Equilibria are a class of attractors that host inherent stability in a dynamic system. Infinite number of equilibria and chaos sometimes coexist in a system with some connections. Hidden chaotic attractors exist independent of any equilibria rather than being excited by them. However, the equilibria can modify, distort, eliminate, or even instead coexist with the chaotic attractor depending on the

The phenomena of bifurcation and chaos are studied for a class of second order nonlinear nonautonomous ordinary differential equations, which may be formulated by the nonlinear radially symmetric motion of the dynamically loaded hyperelastic spherical membrane composed of the Rivlin–Saunders material model with a noninteger power-law exponent. Firstly, based on the variational principle, the governing

Coexisting attractors may arise from many different sources such as hidden basins of attraction or peculiarly organized bifurcation structures. By exploiting the regions of mismatched bifurcations between the system and its fixed points, this study investigates coexisting attractors in a six-dimensional extension of the Lorenz system. This six-dimensional extension takes into account additional physical

In the last few decades, the discrete chaotification of difference equations has gained the considerable attention of academicians and scholars due to its tremendous applications in many branches of science, such as cryptography, traffic control models, secure communications, weather forecasting and engineering. In this article, a modulated logistic system is studied and superior chaos is reported

A new 4D memristive chaotic system with an infinite number of equilibria is proposed via exhaustive computer search. Interestingly, such a new memristive system has a plane of equilibria and two other lines of equilibria. Lyapunov exponent and bifurcation analysis show that this system has chaotic solutions with coexisting attractors. The basins of attraction of the coexisting attractors show chaos

For two families of planar piecewise smooth polynomial differential systems, whose unperturbed system has a degenerate center at the origin, we study the biggest lower bound for the maximum number of limit cycles bifurcating from the periodic orbits of the center. These results are extensions of the known ones on unperturbed nondegenerate Σ-center, derived from a nonsmooth harmonic oscillator model

We prove the L1-norm and bounded variation norm convergence of a piecewise linear least squares method for the computation of an invariant density of the Foias operator associated with a random map with position dependent probabilities. Then we estimate the convergence rate of this least squares method in the L1-norm and the bounded variation norm, respectively. The numerical results, which demonstrate

In this paper, a chaotic communication system based on the combination of Improved–Differential chaos-shift keying (IDCSK) scheme and 2×2 MIMO technique (2T2R), namely 2T2R-IDCSK, is investigated. We point out that there are differences in correlation characteristics between the conventional and time-reversed chaotic sequences. From this aspect, bit-error-rate (BER) performance of the system over an

The relationship between currencies in foreign exchange markets has been a topic of significance in economics. Previous studies have focused more on correlations between currencies. However, the detection of causality can reveal their inherent laws. Although the traditional Granger causality test can identify causality, it cannot take into account the nature and intensity of the causality. Thus, the

We prove that all Hopf bifurcations in the Nicholson’s blowfly equation are supercritical as we increase the delay. Earlier results treated only the first bifurcation point, and to determine the criticality of the bifurcation, one needed to substitute the parameters into a lengthy formula of the first Lyapunov coefficient. With our result, there is no need for such calculations at any bifurcation point

An effort to study one-dimensional nonuniform elementary number conserving cellular automata (NCCA) rules from an exponential order rule space of cellular automata is an excellent computational task. To perform this task effectively, a mathematical heritage under the number of conserving functions over binary strings of length n has been highlighted along with their number conserving cellular automata

Hidden symmetries have been previously explored in the context of coupled cell networks and coupled cell systems. These include interior symmetry, quotient symmetry and quotient interior symmetry. We introduce here an equivariant degree theory that incorporates these different forms of hidden symmetry based on lattice structures of synchrony subspaces. The result is a unified theory capable of treating

In order to avoid high extinction risks of prey and keep the stability of the three-species food chain model, we introduce a Filippov food chain model (FFCM) with Holling type II under threshold policy control. The threshold policy is designed to play a pivotal strategy for controlling the three species in the FFCM. With this strategy, no control is applied if the density of the prey population is

In this paper, the Jacobi stability of a two-degree-of-freedom mechanical system is studied by the innovative application of KCC-theory, namely differential geometric methods. We discuss the Jacobi stability of two equilibria and a periodic orbit by constructing geometric invariants. Both the regions of Jacobi stability and Lyapunov stability are presented to show the difference. We draw the phase

For a family of planar piecewise linear differential systems with two zones both having virtual foci, we investigate the appearance of limit cycles bifurcated from a global center (i.e. Poincaré bifurcations of limit cycles) when the discontinuity boundary is perturbed by the appearance of a nonregular point. Precisely, when the discontinuity boundary, which is a straight line, becomes two rays starting

