Narrow Results

A microcrystalline material refers to a crystallized substance or rock comprised of tiny crystals that can only be observed under a microscope. The strain wave equation is a fourth-order nonlinear partial differential equation encountered in the examination of non-dissipative strain wave propagation within microstructured solids. In this paper, the transmission of waves in microcrystalline materials is dictated by the non-dissipative case of strain wave equation’s structure, accounting for multiple dimensions within microcrystalline structures. The simplest equation method is employed to extract multi-soliton solutions, while the modified Sardar subequation method is applied to identify additional soliton solutions, including bright, combined dark–bright, combined dark-singular, periodic singular, and singular solitons. Furthermore, the dynamical system bifurcation theory approach is utilized to investigate the phase diagrams of the governing equation. Further elaboration on the physical dynamical representation of the presented solutions is provided through profile illustrations. A comparison with the existing literature is also provided, highlighting the efficacy of our work. The significance of the acquired outcomes lies in their capacity to portray a wide array of intricate and diverse phenomena observed in both mathematical and physical systems.

In this paper, we study the traveling wave solutions of a generalized reaction–diffusion system based on the classical Fisher-type system. Through qualitative analysis and blow-up techniques, we prove the existence of traveling fronts of the system. Moreover, we detect the limit points (i.e. $\omega$-limit points or $\alpha$-limit points) of all traveling wave solutions by analyzing the global topological phase portraits of the equivalent system.

In this work a memristive circuit consisting of a first-order memristive diode bridge is presented. The proposed circuit is the simplest memristive circuit containing the specific circuitry realization of the memristor to be so far presented in the literature. Characterization of the proposed circuit confirms its complex dynamic behavior, which is studied by using well-known numerical tools of nonlinear theory, such as bifurcation diagram, Lyapunov exponents and phase portraits. Various dynamical phenomena concerning chaos theory, such as antimonotonicity, which is observed for the first time in this type of memristive circuits, crisis phenomenon and multiple attractors, have been observed. An electronic circuit to reproduce the proposed memristive circuit was designed, and experiments were conducted to verify the results obtained from the numerical simulations.

The presence of hidden attractors in dynamical systems has received considerable attention recently both in theory and applications. A novel three-dimensional autonomous chaotic system with hidden attractors is introduced in this paper. It is exciting that this chaotic system can exhibit two different families of hidden attractors: hidden attractors with an infinite number of equilibrium points and hidden attractors without equilibrium. Dynamical behaviors of such system are discovered through mathematical analysis, numerical simulations and circuit implementation.

The chaotic systems with hidden attractors, such as chaotic systems with a stable equilibrium, chaotic systems with infinite equilibria or chaotic systems with no equilibrium have been investigated recently. However, the relationships between them still need to be discovered. This work explains how to transform a system with one stable equilibrium into a new system with an infinite number of equilibrium points by using a memristive device. Furthermore, some other new systems with infinite equilibria are also constructed to illustrate the introduced methodology.

The Chua Corsage Memristor is the simplest example of a passive but locally active memristor endowed with two asymptotically stable equilibrium points $Q_0$ and $Q_1$ when powered by an E-volt battery, where $-10\mathrm{\text{V}}<E<10\mathrm{\text{V}}$. The basin of attraction is defined by $x(0)<30-E$, $E<10\mathrm{\text{V}}$ for $Q_0$, and $x(0)>30-E$, $E>-10\mathrm{\text{V}}$ for $Q_1$. By adding an inductor of appropriate value $L>0\mathrm{\text{H}}$ in series with the battery, the resulting circuit undergoes a supercritical Hopf bifurcation and becomes an oscillator for $-10\mathrm{\text{V}}<E<-3.334\mathrm{\text{V}}$. Applying a sinusoidal voltage source $v(t)=Asin(2\pi ft)$ across the Chua corsage memristor, one finds two distinct coexisting stable periodic responses, depicted by their associated pinched hysteresis loops, of the same frequency $f$ whose basin of attraction is defined by $x(0)\le x^{\ast }(0)$, and $x(0)>x^{\ast }(0)$, respectively, where $x^{\ast }(0)$ depends on both amplitude A and frequency f.

