Narrow Results

The growing demand for maintaining secure communication channels with the advancements in digital technologies has led to an intensified interest in designing reliable image encryption schemes. Despite various encryption schemes that have been used, some are considered to have insecurity regarding data transmission and multimedia. Motivated by this concern, this paper proposes a new color image encryption algorithm of a multi-key generation-based nD-Hyperchaotic (HC) system. The new algorithm achieves Shannon’s confusion and diffusion principles using a two S-box generation approach, where the first S-box is generated from the proposed nD-HC system and the resulting sequence is used to increase encryption complexity, while the second S-box is generated from ($n+i$)D-HC system. Afterward, mathematical analysis is carried out to showcase the robustness and efficiency of the proposed algorithm, as well as its resistance to visual, statistical, differential, and brute-force attacks. The proposed scheme successfully passes all NIST SP 800 suite tests. The cryptographic system demonstrated by the proposed scheme has proven to have outstanding performance through simulation tests, which indicates promising potential applicability aspects in secure and real-time image communication settings.

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 paper, the chaotic phenomenon is reported in a bearing system. The chaotic dynamics of the new system is first derived from an available bearing system. The proposed system is then analyzed using various tools to observe its dynamical behaviors. Theoretical and simulation results confirm the presence of chaotic and periodic behaviors in the new bearing system.

This paper constructs a new 4D chaotic system from the Sprott B system. The system is dissipative, chaotic with two saddle foci. The bifurcation diagrams verify that the system exists multiple attractors with different initial values, including two strange attractors, two periodic attractors. Furthermore, we apply the passive control to control the system. A controller is designed for driving the system to the origin. The simulations show our theoretical results visually.

The purpose of this work is to introduce a novel 4D chaotic system and investigate its multistability. The novel system has an unstable origin and two stable symmetrical hyperbolic equilibria. When the parameter increases across a critical value, the equilibria lose their stability and double Hopf bifurcations occur with the appearance of limit cycles. A pair of point, periodic, chaotic attractors are observed in the system from different initial values for given parameters. The chaos of the system is yielded via period-doubling bifurcation. A double-scroll chaotic attractor is numerically observed as well. By using the electronic circuit, the chaotic attractor of the system is realized. The control problem of the system is reported. An effective controller is designed to stabilize the system.

This paper introduces a new four-dimensional chaotic system with a unique unstable equilibrium and multiple coexisting attractors. The dynamic evolution analysis shows that the system concurrently generates two symmetric chaotic attractors for fixed parameter values. Based on this system, an effective method is established to construct an infinite number of coexisting chaotic attractors. It shows that the introduction of some non-linear functions with multiple zeros can increase the equilibria and inspire the generation of coexisting attractor of the system. Numerical simulations verify the availability of the method.

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.

We present and illustrate a feedback control-based framework that enables microscopic/stochastic simulators to trace their "coarse" bifurcation diagrams, characterizing the dependence of their expected dynamical behavior on parameters. The framework combines the so-called "coarse time stepper" and arc-length continuation ideas from numerical bifurcation theory with linear dynamic feedback control. An augmented dynamical system is formulated, in which the bifurcation parameter evolution is linked with the microscopic simulation dynamics through feedback laws. The augmentation stably steers the system along both stable and unstable portions of the open-loop bifurcation diagram. The framework is illustrated using kinetic Monte Carlo simulations of simple surface reaction schemes that exhibit both coarse regular turning points and coarse Hopf bifurcations.

Typically, online forecasting of pulse system dynamics assumes the usage of certain kinds of relevant a priori information. A bifurcation (or a parametric) diagram is one of the most informative forms of presentation of dynamics evolution. However, the data represented in this form is usually insufficient for one-to-one identification of the present system state and dynamics evolution direction, especially when online decision-making is necessary. In this paper two additional modified bifurcation diagrams are introduced. These diagrams provide a framework for deriving an algorithm which can be used to solve, in a step-by-step manner, a complex problem that consists in dynamics monitoring, identification and forecasting.

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers were written on these systems, a complete understanding of this class is still missing. Classical problems, and in particular, Hilbert's 16th problem [Hilbert, 1900, 1902], are still open for this class. In this article we make an interdisciplinary global study of the subclass which is the closure within real quadratic differential systems, of the family QW2 of all such systems which have a weak focus of second order. This class also includes the family of all quadratic differential systems possessing a weak focus of third order and topological equivalents of all quadratic systems with a center.