It was shown previously that the synergy of the multi-degree-of-freedom (multi-DOF) and the mechanical stopper techniques is an effective way to further broaden the operation bandwidth of vibration energy harvesters. Considering the stochastic factor at the same time, this paper is devoted to developing a theoretical method for analyzing the dynamical characteristics of an impact-engaged 2DOF energy

Interaction among different units in a network of oscillators may often lead to quenching of oscillations and the importance of oscillation quenching can be found in controlling the dynamics of many real world systems. But there are also many real life phenomena where suppression of oscillation should be avoided for maintaining the sustained evolution of the system. In this work, we propose a self-feedback

This paper completes the description of the generalized Lorenz system (GLS) and hyperbolic generalized Lorenz system (HGLS) along with their canonical forms (GLCF, HGLCF), mostly presented earlier, by deriving explicit state transformation formulas to prove the equivalence between GLS and GLCF, as well as between HGLS and HGLCF. Consequently, complete formulations of the generalized Lorenz canonical

We focus on the two-dimensional stable manifold of Chen system. Based on the Chen system, a new system with nonlinear control is put forward, which exhibits brand new dynamical phenomena such as the coexistence of heteroclinic orbits with periodic solutions, or two new attractors. The emergence of manifold surface is found by tangential self-intersection. The two-dimensional unstable manifold of saddle-focus

In modern times, satellite-based global positioning and navigation systems, such as the GPS, include precise time-keeping devices, e.g. atomic clocks, which are crucial for navigation and for a wide range of economic and industrial applications. However, precise timing might not be available when the environment renders satellite equipment inoperable. In response to this critical need, we have been

Various spontaneous oscillations and Hopf bifurcation have been observed in hair bundles of auditory hair cells, which play very important roles in the auditory function. In the present paper, the bifurcations and chaos of spontaneous oscillations of hair bundles are investigated in a theoretical model to explain the experimental observations. Firstly, the equivalent negative stiffness and symmetrical

In this paper, a new memristor model and a new memcapacitor model are proposed. Based on the two models, a simple chaotic circuit is constructed. Due to the special characteristics of the memristor and memcapacitor, the proposed circuit has two-dimensional normally hyperbolic manifolds of equilibria, and nonparametric bifurcation can occur when the conditions supporting the normal hyperbolicity of

We revisit the model of the laser with feedback and the minimal nonlinearity leading to chaos. Although the model has its origin in laser physics, with peculiarities related to the CO2 laser, it belongs to the class of the three-dimensional paradigmatic nonlinear oscillator models giving chaos. The proposed model contains three key nonlinearities, two of which are of the type xy, where x and y are

Gray–Scott model is one of the most well-known reaction–diffusion models which has a wealth of spatiotemporal chaos behavior. It is commonly used to study spatiotemporal chaos. In the paper, a novel method is proposed for the identification of the Gray–Scott model via deterministic learning and interpolation. The method mainly consists of two phases: the local identification phase and the global identification

Chaotic systems have high potential for engineering applications due to their extremely complex dynamics. In the paper, a five-dimensional (5D) Kolmogorov-like hyperchaotic system is proposed. First, the hyperchaotic property is uncovered, and numerical analysis shows that the system displays the coexistence of different kinds of attractors. This system presents a generalized form of fluid and forced-dissipative

In this paper, a new bifurcation phenomenon of nilpotent singular point is analyzed. A nilpotent focus or center of the planar systems with 3-multiplicity can be broken into two complex singular points and a second order elementary weak focus. Then, two more limit cycles enclosing the second order elementary weak focus can bifurcate through the multiple Hopf bifurcation.

In this paper, we complete the remaining investigation of local bifurcations in a predator–prey model of Leslie-type with simplified Holling type IV functional response. The system has at most three equilibria, and local bifurcations were completely investigated in the cases of one and three equilibria, but in the case of two equilibria the previous study was only on a fixed parameter. We extend the

This paper presents the formulation of Lagrangian function for Lorenz, Modified Lorenz and Chen systems using Lagrangian functions depending on fractional derivatives of differentiable functions, and the estimation of the conserved quantity associated with the respective systems.

We analyze the collective dynamics of an ensemble of globally coupled, externally forced, identical mechanical oscillators with cubic nonlinearity. Focus is put on solutions where the ensemble splits into two internally synchronized clusters, as a consequence of the bistability of individual oscillators. The multiplicity of these solutions, induced by the many possible ways of distributing the oscillators

First, we characterize all the polynomial vector fields in ℝ4 which have the Clifford torus as an invariant surface. Then we study the number of invariant meridians and parallels that such polynomial vector fields can have on the Clifford torus as a function of the degree of these vector fields.

