Narrow Results

This study seeks to explore the integrable dynamics of induced curves through the utilization of the complex-coupled Kuralay system. The importance of the coupled complex Kuralay equation lies in its role as a fundamental model that contributes to the understanding of intricate physical and mathematical concepts, making it a valuable tool in scientific research and applications. The soliton solutions originating from the Kuralay equations are believed to encapsulate cutting-edge research in various important domains such as optical fibers, nonlinear optics, and ferromagnetic materials. Analytical techniques are employed to derive traveling wave solutions for this model, given that the Cauchy problem cannot be resolved using the inverse scattering transform. In the quest for solitary wave solutions, the extended modified auxiliary equation mapping (EMAEM) method is employed. We derive several novel families of precise traveling wave solutions, encompassing trigonometric, hyperbolic, and exponential forms. Moreover, the planar dynamical system of the concerned equation is created, all probable phase portraits are given, and sensitive inspection is applied to check the sensitivity of the considered equation. Furthermore, after adding a perturbed term, chaotic and quasi-periodic behaviors have been observed for different values of parameters, and multistability is reported at the end. Numerical simulations of the solutions are incorporated alongside the analytical results to enhance comprehension of the dynamic characteristics of the solutions obtained. This study’s outcomes can offer valuable insights for addressing other nonlinear partial differential equations (NLPDEs). The soliton solutions obtained in this study offer important insights into the intricate nonlinear equation being examined.

The perturbed nonlinear Schrödinger equation plays a crucial role in various scientific and technological fields. This equation, an extension of the classical nonlinear Schrödinger equation, incorporates perturbative effects that are essential for modeling real-world phenomena more accurately. In this paper, we investigate the traveling wave solutions of the perturbed nonlinear Schrödinger equation using the bifurcation theory of dynamical systems. Graphical presentations of the phase portrait are provided, revealing the traveling wave solutions under various conditions. By employing the auxiliary equation method, we derive a variety of solutions including periodic, dark, singular and bright optical solitons. To provide comprehensive and clearer depiction of the model’s behavior 2D, contour and 3D graphical representations are offered. We also highlight specific constraint conditions that ensure the presence of these obtained solutions. This study expands the scope of known exact solutions and their stability qualities which is offering an extensive analytical technique which enhances previous research. The novelty of our research lies in its examination of bifurcation analysis and the auxiliary equation method within the context of a perturbed nonlinear Schrödinger wave equation for the first time. By integrating these two perspectives, this paper contributes to establishing the complex dynamics and stability characteristics of soliton solutions under perturbations.

This paper discusses a fractional-order prey–predator system with Gompertz growth of prey population in terms of the Caputo fractional derivative. The non-negativity and boundedness of the solutions of the considered model are successfully analyzed. We utilize the Mittag-Leffler function and the Laplace transform to prove the boundedness of the solutions of this model. We describe the topological categories of the fixed points of the model. It is theoretically demonstrated that under certain parametric conditions, the fractional-order prey–predator model can undergo both Neimark–Sacker and period-doubling bifurcations. The piecewise constant argument approach is invoked to discretize the considered model. We also formulate some necessary conditions under which the stability of the fixed points occurs. We find that there are two fixed points for the considered model which are semi-trivial and coexistence fixed points. These points are stable under some specific constraints. Using the bifurcation theory, we establish the Neimark–Sacker and period-doubling bifurcations under certain constraints. We also control the emergence of chaos using the OGY method. In order to guarantee the accuracy of the theoretical study, some numerical investigations are performed. In particular, we present some phase portraits for the stability and the emergence of the Neimark–Sacker and period-doubling bifurcations. The biological meaning of the given bifurcations is successfully discussed. The used techniques can be successfully employed for other models.

The fractional Klein–Gordon equation (fKGE) holds a crucial position in various fields of theoretical and applied physics, with wide applications covering different areas such as nonlinear optics, condensed matter physics, and quantum mechanics. In this paper, we carry out analytical investigation to the fKGE with beta fractional derivative by using the Bernoulli $(G^{\prime }∕G)$-expansion method and improved $tan(\varphi ∕2)$-expansion method. In order to better comprehend the physical structure of the obtained solutions, three-dimensional visualizations, contour diagrams, and line graphs of the exponent function solutions are depicted with the aid of Matlab. Moreover, the phase portraits and bifurcation behaviors of the fKGE under transformation are studied. Sensitivity and chaotic behaviors are analyzed in specific conditions. The phase plots and time series map are exhibited through sensitivity analysis and perturbation factors. These studies enhance our understanding of practical phenomena governed by the model of fKGE, and are crucial for examining the dynamic behaviors and phase portraits of the fKGE system. The strategies utilized here are more direct and effective, which can be effortlessly utilized to various fractional-order differential equations arising in nonlinear optics and quantum mechanics.