The bifurcation diagram for this class, done in the adequate parameter space which is the three-dimensional real projective space, is quite rich in its complexity and yields 373 subsets with 126 phase portraits for , 95 for QW2, 20 having limit cycles but only three with the maximum number of limit cycles (two) within this class. The phase portraits are always represented in the Poincaré disc. The bifurcation set is formed by an algebraic set of bifurcations of singularities, finite or infinite and by a set of points which we suspect to be analytic corresponding to global separatrices which have connections.

Algebraic invariants were needed to construct the algebraic part of the bifurcation set, symbolic computations to deal with some quite complex invariants and numerical calculations to determine the position of the analytic bifurcation set of connections.

The global geometry of this class reveals interesting bifurcations phenomena; for example, all phase portraits with limit cycles in this class can be produced by perturbations of symmetric (reversible) quadratic systems with a center. Many other nonlinear phenomena displayed here form material for further studies.

The results of a study of the bifurcation diagram of the Hindmarsh–Rose neuron model in a two-dimensional parameter space are reported. This diagram shows the existence and extent of complex bifurcation structures that might be useful to understand the mechanisms used by the neurons to encode information and give rapid responses to stimulus. Moreover, the information contained in this phase diagram provides a background to develop our understanding of the dynamics of interacting neurons.

The "Q-curves have long been observed and studied as the shadowy curves which appear illusively — not explicitly drawn — in the familiar orbit diagram of Myrberg's map f_c(x) = x² + c. We illustrate that Q-curves also appear implicitly, for a different reason, in a computer-drawn bifurcation diagram of x² + c as well — by "bifurcation diagram" we mean the collection of all periodic points of f_c (attracting, indifferent and repelling) — these collections form what we call "P-curves". We show Q-curves and P-curves intersect in one of two ways: At a superattracting periodic point on a P-curve, the infinite family of Q-curve s which intersect there are all tangent to the P-curve. At a Misiurewicz point, no tangencies occur at these intersections; the slope of the P-curve is the fixed point of a linear system whose iterates give the slopes of the Q-curves.

We also introduce some new phenomena associated with c sin x illustrating briefly how its two different families of Q-curves interact with P-curves.

Our algorithm for finding and plotting all periodic points (up to any reasonable period) in the bifurcation diagram is reviewed in an Appendix.

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than a thousand papers were written on these systems, a complete understanding of this family is still missing. Classical problems, and in particular, Hilbert's 16th problem [Hilbert, 1900, 1902], are still open for this family. In this article, we conduct a global study of the class QW^I of all real quadratic differential systems which have a weak focus and invariant straight lines of total multiplicity of at least two. This family modulo the action of the affine group and time homotheties is three-dimensional and we give its bifurcation diagram with respect to a normal form, in the three-dimensional real projective space of the parameters of this form. The bifurcation diagram yields 73 phase portraits for systems in QW^I plus 26 additional phase portraits with the center at its border points. Algebraic invariants are used to construct the bifurcation set. We show that all systems in QW^I necessarily have their weak focus of order one and invariant straight lines of total multiplicity exactly two. The phase portraits are represented on the Poincaré disk. The bifurcation set is algebraic and all points in this set are points of bifurcation of singularities. We prove that there is no phase portrait with limit cycles in this class but that there is a total of five phase portraits with graphics, four having the invariant line as a regular orbit and one phase portrait with an infinity of graphics which are all homoclinic loops inside a heteroclinic graphic with two singularities, both at infinity.

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers have been written on these systems, a complete understanding of this family is still missing. Classical problems, and in particular, Hilbert's 16th problem [Hilbert, 1900, 1902], are still open for this family. In this article, we make a global study of the family of all real quadratic polynomial differential systems which have a semi-elemental triple node (triple node with exactly one zero eigenvalue). This family modulo the action of the affine group and time homotheties is three-dimensional and we give its bifurcation diagram with respect to a normal form, in the three-dimensional real space of the parameters of this form. This bifurcation diagram yields 28 phase portraits for systems in counting phase portraits with and without limit cycles. Algebraic invariants are used to construct the bifurcation set. The phase portraits are represented on the Poincaré disk. The bifurcation set is not only algebraic due to the presence of a surface found numerically. All points in this surface correspond to connections of separatrices.

