational cost. The idea is to extract the superior accuracy and smoothness of B-spline functions with the low computational cost of collocation scheme. In the vision of a superficial periodic force, we obtain the quasiperiodic and chaotic behaviors of the forced Gilson–Pickering equation by perusing phase portrait technique, time series and Poincaré section. The sequel of frequency ($\omega$) of the superficial periodic force is very prime in case of transition from weakly chaotic feature to quasiperiodic feature of the forced Gilson–Pickering equation.

Dynamical behavior of nonlinear wave solutions of the perturbed and unperturbed generalized Newell–Whitehead–Segel (GNWS) equation is studied via analytical and computational approaches for the first time in the literature. Bifurcation of phase portraits of the unperturbed GNWS equation is dispensed using phase plane analysis through symbolic computation and it shows stable oscillation of the traveling waves. Chaotic behavior of the perturbed GNWS equation is obtained by applying different computational tools, like phase plot, time series plot, Poincare section, bifurcation diagram and Lyapunov exponent. A period-doubling bifurcation behavior to chaotic behavior is shown for the perturbed GNWS equation and again it shows chaotic to periodic motion through inverse period-doubling bifurcation. The perturbed GNWS equation also shows chaotic motion through a sequence of periodic motions (period-1, period-3 and period-5) depending on the variation of the parameter of linear coefficient. Thus, the parameter of linear coefficient plays the role of a controlling parameter in the chaotic dynamics of the perturbed GNWS equation.

In this research, the authors precisely focus on the analysis of the chaotic behavior in an Oldroyd-B fluid saturated anisotropic porous medium via a feedback control technique. A four-dimensional (4D) weakly nonlinear system emerging from a Galerkin method of the constitutive and preservation equations is developed to accord with a convective stabilization with various Darcy numbers (Da) and feedback control gain value $({\widehat{F}}_g)$. The chaotic dynamic convection is governed by the Darcy–Rayleigh number ($R_n$) and feedback control, while the Da has a significant impact on system stabilization. Their results reveal the effects of the feedback gain parameter $({\widehat{F}}_g)$, stress relaxation parameter ($\mathrm{\Gamma }$), strain retardation parameter ($\mathrm{\Lambda }$), Darcy number (Da), mechanical ($\zeta$) and thermal ($\chi$) anisotropy parameter on the stability and destabilization of thermal convection. Stabilization of thermal convection are important in cooling, nuclear power, and a range of technical, biological and engineering processes. In particular, feedback control gain values are discovered to be the preferred mode for the controlled onset of oscillatory convection. Finally, a graphical representation is presented to demonstrate that the feedback control approach is more effective in regulating the entire system when aperiodic external disturbances occur.

In this paper, we consider an environmental interface as a complex system, in which difference equations for calculating the environmental interface temperature and deeper soil layer temperature are represented by the coupled maps. First equation has its background in the energy balance equation while the second one in the prognostic equation for deeper soil layer temperature, commonly used in land surface parametrization schemes. Nonlinear dynamical consideration of this coupled system includes: (i) examination of period one fixed point and (ii) bifurcation analysis. Focusing part of analysis is calculation of the Lyapunov exponent for a specific range of values of system parameters and discussion about domain of stability for this coupled system. Finally, we calculate Kolmogorov complexity of time series generated from the coupled system.

The famous chaotic circuit known as Chua's oscillator, despite the robustness of the chaotic behavior to parametric mismatches, requires the construction of noncommercial valued inductor. A low cost inductorless version of Chua's oscillator is presented. The chaotic behavior of the circuit is verified by PSpice simulation and also by experimental study on a circuit breadboard. The results lead to excellent agreement with each other and with the results of previous investigators. Experimental results on the possibility of controlling chaos in the modified Chua's oscillator by the inherent feedback mechanism are also reported.

In this paper, we report a new third-order chaotic jerk system with double-hump (bimodal) nonlinearity. The bimodal nonlinearity is of basic interest in biology, physics, etc. The proposed jerk system is able to exhibit chaotic response with proper choice of parameters. Importantly, the chaotic response is also obtained from the system by tuning the nonlinearity preserving its bimodal form. We analytically obtain the symmetry, dissipativity and stability of the system and find the Hopf bifurcation condition for the emergence of oscillation. Numerical investigations are carried out and different dynamics emerging from the system are identified through the calculation of eigenvalue spectrum, two-parameter and single parameter bifurcation diagrams, Lyapunov exponent spectrum and Kaplan–Yorke dimension. We identify that the form of the nonlinearity may bring the system to chaotic regime. Effect of variation of parameters that controls the form of the nonlinearity is studied. Finally, we design the proposed system in an electronic hardware level experiment and study its behavior in the presence of noise, fluctuations, parameter mismatch, etc. The experimental results are in good analogy with that of the analytical and numerical ones.