The nature of the fixed points of the Carotid–Kundalini (C–K) map was studied and the boundary equation of the first bifurcation of the C–K map in the parameter plane is presented. Using the quantitative criterion and rule of chaotic system, the paper reveals the general features of the C–K Map transforming from regularity to chaos. The following conclusions are obtained: (i) chaotic patterns of the C–K map may emerge out of double-periodic bifurcation; (ii) the chaotic crisis phenomena are found. At the same time, the authors analyzed the orbit of critical point of the complex C–K Map and put forward the definition of Mandelbrot–Julia set of the complex C–K Map. The authors generalized the Welstead and Cromer's periodic scanning technique and using this technology constructed a series of the Mandelbrot–Julia sets of the complex C–K Map. Based on the experimental mathematics method of combining the theory of analytic function of one complex variable with computer aided drawing, we investigated the symmetry of the Mandelbrot–Julia set and studied the topological inflexibility of distribution of the periodic region in the Mandelbrot set, and found that the Mandelbrot set contains abundant information of the structure of Julia sets by finding the whole portray of Julia sets based on Mandelbrot set qualitatively.

We theoretically studied current oscillations and the dynamics of the modulation-doped GaAs/AlGaAs heterostructure under the action of electric fields and a perpendicular magnetic field. The results show that the current oscillations and hysteresis in the system can be found under the DC bias voltage. Under the effect of the increasing magnetic fields, the oscillations will disappear and the width of the hysteresis is broader. Considering the AC part, the system shows interesting nonlinear behaviors like the route of an inverse period-doubling to chaos, quasiperiodicity, and frequency-locking.

With the occasional feedback method, the chaotic logistic map is stabilized at an unstable low-periodic orbit. In our method, the qualified feedback coefficient can be obtained through calculation instead of through simulation. Besides, the bifurcation control of the logistic map is studied, and a new scheme is proposed to change the parameter value of any one bifurcation point of this dynamic system optionally. Simulation results show the effectiveness of the methods.

This paper reports a novel four-dimensional hyperchaos generated from Qi system, obtained by adding nonlinear controller to Qi chaos system. The novel hyperchaos is studied by bifurcation diagram, Lyapunov exponent spectrum and phase diagram. Numerical simulations show that the new system's behavior can be periodic, chaotic and hyperchaotic as the parameter varies. Based on the time-domain approach, a simple observer for the hyperchaotic is proposed to guarantee the global exponential stability of the resulting error system. The scheme is easy to implement and different from the other observer design since it does not need to transmit all signals of the dynamical system.

The nonlinear dynamics of a vibration-controlled magnetic system are studied via a three-mode discretization of the governing partial differential equations. The analysis focuses specifically on the effects of modal coupling through the nonlinear terms of the system equation. A bifurcation analysis of the system is performed using sophisticated nonlinear theories, including the center manifold theory and the normal form theorem. The results show that when the first mode and the higher modes are excited simultaneously by the control forces, the three-mode approximation method predicts the existence of a triple zero degeneracy accompanied by complicated bifurcation phenomena. Comparing the dynamics structure predicted by the three-mode approximation model with that obtained from a single-mode approach, it is found that if the higher modes are excited by the control forces, the effects of modal coupling should be taken into consideration since a complicated dynamics structure may exist as a result.

A four-variable dynamical system composed of memristor is proposed to investigate the dependence of multi-scroll attractor on initial setting for one variable with memory, and the description for physical background is supplied. It is found that appropriate setting of initial values for the memory variable can induce different numbers of attractor, as a result, resetting initials can change the profile of attractors which is also dependent on the calculating period. Time-delayed feedback is used to stabilize the dynamical system thus the effect of initial dependence is suppressed and multi-scroll attractors are controlled by applying appropriate time delay and feedback gain in the controller. Furthermore, the system is verified on FPGA circuit platform and memristor is used to describe the memory effect of variable related to magnetic flux. It is confirmed that multi-scroll attractors can be stabilized and the dependence of initials setting is suppressed in experiment way.

