Quadratic Autonomous System Research Articles

Integrability and linearizability of symmetric three-dimensional quadratic systems

<p style='text-indent:20px;'>We study local integrability and linearizability of polynomial and analytic systems of ODEs. It is proven that in the case of non-degenerate singularity if an analytic system is completely integrable and one equation is linearizable, then the system is linearizable in a neighborhood of the singularity. Some integrable and linearizable systems in a family of three-dimensional quadratic autonomous systems of ODEs depending on ten parameters are found.</p>

A real autonomous quadratic system of three differential equations with an infinite number of limit cycles

In this paper, we consider the problem of construction of real autonomous quadratic systems of three differential equations with the nonlocal existence of an infinite number of limit cycles. This means that an infinite number of limit cycles, emerging from the focus due to the Andronov – Hopf bifurcation, can exist in the phase space not only in the vicinity of the focus and not only for parameter values close to the bifurcation value. To solve this problem we use the method of determination of limit cycles as the curves of intersection of an invariant plane with a family of invariant elliptic paraboloids. Then the study of the limit cycles of the constructed system of the third order is reduced to the study of the corresponding system of the second order on each of the invariant elliptic paraboloids. The proof of the nonlocal existence of the limit cycle and the investigation of its stability for such a second-order system is carried out by constructing a topographic system of Poincaré functions or by transforming to polar coordinates.

Local Dynamics of a New Four-Dimensional Quadratic Autonomous System

In this manuscript, Local dynamic behaviors including stability and Hopf bifurcation of a new four-dimensional quadratic autonomous system are studied both analytically and numerically. Determining conditions of equilibrium points on different parameters are derived. Next, the stability conditions are investigated by using Routh-Hurwitz criterion and bifurcation conditions are investigated by using Hopf bifurcation theory, respectively. It is found that Hopf bifurcation on the initial point is supercritical in this four-dimensional autonomous system. The theoretical results are verified by numerical simulation. Besides, the new four-dimensional autonomous system under the parametric conditions of hyperchaos is studied in detail. It is also found that the system can enter hyperchaos, first through Hopf bifurcation and then through periodic bifurcation.

Vector Map Encryption Algorithm Based on Double Random Position Permutation Strategy

Encryption of vector maps, used for copyright protection, is of importance in the community of geographic information sciences. However, some studies adopt one-to-one mapping to scramble vertices and permutate the coordinates one by one according to the coordinate position in a plain map. An attacker can easily obtain the key values by analyzing the relationship between the cipher vector map and the plain vector map, which will lead to the ineffectiveness of the scrambling operation. To solve the problem, a vector map encryption algorithm based on a double random position permutation strategy is proposed in this paper. First, the secret key sequence is generated using a four-dimensional quadratic autonomous hyperchaotic system. Then, all coordinates of the vector map are encrypted using the strategy of double random position permutation. Lastly, the encrypted coordinates are reorganized according to the vector map structure to obtain the cipher map. Experimental results show that: (1) one-to-one mapping between the plain vector map and cipher vector map is prevented from happening; (2) scrambling encryption between different map objects is achieved; (3) hackers cannot obtain the permutation key value by analyzing the pairs of the plain map and cipher map.

Coexistence of multi-scroll chaotic attractors for a three-dimensional quadratic autonomous fractional system with non-local and non-singular kernel

In this paper, we present a numerical scheme and mathematical analysis for the famous three-dimensional quadratic autonomous self-govern system that happens to be chaotic with the coexistence of multi-scroll attractors. The scheme is based on the Atangana-Baleanu fractional derivative in the Caputo sense. The formulation of these schemes introduces the non-local and non-singular kernel to the fractional derivatives. The fractional derivative is then approximated using the family of the Adams–Bashforth schemes. The results are presented in both numerical and graphical as the fractional order β varies between 0<β⩽1. We study the proposed model in the both generalized case that is 0<β<1 and the case where β=1, which is the integer standard case. Due to the impact of the generalized case, the proposed model is able to maintain the coexistence of multi-scroll attractors.

