Saddle-focus Equilibrium Points Research Articles

A Cubic Memristive System with Two Twin Rössler-Type Chaotic Attractors Symmetrical About an Invariant Plane

Memristive circuits and systems have been widely studied in the last years due to their potential applications in several technological areas. They are capable of producing nonlinear periodic and chaotic oscillations, due to their locally-active characteristics. In this paper, we consider a cubic four-parameter differential system which models a memristive circuit consisting of three elements: a passive linear inductor, a passive linear capacitor and a locally-active current-controlled generic memristor. This system has a saddle-focus equilibrium point at the origin, whose global stable and unstable manifolds are, respectively, the [Formula: see text]-axis and the plane [Formula: see text], which are invariant sets where the dynamic is linear. We show that this structure can generate two twin Rössler-type chaotic attractors symmetrical with respect to the plane [Formula: see text]. We describe the mechanism of creation of these chaotic attractors, showing that, although being similar to the Rössler attractor, the twin attractors presented here have simpler structural mechanism of formation, since the system has no homoclinic or heteroclinic orbits to the saddle-focus, as presented by the Rössler system. The studied memristive system has the rare property of having chaotic dynamics and an invariant plane with linear dynamic, which is quite different from other chaotic systems presented in the literature that have invariant surfaces filled by equilibrium points. We also present and discuss the electronic circuit implementation of the considered system and study its dynamics at infinity, via the Poincaré compactification, showing that all the solutions, except the ones contained in the plane [Formula: see text], are bounded and cannot escape to infinity.

Homoclinic Orbits in Several Classes of Three-Dimensional Piecewise Affine Systems with Two Switching Planes

The existence of homoclinic orbits or heteroclinic cycle plays a crucial role in chaos research. This paper investigates the existence of the homoclinic orbits to a saddle-focus equilibrium point in several classes of three-dimensional piecewise affine systems with two switching planes regardless of the symmetry. An analytic proof is provided using the concrete expression forms of the analytic solution, stable manifold, and unstable manifold. Meanwhile, a sufficient condition for the existence of two homoclinic orbits is also obtained. Furthermore, two concrete piecewise affine asymmetric systems with two homoclinic orbits have been constructed successfully, demonstrating the method’s effectiveness.

A simple multi-stable chaotic jerk system with two saddle-foci equilibrium points: analysis, synchronization via backstepping technique and MultiSim circuit design

<span>This paper announces a new three-dimensional chaotic jerk system with two saddle-focus equilibrium points and gives a dynamic analysis of the properties of the jerk system such as Lyapunov exponents, phase portraits, Kaplan-Yorke dimension and equilibrium points. By modifying the Genesio-Tesi jerk dynamics (1992), a new jerk system is derived in this research study. The new jerk model is equipped with multistability and dissipative chaos with two saddle-foci equilibrium points. By invoking backstepping technique, new results for synchronizing chaos between the proposed jerk models are successfully yielded. MultiSim software is used to implement a circuit model for the new jerk dynamics. A good qualitative agreement has been shown between the MATLAB simulations of the theoretical chaotic jerk model and the MultiSIM results</span>

Design and FPGA implementation of multi-wing chaotic switched systems based on a quadratic transformation* *Project supported by the National Natural Science Foundation of China (Grant Nos. 61973199 and 61973200) and the Taishan Scholar Project of Shandong Province of China.

Based on a quadratic transformation and a switching function, a novel multi-wing chaotic switched system is proposed. First, a 4-wing chaotic system is constructed from a 2-wing chaotic system on the basis of a quadratic transformation. Then, a switching function is designed and by adjusting the switching function, the number and the distribution of the saddle-focus equilibrium points of the switched system can be regulated. Thus, a set of chaotic switched systems, which can produce 6-to-8-12-16-wing attractors, are generated. The Lyapunov exponent spectra, bifurcation diagrams, and Poincaré maps are given to verify the existence of the chaotic attractors. Besides, the digital circuit of the multi-wing chaotic switched system is designed by using the Verilog HDL fixed-point algorithm and the state machine control. Finally, the multi-wing chaotic attractors are demonstrated via FPGA platform. The experimental results show that the number of the wings of the chaotic attractors can be expanded more effectively with the combination of the quadratic transformation and the switching function methods.