Circuitry and chemistry are applied in such fields as communication engineering and automatic control, environmental protection and material/medicine sciences, respectively. Biology works as the basis of agriculture and medicine. Studied in this paper is a nonlinear space-fractional Kolmogorov–Petrovskii–Piskunov equation for the electronic circuitry, chemical kinetics, population dynamics, neurophysiology, population genetics, mutant gene propagation, nerve impulses transmission or molecular crossbridge property in living muscles. Kink soliton solutions are obtained via the fractional sub-equation method. Change of the fractional order does not affect the amplitudes of the kink solitons. Via the traveling transformation, the original equation is transformed into the ordinary differential equation, while we obtain two equivalent two-dimensional planar dynamic systems of that ordinary differential equation. According to the bifurcation and qualitative considerations of the planar dynamic systems, we display the corresponding phase portraits when the traveling-wave velocity is nonzero or zero. Nonlinear periodic waves of the original equation are obtained when the traveling-wave velocity is zero.

The survival and occurrence of chaos are much dependent on the intrinsic nonlinearity and parameters region for deterministic nonlinear systems, which are often represented by ordinary differential equations and maps. When nonlinear circuits are mapped into dimensional dynamical systems for further nonlinear analysis, the physical parameters of electric components, e.g. capacitor, inductor, resistance, memristor, can also be replaced by dynamical parameters for possible adjustment. Slight change for some bifurcation parameters can induce distinct mode transition and dynamics change in the chaotic systems only when the parameter is adjustable and controllable. In this paper, a thermistor is included into the chaotic Chua circuit and the temperature effect is considered by investigating the mode transition in oscillation and the dependence of Hamilton energy on parameters setting in thermistor. Furthermore, the temperature of thermistor is adjusted for finding possible synchronization between two chaotic Chua circuits connected by a thermistor. When the coupling channel via thermistor connection is activated, two identical Chua circuits (periodical or chaotic oscillation) can reach complete synchronization. In particular, two periodical Chua circuits can be coupled to present chaotic synchronization by taming parameters in thermistor of coupling channel. However, phase synchronization is reached while complete synchronization becomes difficult when the coupling channel is activated to coupling a periodical Chua circuit and a chaotic Chua circuit. It can give guidance for further control of firing behaviors in neural circuits when the thermistor can capture the heat effectively.

Smart nonlinear circuits can be tamed to reproduce the main dynamical properties in neural activities and thus neural circuits are built to estimate the occurrence of multiple modes in electric activities. In the presence of electromagnetic radiation, the cardiac tissue, brain and neural circuits are influenced because field energy is injected and captured when induction field and current are generated in the media and system. In this paper, an isolated Chua circuit is exposed to external electromagnetic field and energy capturing is estimated for nonlinear analysis from physical viewpoint. Furthermore, two Chua circuits without direct variable coupling are exposed to the same electromagnetic field for energy capturing. Periodical and noise-like radiations are imposed on the Chua circuits which can capture the magnetic field energy via the induction coil. It is found that the two Chua circuits (periodical or chaotic) can reach phase synchronization and phase lock in the presence of periodical radiation. On the other hand, noise-like radiation can realize complete synchronization between two chaotic Chua circuits while phase lock occurs between two Chua circuits in periodical oscillation. It gives some important clues to control the collective behaviors of neural activities under external field.

The nonlinear vibration of axially moving nanobeams at the microscale exhibits remarkable scale effects. A model of an axially moving nanobeam is established based on non-local strain gradient theory and considering two scale effects. The discrete equation of a non-autonomous planar system is obtained using the Galerkin method. The response characteristics of the system are determined using phase diagrams and Poincaré sections, and the effects of the scale parameters on the form of the motion are analyzed. The results show that as the non-local parameter and the material characteristic length parameter vary, the system undergoes multiple forms of motion, including periodic, period-doubling and chaotic motions. Two routes to chaos — period-doubling bifurcation and intermittent chaos — are identified in the variation ranges of the two scale parameters.

In this paper, we have proposed a new chaotic megastable oscillator which has both conservative and dissipative characters depending on the selection of parameters. Various dynamical characteristics including megastability of the new system are investigated and presented. The bifurcation plots and the corresponding Lyapunov exponents (LEs) confirm the existence of both dissipative and conservative oscillations in the system. The proposed megastable oscillator is used as a carrier generator in a differential chaos shift keying (DCSK). Another application of the new chaotic oscillator is shown by using it in developing a random number generator (RNG) and the NIST test results are presented to show the statistical complexity of the new system.