A dynamical system approach to a class of radial weighted fully nonlinear equations

In this paper we study existence, nonexistence and classification of radial positive solutions of some weighted fully nonlinear equations involving Pucci extremal operators. Our results are entirely based on the analysis of the dynamics induced by an autonomous quadratic system which is obtained after a suitable transformation. This method allows to treat both regular and singular solutions in a unified way, without using energy arguments. In particular we recover known results on regular solutions for the fully nonlinear non weighted problem by alternative proofs. We also slightly improve the classification of the solutions for the extremal operator

Periodic solutions for a four-dimensional hyperchaotic system

In this paper, we show a zero-Hopf bifurcation in a four-dimensional smooth quadratic autonomous hyperchaotic system. Using averaging theory, we prove the existence of periodic orbits bifurcating from the zero-Hopf equilibrium located at the origin of the hyperchaotic system, and the stability conditions of periodic solutions are given.

New dynamics coined in a 4-D quadratic autonomous hyper-chaotic system

This note revisits a 4-D quadratic autonomous hyper-chaotic system in Zarei and Tavakoli (2016) and mainly considers some of its rich dynamics not yet investigated: global boundedness, invariant algebraic surface, singularly degenerate heteroclinic cycle and limit cycle. The main contributions of the work are summarized as follows: Firstly, we prove that for 4a ≥ c > 2a > 0, d > 0 and e > 0 the solutions of that system are globally bounded by constructing a suitable Lyapunov function. Secondly, Q=x3−12ax12=0 is found to be one of invariant algebraic surfaces with the cofactor -4a for the model. Thirdly, numerical simulations for c=0 not only illustrate different types of infinitely many singularly degenerate heteroclinic cycles near which chaotic attractors or limit cycles generate, but also that some of more degenerate (in term of a pure imaginary pair, one zero and one negative eigenvalue) or stable (in sense of three negative eigenvalues and one null eigenvalue) non-isolated equilibria (0,0,x3,0)(x3∈R) directly change into the limit cycles or chaotic attractors with a small perturbation of c > 0, which is in the absence of singularly degenerate heteroclinic cycles and degenerate pitchfork bifurcation at the non-isolated equilibria. In particular, some kind of forming mechanism of Lorenz attractor and the hyper-chaotic attractor of that system with (a,b,c,d,e)=(10,28,83,1,16) is revealed, which are collapses of singularly degenerate heteroclinic cycles and explosions of stable non-isolated equilibria. Finally, circuit experiment implements the aforementioned hyper-chaotic attractor, showing very good agreement with the simulation results.

Local Bifurcation Analysis and Global Dynamics Estimation of a Novel 4-Dimensional Hyperchaotic System

In this paper, we present a novel 4-dimensional (4D) smooth quadratic autonomous hyperchaotic system with complex dynamics. In order to investigate the dynamics evolution of the system, the Lyapunov exponent spectrum, bifurcation diagram and various phase portraits are provided. The local dynamics of this hyperchaotic system, such as the stability, pitchfork bifurcation, and Hopf bifurcation of equilibrium point, are analyzed by using the center manifold theorem and bifurcation theory. About the global dynamics, the ultimate bound sets of the system are found by combining the Lyapunov function method and appropriate optimization method. Numerical simulations are given to demonstrate the emergence of the two bifurcations and show the ultimate boundary regions.