The results in this paper show that the cubic vector fields ẋ = -y + M(x, y) - y(x² + y²), ẏ = x + N(x, y) + x( x² + y²), where M, N are quadratic homogeneous polynomials, having simultaneously a center at the origin and at infinity, have at least 61 and at most 68 topologically different phase portraits. To this end, the reversible subfamily defined by M(x, y) = -γxy, N(x, y) = (γ - λ)x² + α²λy² with α, γ ∈ ℝ and λ ≠ 0, is studied in detail and it is shown to have at least 48 and at most 55 topologically different phase portraits. In particular, there are exactly five for γλ < 0 and at least 46 for γλ > 0. Furthermore, the global bifurcation diagram is analyzed.

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers have been written on these systems, a complete understanding of this family is still missing. Classical problems, and in particular, Hilbert's 16th problem [Hilbert, 1900, 1902], are still open for this family. In this paper, we study the bifurcation diagram of the family QsnSN which is the set of all quadratic systems which have at least one finite semi-elemental saddle-node and one infinite semi-elemental saddle-node formed by the collision of two infinite singular points. We study this family with respect to a specific normal form which puts the finite saddle-node at the origin and fixes its eigenvectors on the axes. Our aim is to make a global study of the family which is the closure of the set of representatives of QsnSN in the parameter space of that specific normal form. This family can be divided into three different subfamilies according to the position of the infinite saddle-node, namely: (A) with the infinite saddle-node in the horizontal axis, (B) with the infinite saddle-node in the vertical axis and (C) with the infinite saddle-node in the bisector of the first and third quadrants. These three subfamilies modulo the action of the affine group and times homotheties are four-dimensional. Here, we provide the complete study of the geometry with respect to a normal form of the first two families, (A) and (B). The bifurcation diagram for the subfamily (A) yields 38 phase portraits for systems in (29 in QsnSN(A)) out of which only three have limit cycles and 13 possess graphics. The bifurcation diagram for the subfamily (B) yields 25 phase portraits for systems in (16 in QsnSN(B)) out of which 11 possess graphics. None of the 25 portraits has limit cycles. Case (C) will yield many more phase portraits and will be written separately in a forthcoming new paper. Algebraic invariants are used to construct the bifurcation set. The phase portraits are represented on the Poincaré disk. The bifurcation set of is the union of algebraic surfaces and one surface whose presence was detected numerically. All points in this surface correspond to connections of separatrices. The bifurcation set of is formed only by algebraic surfaces.

This paper introduces a new no-equilibrium chaotic system that is constructed by adding a tiny perturbation to a simple chaotic flow having a line equilibrium. The dynamics of the proposed system are investigated through Lyapunov exponents, bifurcation diagram, Poincaré map and period-doubling route to chaos. A circuit realization is also represented. Moreover, two other new chaotic systems without equilibria are also proposed by applying the presented methodology.

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers have been written on these systems, a complete understanding of this family is still missing. Classical problems, and in particular, Hilbert's 16th problem [Hilbert, 1900, 1902], are still open for this family. Our goal is to make a global study of the family QsnSN of all real quadratic polynomial differential systems which have a finite semi-elemental saddle-node and an infinite saddle-node formed by the collision of two infinite singular points. This family can be divided into three different subfamilies, all of them with the finite saddle-node in the origin of the plane with the eigenvectors on the axes and with the eigenvector associated with the zero eigenvalue on the horizontal axis and (A) with the infinite saddle-node in the horizontal axis, (B) with the infinite saddle-node in the vertical axis and (C) with the infinite saddle-node in the bisector of the first and third quadrants. These three subfamilies modulo the action of the affine group and time homotheties are three-dimensional and we give the bifurcation diagram of their closure with respect to specific normal forms, in the three-dimensional real projective space. The subfamilies (A) and (B) have already been studied [Artés et al., 2013b] and in this paper we provide the complete study of the geometry of the last family (C). The bifurcation diagram for the subfamily (C) yields 371 topologically distinct phase portraits with and without limit cycles for systems in the closure within the representatives of QsnSN(C) given by a chosen normal form. Algebraic invariants are used to construct the bifurcation set. The phase portraits are represented on the Poincaré disk. The bifurcation set of is not only algebraic due to the presence of some surfaces found numerically. All points in these surfaces correspond to either connections of separatrices, or the presence of a double limit cycle.

This paper develops topological methods for qualitative analysis of the behavior of nonholonomic dynamical systems. Their application is illustrated by considering a new integrable system of nonholonomic mechanics, called a nonholonomic hinge. Although this system is nonholonomic, it can be represented in Hamiltonian form with a Lie–Poisson bracket of rank two. This Lie–Poisson bracket is used to perform stability analysis of fixed points. In addition, all possible types of integral manifolds are found and a classification of trajectories on them is presented.

We consider the family of planar differential systems depending on two real parameters