The bouncing ball system with two rigidly connected harmonic limiters is revisited in order to further analyze its periodic movement and bifurcation dynamics. By using the impulsive impact maps, we obtain several sufficient conditions for the existence and local stability of three different types of periodic orbits, respectively, and then plot the bifurcation diagrams in the space of the relative velocity and the restitution coefficient for different parameters of the limiter. The numerical simulation results are consistent with those of the theoretical analysis.

Time-dependent electron current response of GaAs-based miniband superlattice under dual ac electric fields and a magnetic field is studied using balance equation approach. The space charge-induced self-consistent electric field is taken into account in the model. The miniband superlattice operates in the diffusive regime without electric field domain formation. Electron current displays very complicated oscillating behavior with the influence of external fields. The effect of dissipation on nonlinear electron transport is carefully studied based on Poincaré bifurcation diagram and power spectrum. The exhibition of complicate nonlinear oscillation in superlattice is attributed to the nonlinearity induced by self-consistent field and interaction between external radiation and internal cooperative oscillating mode relative to Bloch oscillation and cyclotron oscillation.

In this paper, modulational instability (MI) of information via membrane potential is studied analytically and numerically in an improved Izhikevich neural network under electromagnetic induction. By applying the powerful discrete multiple scale expansion method, a spatiotemporal nonlinear amplitude differential-difference equation governing the information dynamics is derived from the generic model. Linear stability of plane impulse wave solution is then performed on the latter and the impact of electromagnetic induction feedback through the memristor couplings is portrayed on the growth rate diagram. From the diagram, it is found that negative memristor coupling parameter decreases the critical amplitude while positive parameter increases the critical amplitude. To support our analytical predictions, numerical simulations are performed and data selected from the unstable zone of MI lead to the formation of localized solitonic energy patterns, related to the energy coding patterns in the nervous system. Furthermore, the sampled time series for membrane potential under the influence of memristor coupling revealed the breakdown of action potential into multiple impulse-wave trains for high parameter values thus confirming an analytical prediction. Our results provide a potential way to manipulate information coding in the brain.

Neuron exhibits nonlinear dynamics such as excitability transition and post-inhibitory rebound (PIR) spike related to bifurcations, which are associated with information processing, locomotor modulation, or brain disease. PIR spike is evoked by inhibitory stimulation instead of excitatory stimulation, which presents a challenge to the threshold concept. In this paper, 7 codimension-2 or degenerate bifurcations related to 10 codimension-1 bifurcations are acquired in a neuronal model, which presents the bifurcations underlying the excitability transition and PIR spike. Type I excitability corresponds to saddle-node bifurcation on an invariant cycle (SNIC) bifurcation, and type II excitability to saddle-node (SN) bifurcation or sub-critical Hopf (SubH) bifurcation or sup-critical Hopf (SupH) bifurcation. The excitability transition from type I to II corresponds to the codimension-2 bifurcation, Saddle-Node Homoclinic orbit (SNHO) bifurcation, via which SNIC bifurcation terminates and meanwhile big homoclinic orbit (BHom) bifurcation and SN bifurcation emerge. A degenerate bifurcation via which BHom bifurcation terminates and limit point of cycle (LPC) bifurcation emerges is responsible for spiking transition from type I to II, and the roles of other codimension-2 bifurcations (Cusp, Bogdanov-Takens, and Bautin) are discussed. In addition, different from the widely accepted viewpoint that PIR spike is mainly evoked near Hopf bifurcation rather than SNIC bifurcation, PIR spike is identified to be induced near SNIC or BHom or LPC bifurcations, and threshold curves resemble that of Hopf bifurcation. The complex bifurcations present comprehensive and deep understandings of excitability transition and PIR spike, which are helpful for the modulation to neural firing activities and physiological functions.

In this paper, modified equal width Burgers’ equation has been investigated with the aid of unified method and bifurcation. This model has many applications in long wave transmission with dispersion and dissipation in nonlinear medium. The applied technique is efficient to retrieve exact solutions and their dynamic behaviors. The obtained solutions are polynomial and rational function solutions. The behavior of dynamical planer system has been analyzed by assigning different values to the parameters, also each possible case has been shown as phase portraits in this research paper. The estimated solutions demonstrate that the proposed approaches are simple, practical, and promising for investigating further equal width equation’s soliton wave solutions and phase portraits.