An in-depth and comprehensive analysis of the above global nonlinear phenomena is presented using tools from nonlinear circuit theory, such as Chua’s dynamic route method, and from nonlinear dynamics, such as phase portrait analysis and bifurcation theory.

Many nonlinear dynamic systems have a rotating behavior where an angle defining its state may extend to more than 360${}^{\circ }$. In these cases the use of the phase portrait does not properly depict the system’s evolution. Normalized phase portraits or cylindrical phase portraits have been extensively used to overcome the original phase portrait’s disadvantages. In this research a new graphic representation is introduced: the phase shadow. Its use clearly reveals the system behavior while overcoming the drawback of the existing plots. Through the paper the method to obtain the graphic is stated. Additionally, to show the phase shadow’s expressiveness, a rotating pendulum is considered. The work exposes that the new graph is an enhanced representational tool for systems having equilibrium points, limit cycles, chaotic attractors and/or bifurcations.

In this paper, we study the global dynamical behavior of the Hamiltonian system $ẋ=H_y(x,y)$, $ẏ=-H_x(x,y)$ with the rational potential Hamiltonian $H(x,y)=y^2/2+P(x)/Q(y)$, where $P(x)$ and $Q(y)$ are polynomials of degree 1 or 2. First we get the normal forms for these rational Hamiltonian systems by some linear change of variables. Then we classify all the global phase portraits of these systems in the Poincaré disk and provide their bifurcation diagrams.

We characterize the phase portraits in the Poincaré disk of all planar polynomial Hamiltonian systems of degree three with a nilpotent saddle at the origin and ${\mathbb{Z}}_2$-symmetric with $(x,y)\mapsto (-x,y)$.

We provide normal forms and the global phase portraits on the Poincaré disk of all planar Kukles systems of degree $3$ with ${\mathbb{Z}}_2$-equivariant symmetry. Moreover, we also provide the bifurcation diagrams for these global phase portraits.

This paper presents a global study of the class $QsnS{N}_{11}$ of all real quadratic polynomial differential systems which have a finite semi-elemental saddle-node and an infinite saddle-node formed by the coalescence of a finite singularity and an infinite singularity. This class can be divided into two different families, namely, $QsnS{N}_{11}(A)$ phase portraits possessing a finite saddle-node as the only finite singularity and $QsnS{N}_{11}(B)$ phase portraits possessing a finite saddle-node and also a simple finite elemental singularity. Each one of these two families is given by a specific normal form. The study of family $QsnS{N}_{11}(A)$ was reported in [Artés et al., 2020b] where the authors obtained $36$ topologically distinct phase portraits for systems in the closure $\overline{QsnS{N}_{11}(A)}$. In this paper, we provide the complete study of the geometry of family $QsnS{N}_{11}(B)$. This family which modulo the action of the affine group and time homotheties is three-dimensional and we give the bifurcation diagram of its closure with respect to a specific normal form, in the three-dimensional real projective space. The respective bifurcation diagram yields 631 subsets with 226 topologically distinct phase portraits for systems in the closure $\overline{QsnS{N}_{11}(B)}$ within the representatives of $QsnS{N}_{11}(B)$ given by a specific normal form. Some of these phase portraits are proven to have at least three limit cycles.

We provide the normal forms, the bifurcation diagrams and the global phase portraits on the Poincaré disk of all planar Kukles systems of degree $3$ with ${\mathbb{Z}}_2$-symmetries.

In this paper, we classify the phase portraits in the Poincaré disc of all the Kolmogorov systems

We classify the global dynamics of the five-parameter family of planar Kolmogorov systems

In this paper, we study the global dynamics of continuous piecewise quadratic Hamiltonian systems separated by the straight line $x=0$, where these kinds of systems have a nilpotent center at $(0,0)$, which comes from the combination of two cusps of both Hamiltonian systems. By the Poincaré compactification we classify the global phase portraits of these systems. We must mention that it is extremely rare to find works studying the center-focus problem in piecewise smooth systems with nonelementary singular points as we did here.