Emergence of a square chaotic attractor through the collision of heteroclinic orbits

In this work we introduce a square chaotic attractor based on the collision of two heteroclinic orbits. Before the collision, the system presents the coexistence of two double scroll attractors that are generated via piecewise linear (PWL) systems that deal with two saddle-foci equilibria of different classes, i.e. the dimensions of the unstable and stable manifolds of one of the equilibrium points are one and two, respectively, and vice versa for the unstable and stable manifolds associated with the second equilibrium point. The new class is constructed based on the existence of a heteroclinic loop by linear affine systems with two saddle-focus equilibrium points of different types. The chaotic behavior of the proposed system is tested by the maximum Lyapunov exponent and the 0–1 chaos test. This new class of PWL dynamical systems exhibits different behaviors and allows the generation of different basins of attraction for the coexistence of two or more attractors. Therefore, a mechanism to establish bistability is provided and how it can be controlled (i.e. bistability annihilation) via varying system’s parameters is demonstrated.

Generation and application of novel Chua multi-scroll chaotic attractors

Chaos has great potential applications in engineering fields, such as secure communication and digital encryption. Since the double-scroll Chua’s circuit was developed first by Chua, it has quickly become a paradigm to study the double-scroll chaotic attractors. Compared with the conventional double-scroll chaotic attractors, the multi-scroll chaotic attractors have complex structures and rich nonlinear dynamical behaviors. The multi-scroll chaotic attractors have been applied to various chaos-based information technologies, such as secure communication and chaotic cryptanalysis. Hence, the generation of the multi-scroll chaotic attractors has become a hot topic in research field of chaos at present. In this paper, a novel Chua multi-scroll chaotic system is constructed by using a logarithmic function series. The nonlinear dynamical behaviors of the novel Chua multi-scroll chaotic system are analyzed, including symmetry, invariance, equilibrium points, the largest Lyapunov exponent, etc. The existence of chaos is confirmed by theoretical analyses and numerical simulations. The results show that the rich Chua multi-scroll chaotic attractors can be generated by combining the logarithmic function series with the novel Chua double-scroll chaotic system. The generation mechanism of the Chua multi-scroll chaotic attractors is that the saddle-focus equilibrium points of index 2 are used to generate the scrolls, and the saddle-focus equilibrium points of index 1 are used to connect these scrolls. Then, three recursive back-stepping controllers are designed to control the chaotic behavior in the novel Chua multi-scroll chaotic system. The recursive back-stepping controllers can control the novel Chua multi-scroll chaotic system to a fixed point or a given sinusoidal function. Finally, a new method of detecting a weak signal embedded in the Gaussian noise is proposed on the basis of the novel Chua multi-scroll chaotic system and the recursive back-stepping controllers. The immunity of the novel Chua multi-scroll chaotic system to the Gaussian noise with the zero mean is analyzed by using the stochastic differential equation theory. The results show that the proposed new method of detecting the weak signal can detect the frequencies of the multi-frequency weak periodic signal embedded in the Gaussian noise. In addition, the novel Chua multi-scroll chaotic system has strong immunity to any Gaussian noise with the zero mean. The proposed method provides a new thought for detecting the weak signal.

A novel grid multiwing chaotic system with only non-hyperbolic equilibria

The structure of the chaotic attractor of a system is mainly determined by the nonlinear functions in system equations. By using a new saw-tooth wave function and a new stair function, a novel complex grid multiwing chaotic system which belongs to non-Shil’nikov chaotic system with non-hyperbolic equilibrium points is proposed in this paper. It is particularly interesting that the complex grid multiwing attractors are generated by increasing the number of non-hyperbolic equilibrium points, which are different from the traditional methods of realising multiwing attractors by adding the index-2 saddle-focus equilibrium points in double-wing chaotic systems. The basic dynamical properties of the new system, such as dissipativity, phase portraits, the stability of the equilibria, the time-domain waveform, power spectrum, bifurcation diagram, Lyapunov exponents, and so on, are investigated by theoretical analysis and numerical simulations. Furthermore, the corresponding electronic circuit is designed and simulated on the Multisim platform. The Multisim simulation results and the hardware experimental results are in good agreement with the numerical simulations of the same system on Matlab platform, which verify the feasibility of this new grid multiwing chaotic system.

A class of Chua-like systems with only two saddle-foci of different type

Abstract Since the reported Chua’s system, several generalizations of this system have been presented, these approaches include new equilibria in order to obtain three or more scrolls in the attractor. One of these generalizations requires at least the same number of saddle-foci with local two-dimensional unstable manifolds as the desired number of scrolls. In this work, we present the generation of a double-scroll chaotic attractor called Chua-like system. Once that an equilibrium point has been removed from the Chua’s system and there are only two saddle-foci of different class, i.e. the dimension of one of the local unstable manifolds is one while the other is of dimension two. The new class is constructed based on the existence of a heteroclinic loop by linear affine systems with two saddle-focus equilibrium points of different type. Furthermore, the chaotic behavior of the proposed system is tested by the maximum Lyapunov exponent and the 0 — 1 chaos test.