COMPLEX DYNAMICS OF A SIMPLE 3D AUTONOMOUS CHAOTIC SYSTEM WITH FOUR-WING

The present paper revisits a three dimensional (3D) autonomous chaotic system with four-wing occurring in the known literature[Nonlinear Dyn (2010) 60(3):443-457] with the entitle "A new type of four-wing chaotic attractors in 3-D quadratic autonomous systems" and is devoted to discussing its complex dynamical behaviors, mainly for its non-isolated equilibria, Hopf bifurcation, heteroclinic orbit and singularly degenerate heteroclinic cycles, etc. Firstly, the detailed distribution of its equilibrium points is formulated. Secondly, the local behaviors of its equilibria, especially the Hopf bifurcation, are studied. Thirdly, its such singular orbits as the heteroclinic orbits and singularly degenerate heteroclinic cycles are exploited. In particular, numerical simulations demonstrate that this system not only has four heteroclinic orbits to the origin and other four symmetry equilibria, but also two different kinds of infinitely many singularly degenerate heteroclinic cycles with the corresponding two-wing and four-wing chaotic attractors nearby.

A fully qualified four-wing four-dimensional autonomous chaotic system and its synchronization

In order to generate a real four-wing chaotic attractor, a new four-dimensional (4D) smooth quadratic autonomous system which is constructed by a linear controller to a three-dimensional (3D) pseudo four-wing system is proposed in this paper. Some complex dynamical behaviors such as equilibrium points and chaos are investigated and analyzed. The corresponding four-wing chaotic attractor is first numerically verified through investigating phase trajectories, Lyapunove exponents, bifurcation diagram, analysis of power spectrum and Poincaré maps. Furthermore, an oscillator circuit is designed for implementation, with Multisim observations for verification and demonstration. Finally, an extended synchronization approach is proposed for synchronizing the four-wing chaotic system. On the basis of Jacobin matrix method, it is not necessary to construct the Lyapunov functions, which makes this scheme simpler. Theoretical analysis and numerical simulations demonstrate the validity and feasibility of the proposed control method.

Chaos in Essentially Singular 3D Dynamical Systems with Two Quadratic Nonlinearities

A new class of 3D autonomous quadratic systems, the dynamics of which demonstrate a chaotic behavior, is found. This class is a generalization of the well-known class of Lorenz-like systems. The existence conditions of limit cycles in systems of the mentioned class are found. In addition, it is shown that, with the change of the appropriate parameters of systems of the indicated class, chaotic attractors different from the Lorenz attractor can be generated (these attractors are the result of the cascade of limit cycles bifurcations). Examples are given.

Hopf bifurcation analysis and ultimate bound estimation of a new 4-D quadratic autonomous hyper-chaotic system

Based on Lorenz system, a new four-dimensional quadratic autonomous hyper-chaotic attractor is presented in this paper. It can generate double-wing chaotic and hyper-chaotic attractors with only one equilibrium point. Several properties of the new system are investigated using Lyapunov exponents spectrum, bifurcation diagram and phase portraits. Using the center manifold and normal form theories, the local dynamics, the stability and Hopf bifurcation at the equilibrium point are analyzed. To obtain the ellipsoidal ultimate bound, the ultimate bound of the proposed system is theoretically estimated using Lagrange multiplier method, Lagrangian function and local maximizer point. By properly choosing P and Q matrices, an estimation of the ultimate bound region, as a function of the Lagrange coefficient, is obtained using local maximizer point and reduced Hessian matrix. To demonstrate the evolution of the bifurcation and to show the ultimate bound region, numerical simulations are provided.

On the analysis of local bifurcation and topological horseshoe of a new 4D hyper-chaotic system

In this paper, a new four-dimensional (4D) smooth quadratic autonomous system with complex hyper-chaotic dynamics is presented and analyzed. The Lyapunov exponent (LE) spectrum, bifurcation diagram and various phase portraits of the system are provided. The stability, Hopf bifurcation and pitchfork bifurcation of equilibrium point are discussed by using the center manifold theorem and bifurcation theory. Numerical simulation results are consistent with the theoretical analysis. Besides, by combining the topological horseshoe theory with a computer-assisted method of Poincaré maps and utilizing the algorithm for finding horseshoes in 3D hyper-chaotic maps, a horseshoe with two-directional expansions in the 4D hyper-chaotic system is successfully found, which rigorously proves the existence of hyper-chaos in theory.

