Narrow Results

In this paper, the periodic solutions and stability in the dry friction dynamic system are researched by two-dimensional Poincaremap when the fault has happened in the system because of the spring failure. As a result of numerical computation and theoretical analysis, the bifurcation and chaos will arise when the parameters of system are changed. It is very important to analyze the dynamic behavior to conduct fault diagnosis.

The nature of the fixed points of the coupled Logistic map is researched, and the boundary equation of the first bifurcation of the coupled Logistic map in the parameter space is given out. Using the quantitative criterion and rule of system chaos, i.e., phase graph, bifurcation graph, power spectra, the computation of the fractal dimension, and the Lyapunov exponent, the paper reveals the general characteristics of the coupled Logistic map transforming from regularity to chaos, the following conclusions are shown: (1) chaotic patterns of the coupled Logistic map may emerge out of double-periodic bifurcation and Hopf bifurcation, respectively; (2) during the process of double-period bifurcation, the system exhibits self-similarity and scale transform invariability in both the parameter space and the phase space. From the research of the attraction basin and Mandelbrot–Julia set of the coupled Logistic map, the following conclusions are indicated: (1) the boundary between periodic and quasiperiodic regions is fractal, and that indicates the impossibility to predict the moving result of the points in the phase plane; (2) the structures of the Mandelbrot–Julia sets are determined by the control parameters, and their boundaries have the fractal characteristic.

The numerical results of a set of nonlinear equations for a modulation-doped GaAs/AlGaAs heterostructure under external fields are presented. For pure dc bias at a fixed magnetic field, we observe the self-sustained oscillations due to the traveling high electric field domain in the system. Varying the magnetic field, the width of the domain and the oscillation amplitude will be changed. By imposing the microwave irradiation, the system shows frequency locking, quasiperiodicity, and chaos depending on the amplitude and the frequency of the microwave. It is shown that these interesting oscillations are connected with the high electric field domains.

In this paper, a new hyperchaotic system is formulated by introducing an additional state into the third-order unified system. Some of its basic dynamical properties, such as Lyapunov exponent, bifurcation diagram and the Poincáre section are investigated. It was found that the system is hyperchaotic in several different regions of the parameters. The analysis of equilibrium points and stability are also given. Two different methods, i.e., nonlinear hyperbolic function feedback control and tracking control methods, are used to control hyperchaos in the new hyperchaotic system. Based on the Routh–Hurwitz criteria, the conditions suppressing hyperchaos to unstable equilibrium point are discussed. A tracking control method is proposed. It is also proved that the strategy can make the system approach any desired smooth orbit at an exponential rate. Numerical results have shown the effectiveness of the control methods.

This paper presents a four-dimensional hyperchaos Qi system, obtained by adding linear term and nonlinear term of nonlinear controller to Qi chaos system. The hyperchaos Qi system 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.

In this paper, we have proposed a novel three-dimensional Lorenz-like chaotic system. Some basic properties of the system, such as dynamical behaviors, bifurcation diagram. Lyapunov exponents and Poincare mapping are investigated either analytically or numerically. Furthermore, the control problem of the new chaotic system was studied via nonlinear backstepping method. The single backstepping control input was designed according to Lyapunov stability criterion. Numerical simulations are carried out in order to demonstrate the effectiveness of the proposed control design.

Pattern estimation and selection in media can give important clues to understand the collective response to external stimulus by detecting the observable variables. Both reaction–diffusion systems (RDs) and neuronal networks can be treated as multi-agent systems from molecular level, intrinsic cooperation, competition. An external stimulus or attack can cause collapse of spatial order and distribution, while appropriate noise can enhance the consensus in the spatiotemporal systems. Pattern formation and synchronization stability can bridge isolated oscillators and the network by coupling these nodes with appropriate connection types. As a result, the dynamical behaviors can be detected and discussed by developing different spatial patterns and realizing network synchronization. Indeed, the collective response of network and multi-agent system depends on the local kinetics of nodes and cells. It is better to know the standard bifurcation analysis and stability control schemes before dealing with network problems. In this review, dynamics discussion and synchronization control on low-dimensional systems, pattern formation and synchronization stability on network, wave stability in RDs and neuronal network are summarized. Finally, possible guidance is presented when some physical effects such as polarization field and electromagnetic induction are considered.