In this paper, we investigate the bifurcations and exact traveling wave solutions of the Lakshmanan–Porsezian–Daniel model with parabolic law nonlinearity. Employing the bifurcation theory of planar dynamical system, we obtain the phase portraits of the corresponding traveling wave system. The existence of the two singular straight lines leads to the dynamical behavior of solutions with two scales. Corresponding to some special level curves, we give the explicit exact solutions under different parameter conditions, containing peakon solutions, solitary wave solutions, kink and anti-kink wave solutions, periodic wave solutions and periodic peakon solutions. Moreover, we collect all these traveling solutions into a theorem. Finally, we discuss a small perturbation of the LPD equation and persistence of the traveling waves.

In this paper, we study global dynamics of a planar piecewise linear refracting system with a straight line of separation of node-node type, improper node-improper node type and improper node-node type, respectively. We obtain the conditions of the uniqueness and nonexistence of limit cycles of the refracting system and provide classifications of the topological phase portraits in the Poincaré disk.

In this work, we design a novel 3D chaotic circuit model and investigate the dynamics of a system without an equilibrium point inspired by Justin’s model. New features are presented by tuning the controlling circuit parameters, including dramatic hysteresis loops, heart bistable hidden attractors, and symmetrical attractors. We surprisingly find that these behaviors indeed lead to switched systems among various oscillators such as “hysteresis loops”, “Van der Pol”, “heart”, “bell” and “butterfly”. Hence, both the voltage’s amplitude and frequency are modulated in proper parameters. It is interesting to find that in the system, it is easy to control the bistable threshold value and the transition trajectory between the chaotic and the periodic states. These characteristics have great potential to dramatically enhance the accuracy and sensitivity of signal detection. A high quality factor circuit is achieved by adjusting the parameters of the chaotic system, so that the influence of noise on the ratio of signal to noise (SNR) of the system is almost negligible. Systematic experiments are carried out to verify the prediction from numerical simulations. To conclude, this system enables a new method to detect weak signals coupled with strong noise.

In this paper, we show that any switching hypersurface of $n$-dimensional continuous piecewise linear systems is an $(n-1)$-dimensional hyperplane. For two-dimensional continuous piecewise linear systems, we present local phase portraits and indices near the boundary equilibria (i.e. equilibria at the switching line) and singular continuum (i.e. continuum of nonisolated equilibria) between two parallel switching lines. The index of singular continuum is defined. Then we show that boundary-equilibria and singular continuums can appear with many parallel switching lines.

Employing the Riccati–Bernoulli sub-ODE method (RBSM) and improved Weierstrass elliptic function method, we handle the proposed $(2+1)$-dimensional nonlinear fractional electrical transmission line equation (NFETLE) system in this paper. An infinite sequence of solutions and Weierstrass elliptic function solutions may be obtained through solving the NFETLE. These new exact and solitary wave solutions are derived in the forms of trigonometric function, Weierstrass elliptic function and hyperbolic function. The graphs of soliton solutions of the NFETLE describe the dynamics of the solutions in the figures. We also discuss elaborately the effects of fraction and arbitrary parameters on a part of obtained soliton solutions which are presented graphically. Moreover, we also draw meaningful conclusions via a comparison of our partially explored areas with other different fractional derivatives. From our perspectives, by rewriting the equation as Hamiltonian system, we study the phase portrait and bifurcation of the system about NFETLE and we also for the first time discuss sensitivity of the system and chaotic behaviors. To our best knowledge, we discover a variety of new solutions that have not been reported in existing articles $V_{1,2}^{\ast \ast },\dots ,V_{7,8}^{\ast \ast }$, $V_{9,10},\dots ,V_{13,14}$. The most important thing is that there are iterative ideas in finding the solution process, which have not been seen before from relevant articles such as [Tala-Tebue et al., 2014; Fendzi-Donfack et al., 2018; Ashraf et al., 2022; Ndzana et al., 2022; Halidou et al., 2022] in seeking for exact solutions about NFETLE.