The methods of recurrence plots (RPs) and recurrence quantification analysis (RQA) have been used to investigate the tribosystem. The morphology of RPs and RQA measures are strongly dependent on the embedding parameters of the recursive matrix and the segment sizes of the time-series. To improve the calculation accuracy of recursive characteristics analysis, the influences of the embedding parameters

The purpose of the present paper is to study the stability of a prey–predator model using KCC theory. The KCC theory is based on the assumption that the second-order dynamical system and geodesics equation, in associated Finsler space, are topologically equivalent. The stability (Jacobi stability) based on KCC theory and linear stability of the model are discussed in detail. Further, the effect of

In this paper, we consider an SI epidemic model incorporating additive Allee effect and time delay. The primary purpose of this paper is to study the dynamics of the above system. Firstly, for the model without time delay, we demonstrate the existence and stability of equilibria for three different cases, i.e. with weak Allee effect, with strong Allee effect, and in the critical case. We also investigate

In this short note, we prove some lower and upper bounds for the interval length function for vertical monotonicity of topological entropy of the Lozi mappings.

In this paper, the dynamics of an economic system with foreign financing, of integer or fractional order, are analyzed. The symmetry of the system determines the existence of two pairs of coexisting attractors. The integer-order version of the system proves to have several combinations of coexisting hidden attractors with self-excited attractors. Because one of the system variables represents the foreign

In this paper, the existence and bifurcations of periodic motions in a discontinuous dynamical system is studied through a discontinuous mechanical model. One can follow the study presented herein to investigate other discontinuous dynamical systems. Such a sampled discontinuous system consists of two subsystems on boundaries and three subsystems in subdomains. From the theory of discontinuous dynamical

We analyze benchmark models for reaction dynamics associated with a time-dependent saddle point. Our model allows us to incorporate time dependence of a general form, subject to an exponential growth restriction. Under these conditions, we analytically compute the time-dependent normally hyperbolic invariant manifold; its time-dependent stable and unstable manifolds; and a time-dependent dividing surface

Based on the theory of symbolic dynamical systems, we propose a novel computation method to locate and stabilize the unstable periodic points (UPPs) in a two-dimensional dynamical system with a Smale horseshoe. This method directly implies a new framework for controlling chaos. By introducing the subset based correspondence between a planar dynamical system and a symbolic dynamical system, we locate

A general formalism describing a type of energy-conservative system is established. Some possible dynamic behaviors of such energy-conservative systems are analyzed from the perspective of geometric invariance. A specific 4D chaotic energy-conservative system with a line of equilibria is constructed and analyzed. Typically, an energy-conservative system is also conservative in preserving its phase

In this paper, we focus on a network system which describes spatiotemporal dynamics of single species population at different patches since species can have different features in various life stages and different behaviors in various spatial environments. With the effect of time delay and spatial dispersion, homogenous, periodic and spatiotemporally nonhomogeneous distributions are identified. The

The paper describes some aspects of sudden transformations of closed invariant curves in a 2D piecewise smooth map. In particular, using detailed numerically calculated phase portraits, we discuss transitions from smooth to piecewise smooth closed invariant curves. We show that such transitions may occur not only when a closed invariant curve collides with a border but also via a homoclinic bifurcation

In this work, we present a monoparametric family of piecewise linear systems to generate multiscroll attractors through a polynomial family defined by path curves that connect to the roots. The idea is to define path curves where the roots of a polynomial can take values by determining an initial and a final polynomial. As a consequence, structural stability and bifurcation of the system can be obtained

In this paper, a reaction–diffusion model with delay effect and Dirichlet boundary condition is considered. Firstly, the existence, multiplicity, and patterns of spatially nonhomogeneous steady-state solution are obtained by using the Lyapunov–Schmidt reduction. Secondly, by means of space decomposition, we subtly discuss the distribution of eigenvalues of the infinitesimal generator associated with

The article is devoted to the study of the global behavior of the generalized Rabinovich system describing the process of interaction between waves in plasma. For this generalized system, we obtain the positive invariant set (ultimate bound) and globally exponential attractive set using the approach where we transform the initial problem of finding the corresponding set to the conditional extremum

This paper investigates some nonlinear dynamical behaviors about domains of attraction, bifurcations, and chaos in an axially accelerating viscoelastic beam under a time-dependent tension and a time-dependent speed. The axial speed and the axial tension are coupled to each other on the basis of a harmonic variation over constant initial values. The transverse motion of the moving beam is governed by