Analysis of a Class of Complex System without Equilibria via Switched Control Law

In this article we introduce a new class of complex system without equilibria which exhibits a chaotic multiscroll attractor in each node. The number of scrolls in the attractor is determined by a switched control law to allow the operation of different linear affine systems. Thus, the system is composed of many subsystems which interact with each other to generate a multiscroll attractor. This new class of piecewise linear (PWL) system presents no positive real part in the eigenvalues of the Jacobian matrix as opposed to the reported systems with multiscrolls. The scrolls present a complex behavior since these don’t unwrap and don’t appear close to an unstable manifold of a saddle-focus equilibrium point. A particular case is taken as case study and simulation plots of the attractor are provided.

A novel multi-scroll chaotic generator: Analysis, simulation, and implementation**Project supported by the National Natural Science Foundation of China (Grant Nos. 51377124 and 51521065), the Foundation for the Author of National Excellent Doctoral Dissertation, China (Grant No. 201337), and the New Star of Youth Science and Technology of Shaanxi Province, China (Grant No. 2016KJXX-40).

Based on a new three-dimensional autonomous linear system and designing a specific form of saturated function series and a sign function with two variables of system, which are employed to increase saddle-focus equilibrium points with index 2, a novel multi-scroll chaotic system is proposed and its typical dynamical characteristics including bifurcation diagram, Poincare map, and the stability of equilibrium points are analyzed. The hardware circuit is designed and the experimental results are presented for confirmation.

Mixed-Mode Oscillations of El Niño–Southern Oscillation

Abstract Very strong El Niño events occur sporadically every 10–20 yr. The origin of this bursting behavior still remains elusive. Using a simplified three-dimensional dynamical model of the tropical Pacific climate system, which captures El Niño–Southern Oscillation (ENSO) combined with recently developed mathematical tools for fast–slow systems, the authors show that decadal ENSO bursting behavior can be explained as a mixed-mode oscillation (MMO), which also predicts a critical threshold for rapid amplitude growth. It is hypothesized that the MMO dynamics of the low-dimensional climate model can be linked to a saddle-focus equilibrium point, which mimics a tropical Pacific Ocean state without ocean circulation.

Theoretical design and circuit realization of complex grid multi-wing chaotic system

This paper further investigates a novel method for generating complex grid multi-wing chaotic attractors by double-mirror symmetry conversion both in z direction and in y direction. First, multi-wing chaotic attractors are obtained from a three-dimensional quadratic chaotic system by designing an even-symmetry multi-segment quadratic function to extend the number of saddle-focus equilibrium points with index 2 in y direction. Second, the principle of mirror symmetry conversion for generating grid multi-wing is given. Third, based on the principle, by the double-mirror symmetry conversion with respect to z-axis and y-axis, respectively, intended grid n×m-wing chaotic systems can be obtained. Finally, improved module-based circuits are designed for generating grid multi-wing chaotic attractors, which has demonstrated the feasibility of the proposed approaches.

Chaos generator design with piecewise affine systems

This paper considers design of chaos generator with a class of three-dimensional two-zone piecewise affine systems. A criterion is established to guarantee the existence of a heteroclinic cycle connecting two saddle-focus equilibrium points located in two zones, respectively, by which the existence of chaotic attractors in a neighborhood of the heteroclinic cycle can be easily obtained by Shil’nikov type theory. Based on the mathematical criteria proposed, we present a methodology for design of chaos generator.

Design and application of multi-scroll chaotic attractors based on simplified Lorenz system

Two linear systems are obtained by employing linearization technique in a simplified Lorenz system, and a two-scroll chaotic attractor is generated via the control method. Multi-scroll chaotic attractors are generated by extending the saddle-focus equilibrium points with index 2. Dynamic characteristics of the multi-scroll chaotic system are analyzed by observing the phase diagrams, bifurcation diagrams, Poincaré sections and calculating the largest Lyapunov exponent. A circuit for the multi-scroll attractor is designed and simulated. The numerical simulation result and the circuit simulation result are consistent with each other. To apply the multi-scroll chaotic systems to image encryption, an improved hybrid encryption algorithm is designed based on the multi-scroll chaotic system and advanced encryption standard (AES), and its encryption performances are analyzed. The results show that the improved hybrid encryption has a higher security.