Exponential-Algebraic Maps and Chaos in 3D Autonomous Quadratic Systems

For some 3D autonomous quadratic dynamical systems an explicit autonomous exponential-algebraic 1D map, generating chaos in mentioned systems, is designed. Examples of the systems, where chaos is generated by such discrete maps, are given. New results about an existence of chaotic dynamics in the quadratic 3D systems are also derived. Besides, for the Lanford system (it is 3D autonomous quadratic dynamical system) the value of some parameter at which the system shows increased chaotic behavior is indicated. This assertion is based on the construction for the Lanford system of 2D exponential-algebraic discrete map which possesses chaotic properties.

Research of Chaotic Dynamics of 3D Autonomous Quadratic Systems by Their Reduction to Special 2D Quadratic Systems

New results about the existence of chaotic dynamics in the quadratic 3D systems are derived. These results are based on the method allowing studying dynamics of 3D system of autonomous quadratic differential equations with the help of reduction of this system to the special 2D quadratic system of differential equations.

Local bifurcation analysis and ultimate bound of a novel 4D hyper-chaotic system

This paper presents a new four-dimensional smooth quadratic autonomous hyper-chaotic system which can generate novel two double-wing periodic, quasi-periodic and hyper-chaotic attractors. The Lyapunov exponent spectrum, bifurcation diagram and phase portrait are provided. It is shown that this system has a wide hyper-chaotic parameter. The pitchfork bifurcation and Hopf bifurcation are discussed using the center manifold theory. The ellipsoidal ultimate bound of the typical hyper-chaotic attractor is observed. Numerical simulations are given to demonstrate the evolution of the two bifurcations and show the ultimate boundary region.

Grid multi-wing butterfly chaotic attractors generated from a new 3-D quadratic autonomous system

Due to the dynamic characteristics of the Lorenz system, multi-wing chaotic systems are still confined in the positive half-space and fail to break the threshold limit. In this paper, a new approach for generating complex grid multi-wing attractors that can break the threshold limit via a novel nonlinear modulating function is proposed from the firstly proposed double-wing chaotic system. The proposed method is different from that of classical multi-scroll chaotic attractors generated by odd-symmetric multi-segment linear functions from Chua system. The new system is autonomous and can generate various grid multi-wing butterfly chaotic attractors without requiring any external forcing, it also can produce grid multi-wing both on the xz-plane and yz-plane. Basic properties of the new system such as dissipation property, equilibrium, stability, the Lyapunov exponent spectrum and bifurcation diagram are introduced by numerical simulation, theoretical analysis and circuit experiment, which confirm that the multi-wing attractors chaotic system has more rich and complicated chaotic dynamics. Finally, a novel module-based unified circuit is designed which provides some principles and guidelines for future circuitry design and engineering application. The circuit experimental results are consistent with the numerical simulation results.

Dynamics of a 3D autonomous quadratic system with an invariant algebraic surface

An invariant algebraic surface is calculated for a 3D autonomous quadratic system. Also, the dynamics near finite singularities and near infinite singularities on the invariant algebraic surface is analyzed. Furthermore, pitchfork bifurcation is analyzed using center manifold theorem and a first integral of this quadratic system for some special parameters is provided. Finally, the dynamics of this system at infinity using the Poincare compactification in \(R^3\) is investigated and the singularly degenerate heteroclinic cycles are presented by a first integral and verified by numerical simulations.

GENERATING CHAOS IN 3D SYSTEMS OF QUADRATIC DIFFERENTIAL EQUATIONS WITH 1D EXPONENTIAL MAPS

New existence conditions of homoclinic orbits for some systems of ordinary quadratic differential equations with singular linear part are found. A realization of these conditions guarantees the existence of chaotic attractors at 3D autonomous quadratic systems. In addition, a chaotic behavior of solutions of these systems is determined by the 1D discrete map [Formula: see text] at some values of parameters r &gt; 0, p ∈ ℝ, and γ ∈ (-d, ∞), where d &gt; 0; n = 0, 1, 2, …. Examples of chaotic attractors are given.