In this paper, a predator–prey model with prey-stage structure and prey-taxis is proposed and studied. Firstly, the local stability of non-negative constant equilibria is analyzed. It is shown that non-negative equilibria have the same stability between ODE system and self-diffusion system, and self-diffusion does not have a destabilization effect. We find that there exists a threshold value ξ0 such

The importance of cooperation is self-evident to humans, yet the existence of corruption where law violators can avoid being punished by paying bribes to corrupt law enforcers may threaten the maintenance of cooperation. Although powerful monitoring has been used to resolve such matters, existing studies show that the effects of such measures are either transient or uncertain. Thus how to efficiently

In this paper, we introduce the concept of stochastic bifurcations of group-invariant solutions for stochastic nonlinear wave equations. The essence of this concept is to display bifurcation phenomena by investigating stochastic P-bifurcation and stochastic D-bifurcation of stochastic ordinary differential equations derived by Lie symmetry reductions of stochastic nonlinear wave equations. Stochastic

A singularity is described that creates a forward time loss of determinacy in a two-timescale system, in the limit where the timescale separation is large. We describe how the situation can arise in a dynamical system of two fast variables and three slow variables or parameters, with weakly coupling between the fast variables. A wide set of initial conditions enters the 𝜖-neighborhood of the singularity

Ising model at zero temperature leads to a ferromagnetic state asymptotically. There are two such possible states linked by symmetry, and Glauber–Ising dynamics are employed to reach them. In some stochastic or deterministic dynamical systems, the same absorbing state with 𝒵2 symmetry is reached. This transition often belongs to the directed Ising (DI) class where dynamic exponents and persistence

Chaotic systems with hidden attractors have been widely studied in recent decades. In this field, systems without equilibrium are of special interest. Since multistable systems can alternate between several hidden attractors, it is difficult to detect their hidden oscillations using known analytical techniques. In this study, we propose to apply recurrence analysis methods as a possible solution to

The so-called Dixon system is often cited as an example of a two-dimensional (continuous) dynamical system that exhibits chaotic behavior, if its two parameters take their values in a certain domain. We provide first a rigorous proof that there is no chaos in Dixon’s system. Then we perform a complete bifurcation analysis of the system showing that the parameter space can be decomposed into 16 different

In this paper, a chaotic duplex H.264-codec-based secure video communication scheme is designed and its smartphone implementation is also carried out. First, an improved self-synchronous chaotic stream cipher algorithm equipped with a sinusoidal modulation, a multiplication, a modulo operation and a round down operation (SCSCA-SMMR) is developed. Using the sinusoidal modulation and multiplication,

The objective of this study is to investigate the complex dynamics of an eco-epidemic predator–prey system where disease is transmitted in prey species and predator population is being provided with alternative food. Holling type-II functional response is taken into consideration for interaction of predator and prey species. The half saturation constant for infected prey, the growth rate of susceptible

This paper reports the complex dynamics of a class of two-dimensional maps containing hidden attractors via linear augmentation. Firstly, the method of linear augmentation for continuous dynamical systems is generalized to discrete dynamical systems. Then three cases of a class of two-dimensional maps that exhibit hidden dynamics, the maps with no fixed point and the maps with one stable fixed point

The study is an attempt to explore the competitive effects on the top predators in the tri-trophic food chain model. Holling types II and IV functional responses are used to investigate the complex behavior of the tri-trophic prey–predator system. Reaction–diffusion systems have been used to represent temporal evolution and spatial interaction among the species. Consequently, local and global stabilities

This paper uncovers some striking and new complex phenomena in a memristive diode bridge-based Murali–Lakshmanan–Chua (MLC) circuit. These striking dynamical behaviors include the coexistence of multiple attractors and double-transient chaos. Also, period-doubling, chaos, crisis scenarios are observed in the system when varying the amplitude of the external excitation. Numerical simulation tools like

Of concern is the global dynamics of a two-species Holling-II amensalism system with nonlinear growth rate. The existence and stability of trivial equilibrium, semi-trivial equilibria, interior equilibria and infinite singularity are studied. Under different parameters, there exist two stable equilibria which means that this model is not always globally asymptotically stable. Together with the existence

The mean Poincaré recurrence time as well as the Lyapunov time are measured for the Fermi–Ulam model. It is confirmed that the mean recurrence time is dependent on the size of the window chosen in the phase space where particles are allowed to return. The fractal dimension of the region is determined by the slope of the recurrence time against the size of the window and two numerical values are measured:

In the last two years, many chaotic or hyperchaotic systems with megastability have been reported in the literature. The reported systems with megastability are mostly developed from their dynamic equations without any reference to the physical systems. In this paper, the dynamics of a single-link manipulator is considered to observe the existence of interesting dynamical behaviors. When the considered