ON CONSTRUCTING COMPLEX GRID MULTIWING CHAOTIC SYSTEM BY SWITCHING CONTROL AND MIRROR SYMMETRY CONVERSION

This paper further investigates a novel method, namely the mirror and double-mirror symmetry conversion both in z direction and in y direction, for generating complex grid multiwing chaotic attractors from a three-dimensional quadratic chaotic system. First, by designing a switching controller with even-symmetry multisegment quadratic function to extend the number of saddle-focus equilibrium points with index 2 in x direction, multiwing chaotic attractors are obtained. Based on this approach, then, different from the methods proposed in the previous literature, by the mirror and double-mirror symmetry conversion with respect to x-axis and y-axis respectively, various intended grid n × m-wing chaotic systems can be obtained. The principle and method for generating grid multiwing are also given. Numerical simulations and circuit realizations have demonstrated the feasibility and effectiveness of the proposed approaches.

Grid Multiscroll Hyperchaotic Attractors Based on Colpitts Oscillator Mode with Controllable Grid Gradient and Scroll Numbers

Through introducing two piecewise-linear triangular wave functions in a three-dimensional spiral chaotic Colpittsoscillator model, a four-dimensional grid multiscroll hyperchaotic system is constructed. Interestingly, by adjusting abuild-in parameter in a variable of one triangle wave function, the control of the gradient of the multiscroll grid isachieved. Whereas by deploying the zero points of the two triangular wave functions to extend the saddle-focusequilibrium points with index-2 in phase space the scroll numbers do not only increase along with the number ofturning points, but they can also generate arbitrary multiples of products. The basic dynamical behaviors of theproposed four-dimensional multiscroll hyperchaotic system are analyzed. Finally, the hardware experimental circuit isdesigned and the interrelated circuit implementation is realized. The experimental results are in agreement with boththeoretical analyses and numerical simulations, which verify the feasibility of the design methods.

Heteroclinic cycles arising in generic unfoldings of nilpotent singularities

In this paper we study the existence of heteroclinic cycles in generic unfoldings of nilpotent singularities. Namely we prove that any nilpotent singularity of codimension four in $\mathbb{R}^4$ unfolds generically a bifurcation hypersurface of bifocal homoclinic orbits, that is, homoclinic orbits to equilibrium points with two pairs of complex eigenvalues. We also prove that any nilpotent singularity of codimension three in $\mathbb{R}^3$ unfolds generically a bifurcation curve of heteroclinic cycles between two saddle-focus equilibrium points with different stability indexes. Under generic assumptions these cycles imply the existence of homoclinic bifurcations. Homoclinic orbits to equilibrium points with complex eigenvalues are the simplest configurations which can explain the existence of complex dynamics as, for instance, strange attractors. The proof of the arising of these dynamics from a singularity is a very useful tool, particularly for applications.

Construction of grid multi-scroll chaotic attractors and its circuit implementation with polynomial and step function

A constructing approach to generating N×M-scroll attractors with polynomial and step function is reported. In Chuas circuit, only two or three scrolls can be generated by traditional polynomial function. On this basis, the multi-scroll of x direction in phase space is obtained by polynomial shift. And then the saddle-focus equilibrium points with index-2 in phase space are extended by combining both polynomial and step function, which makes it possible to extend the multi-scroll in y direction. Then the grid multi-scroll chaotic attractors are generated. The main feature of this constructing approach is generating grid multi-scroll chaotic attractors by combining both smooth curves and non-smooth curves for the first time. And the arbitrary planar grid multi-scroll chaotic attractors array can be generated by adjusting the values of natural numbers N and M. The effectiveness of this method has been verified by theoretical analysis, numerical simulation and circuit simulation.

Heteoclinic orbit and backstepping control of a 3D chaotic system

The existence of heteoclinic loop which connects the saddle focus equilibrium points is analyzed for a three-dimensional differential system based on heteoclinic orbit Shilnikov method, which proves the system possesses "horseshoe" chaos. Then the system bifurcation, Lyapunov exponent, Poincare mapping are studied by numerical analysis. In addition, adaptive backstepping design is used to control this system with three unknown key parameters, and an algorithm of this controller is presented. Finally, we make some numerical simulations of the system in order to verify the analytic results.

Existence of attractor and control of a 3D differential system

This paper analyzes the orbit of a three-dimensional differential system based on the Shilnikov criterion. It also applies the Shilnikov method of constructing a heteroclinic connection between saddle focus equilibrium points of the system, proving that the system possesses “horseshoe” chaos. In addition, adaptive backstepping design is used to control this system with three unknown key parameters, and an algorithm of this controller is presented. Finally, we give some numerical simulation studies of the system in order to verify the analytic results.