Synapse coupling is critical for information encoding of neurons. The effect of electromagnetic induction in cell becomes distinct when the exchange of charged ions across membrane is frequently triggered by external electric field or synapse current from adjacent neurons. In this paper, Gaussian white noise is imposed on a two-layer network composed of neurons with electromagnetic induction. The gap junction coupling is applied to connect the adjacent neurons and external stimulus with diversity is applied to keep different excitabilities of neurons in each layer. Neurons on the second layer are activated and modulated by using field coupling rather than channel coupling. It is found that the pattern formation on the network is much dependent on the initial setting due to the memory effect based on induction current via memristive synapse. Furthermore, field coupling intensity (D₀) and noise intensity (D₁) are changed to detect the development of spiral waves, target waves on the network in presence of noise. It confirms that noise can be helpful for pattern selection and synchronization approach on the two-layer network under field coupling, while field coupling can suppress the self-organization for pattern formation.

We investigate the evolution of the difference in the conventional logistic map and observed complicated but interesting dynamics. Our results show that the bifurcation, the attractor and the asymptotic measure depend on initial values. The reverse bifurcation, the multiple bifurcation and other bifurcation properties obtained here are different from those of the original logistic map. Furthermore, we find that attractors form a structure which looks like a jigsaw puzzle, and the corresponding distributions related to the Feigenbaum constant are also analyzed.

The effect of electromagnetic induction on neural firing activities has been widely investigated and has also been related to neural information coding. In this study, electromagnetic induction and Gaussian white noise (GWN) are introduced to the FitzHugh–Nagumo neuron, and the effect of induction current on GWN-induced coherence resonance (CR) is investigated. For an appropriate value of bias voltage source, the induction current is excitatory (respectively, inhibitory) when the gain of induction current is positive (respectively, negative). When the induction current is excitatory, the degree of CR is enhanced. When the induction current is inhibitory, however, the degree of CR is weakened. Furthermore, by varying the value of bias voltage source, we show that the degree of CR increases with the increase in the gain of induction current. Bifurcation analysis shows that the supercritical Hopf bifurcation point gets close to the resting state with the increase in the gain of induction current, which causes the increase in the degree of CR. The results of this study show that excitatory electromagnetic induction current can enhance noise-induced CR. This finding provides a possible way to modulate CR of the nervous system.

Chaotic systems can be used for secure communication and image encryption by applying a variety of encryption algorithms. While most of the low-dimensional chaotic systems and maps can be estimated by using phase reconstruction and thus the safety in signal processing and propagation is attacked. In this paper, an initial-dependent dynamical system, which is developed from the Rössler system by adding memristive function and disturbance function on the memristive variable $z$, is presented for realizing image encryption and bifurcation analysis is supplied in detail. Time-varying disturbance from sampled variables is applied to control the memristive variable and the dependence of mode oscillation on initial values is enhanced. As a result, the dynamics of this memristive system is switched between different oscillation modes (e.g., periodical to chaotic, chaotic to chaotic) by activating the initial value, memristive gain and disturbance gain, respectively. From a dynamical viewpoint, the involvement of stochastic adjustment on the memristive variable can reset the initial value and then induce time-varying parameter regulation or switch on certain parameter embedded in the memristive nonlinearity and function, and thus the dynamics dependence on the initial setting is enhanced. Standard bifurcation analysis is carried out on this memristive system and then the sampled time series are used for image encryption, furthermore, the reliability for this scheme is discussed and suggestions for further study are supplied in the end.

Dynamical behavior of the optical traveling pulses for the resonant nonlinear Schrödinger (RNS) equation with external periodic force is studied. Using a complex transformation we obtain an unperturbed dynamical system for the RNS equation. Existence of periodic optical pulses, solitary optical pulses of dark and bright types, breaking optical pulses is dispensed using phase plane analysis of the unperturbed dynamical system. Introducing an external perturbation to the unperturbed dynamical system, quasiperiodicity and chaotic features of the nonlinear optical pulses for the perturbed dynamical system are studied by varying the resonance parameter (c) with special values of other system parameters through different computational tools, like time series plot, phase plot, sensitivity plot, Lyapunov exponent, and Poincare section. The resonance parameter (c) acts as a control parameter on qualitative transition of the nonlinear optical pulses for the perturbed dynamical system from quasiperiodic motion to chaotic motion.