This Letter displays, via the numerical simulation of a real digital filter, that a finite-state machine may behave in a near-chaotic way even when its corresponding infinite-state machine does not exhibit chaotic behavior.

Using the theory of Hamiltonian systems and dynamical systems to a class of coupled field equations, the existence of uncountably infinite many solitary wave solutions, arbitrarily many distinct periodic solutions and chaotic behavior is obtained. Some sufficient conditions to guarantee the existence of the above solutions are given.

Chaotic behavior for a periodically forced system is considered. Under certain parameter conditions, the fact that some combination-type of chaotic motions and all possible types of subharmonic solutions occur simultaneously can be proved. The dynamics of stochastic and resonant layer have been discussed.

In this paper, we propose a new autonomous electronic oscillator designed with some modifications of the well-known Wien bridge oscillator. In the mathematical model planned for such a circuit, the nonlinearity in the operational amplifier saturation is considered and reference is made to the only equilibrium point at the origin of phase-space. We show how the relation between the bifurcation parameters starts stable oscillations, providing an example for chaotic behavior and bifurcations diagrams. Finally, we conclude with a brief summary of the oscillators operation using a parameters plane.

The phenomenon of breathing (intermittent operation) is studied in a class of piecewise continuous systems as well as its relation with system parameters.

The class of systems under study comprises a continuous time subsystem and a switching rule that induces an oscillatory path by switching alternately between stable and unstable conditions. An interesting feature of the system is that eigenvalues of linear subsystems play an important role in system evolution.

It is shown that although regular and chaotic phases evolve irregularly for a given system, their average behavior is surprisingly regular with respect to a bifurcation parameter. It is found that the phenomenon of breathing share some structural characteristics with intermittency; i.e. existence of a critical exponent. However, for switched systems, many critical exponents may be required. Bifurcation maps and other analysis tools allow us to gain insight into the origin of breathing. This work constitutes a first step toward the characterization of intermittent operation in piecewise continuous systems.

Certain systems present chaotic dynamics when subjected to a regular periodic input. In a study of a nonlinear model of an electromechanical transducer, its dynamic stability is analyzed and it is observed to present chaotic dynamics when a squared signal is introduced as input to the excitor circuit voltage. It is demonstrated that the chaotic movement is due to the periodic modification in the attraction basin of the state space, caused by the input varying in time. Varying the input causes the system to cross saddle type bifurcation values in which points of equilibrium appear and disappear, periodically modifying the qualitative aspects of the system's phase space. This paper describes the deterministic chaos generation by the regular and periodic modification of the properties of the phase space.

In this paper, a mathematical analysis of a possible way to chaos (in the sense of Li and Yorke) for bounded piecewise smooth systems of dimension three submitted to nonsmooth transitions is proposed. This study is based on period doubling method applied to the relied Poincaré map defined on a Poincaré section chosen in the neighborhood of the bifurcation point and transverse to the switching manifold. This choice permits us to reduce the corresponding discrete system to dimension one and allows us to apply "period three implies chaos".

It is well known that second order lowpass interpolative sigma delta modulators (SDMs) may suffer from instability and limit cycle problems when the magnitudes of the input signals are at large and at intermediate levels, respectively. In order to solve these problems, we propose to replace the second order lowpass interpolative SDMs to a specific class of second order bandpass interpolative SDMs with the natural frequencies of the loop filters very close to zero. The global stability property of this class of second order bandpass interpolative SDMs is characterized and some interesting phenomena are discussed. Besides, conditions for the occurrence of limit cycle and fractal behaviors are also derived, so that these unwanted behaviors will not happen or can be avoided. Moreover, it is found that these bandpass SDMs may exhibit irregular and conical-like chaotic patterns on the phase plane. By utilizing these chaotic behaviors, these bandpass SDMs can achieve higher signal-to-noise ratio (SNR) and tonal suppression than those of the original lowpass SDMs.