Based on the study of the synchronization of two electric synapse-coupled Sherman neuron systems, this paper analyzes the rich discharge behavior of Sherman neurons through the peak-to-peak interval bifurcation diagram, which determines the parameter values for the study of the electrical synapse coupling Sherman neuron system synchronization. By using the synchronization difference and the correlation coefficient value, this paper analyzes the synchronous transition process of the two electrical synapse-coupled Sherman neuron systems with the change of coupling intensity and studies the bifurcation behavior of neurons in the two electrical synapse-coupled Sherman neuron systems. The experimental results show the transition process of two electrical synapse-coupled Sherman neurons from nonsynchronized, peak-independent cluster synchronization to fully synchronized. In addition, we study the synchronization process of the ring-connected electrical synapse-coupled Sherman neuron system. The experimental results show that the two electrical synapse-coupled Sherman neuron systems show a similar synchronous transition process.

This paper mainly studied firing patterns and related bifurcations in the Purkinje cell dendrite model. Based on the methods of equivalent potentials and time scale analysis, the initial six-dimensional (6D) dendrite model is reduced to a 3D form to facilitate the calculation. We numerically show that the dendrite model could exhibit period-adding bifurcation and four bursting patterns for several vital parameters. Then the bifurcation mechanisms and transition of these four bursting patterns are discussed by phase plane analysis, and two-parameter bifurcation analysis of the fast subsystem, respectively. Moreover, we computed the first Lyapunov coefficient to determine the stability of Hopf bifurcation. Ultimately, we analyzed the codimension-two bifurcation of the whole system and gave a detailed theoretical derivation of the Bogdanov–Takens bifurcation.

Since time delays are very vital and inevitable in the transmission of neurons, this paper is mainly concerned with the dynamic analysis of the magnetic flux e-HR neuron model with time delays. Specifically, the stability of the system at the equilibrium point and the conditions for bifurcations are analyzed to explore the impact of time delays on dynamics behavior. Results show that the system exhibits Hopf bifurcation within the specific range of time delay. Moreover, the increase of time delay can induce the tip discharge behavior. In addition, we investigate the stability and bifurcation behavior of the model with dual time delays. Our results indicate that the system displays rich dynamics performance under certain conditions. Furthermore, numerical simulations show time delays play a crucial role in the dynamics process for the magnetic flux e-HR neuron model.

The temporal-second-order KdV equation, which describes the propagation of two wave modes with different phase velocities and same dispersion relation, nonlinearity and dispersion parameters are investigated. The similarity reductions and new exact solutions are obtained via the Kudryashov method and a new version of Kudryashov method. Furthermore, the conservation laws are derived using the new conservation theorem. The bilinear forms and bilinear Bäcklund transformation of the temporal-second-order KdV equation are derived through the binary Bell polynomial. Moreover, the N-soliton solutions of the equation are constructed with the help of the Hirota method. The characteristics and interaction of the solitons are discussed graphically. We discuss the effect of the phase velocities $c_1$ and $c_2$ and the parameters of nonlinearity $a_1$ and $a_2$ on the soliton amplitudes and velocities. Bifurcation method of dynamical systems is employed to investigate bifurcation of solitary waves in the temporal-second-order KdV equation.

Functional neurons built from neural circuits are capable of perceiving and processing external signals such as light illumination and magnetic radiation, by converting the physical signals into modulated bioelectric signals called action potential with diverse forms and shapes. Through modulational instability (MI), modulated wave formation and pattern transition are studied in a chain memristive network of $100$ photosensitive neurons. Memristors and photocells are incorporated in a simple FitzHugh–Nagumo neuron to detect and process the external magnetic flux and light illumination. To determine regions of modulated wave formation, linear stability analysis is performed on a nonlinear envelope equation which resulted from the asymptotic expansion of the generic dynamical equations. The growth rate of MI is plotted and the distinct zones of stable/unstable MI are presented. We confirm the analytical result through numerical simulations whereby the initial plane wave solutions lead to the emergence of localized structures with traits of spiking, bursting and chaotic states. High-frequency photocurrent changes orderly localized patterns to chaotic-like patterns while high-frequency magnetic flux promotes pattern transition from bursting to 2-period spiking state and a 4-period spiking state. This could provide an adequate way to influence the behaviors of artificial neurons as well as potential mechanism of information coding in the nervous system.