Modeling and Analysis of a Three-Terminal-Memristor-Based Conservative Chaotic System

In this paper, a three-terminal memristor is constructed and studied through changing dual-port output instead of one-port. A new conservative memristor-based chaotic system is built by embedding this three-terminal memristor into a newly proposed four-dimensional (4D) Euler equation. The generalized Hamiltonian energy function has been given, and it is composed of conservative and non-conservative parts of the Hamiltonian. The Hamiltonian of the Euler equation remains constant, while the three-terminal memristor’s Hamiltonian is mutative, causing non-conservation in energy. Through proof, only centers or saddles equilibria exist, which meets the definition of the conservative system. A non-Hamiltonian conservative chaotic system is proposed. The Hamiltonian of the conservative part determines whether the system can produce chaos or not. The non-conservative part affects the dynamic of the system based on the conservative part. The chaotic and quasiperiodic orbits are generated when the system has different Hamiltonian levels. Lyapunov exponent (LE), Poincaré map, bifurcation and Hamiltonian diagrams are used to analyze the dynamical behavior of the non-Hamiltonian conservative chaotic system. The frequency and initial values of the system have an extensive variable range. Through the mechanism adjustment, instead of trial-and-error, the maximum LE of the system can even reach an incredible value of 963. An analog circuit is implemented to verify the existence of the non-Hamiltonian conservative chaotic system, which overcomes the challenge that a little bias will lead to the disappearance of conservative chaos.     

Robust tori in a double-waved Hamiltonian model

A Hamiltonian system perturbed by two waves with particular wave numbers can present robust tori, which are barriers created by the vanishing of the perturbed Hamiltonian at some defined positions. When robust tori exist, any trajectory in phase space passing close to them is blocked by emergent invariant curves that prevent the chaotic transport. Our results indicate that the considered particular solution for the two waves Hamiltonian model shows plenty of robust tori blocking radial transport.     

Robust tori-like Lagrangian coherent structures

In general the term “Lagrangian coherent structure” (LCS) is used to make reference about structures whose properties are similar to a time-dependent analog of stable and unstable manifolds from a hyperbolic fixed point in Hamiltonian systems. Recently, the term LCS was used to describe a different type of structure, whose properties are similar to those of invariant tori in certain classes of two-dimensional incompressible flows. A new kind of LCS was obtained. It consists of barriers, called robust tori that block the trajectories in certain regions of the phase space. We used the Double-Gyre Flow system as the model. In this system, the robust tori play the role of a skeleton for the dynamics and block, horizontally, vortices that come from different parts of the phase space.     

Reducing or enhancing chaos using periodic orbits

A method to reduce or enhance chaos in Hamiltonian flows with two degrees of freedom is discussed. This method is based on finding a suitable perturbation of the system such that the stability of a set of periodic orbits changes (local bifurcations). Depending on the values of the residues, reflecting their linear stability properties, a set of invariant tori is destroyed or created in the neighborhood of the chosen periodic orbits. An application on a paradigmatic system, a forced pendulum, illustrates the method.     

Integrable discrete static Hamiltonian with a parametric double-well potential

A one-dimensional discrete conservative Hamiltonian with a generalized form of the Schmidt potential, is constructed with the help of a non-integrable discrete Hamiltonian whose parametrized double-well potential can be reduced to the ?⁴ potential. The new conservative Hamiltonian is completely integrable in the discrete static regime, and the associate exact nonlinear solution is shown to coincide with the continuum nonlinear periodic solution of the non-integrable Hamiltonian. Numerical simulations and nonlinear stability analysis suggest that the discrete mapping derived from the completely integrable Hamiltonian undergoes a bifurcation which does not leads to the chaotic phase with randomly pinned states, but instead to a phase where real solutions become rare forming a cluster of periodic points around an elliptic fixed point.     

Many faces of stickiness in Hamiltonian systems

We discuss the phenomenon of stickiness in Hamiltonian systems. By visual examples of billiards, it is demonstrated that one must make a difference between internal (within chaotic sea(s)) and external (in vicinity of KAM tori) stickiness. Besides, there exist two types of KAM-islands, elliptic and parabolic ones, which demonstrate different abilities of stickiness.     

A dynamical system with a strange attractor and invariant tori

This paper describes a simple three-dimensional time-reversible system of ODEs with quadratic nonlinearities and the unusual property that it is exhibits conservative behavior for some initial conditions and dissipative behavior for others. The conservative regime has quasi-periodic orbits whose amplitude depend on the initial conditions, while the dissipative regime is chaotic. Thus a strange attractor coexists with an infinite set of nested invariant tori in the state space.     

Chaotic behaviors and toroidal/spherical attractors generated by discontinuous dynamics

This paper reports a class of chaotic attractors with toroidal or spherical patterns. These attractors look like spheres or tori, but they are not exactly on the two-dimensional toroidal/spherical surfaces. Discontinuous structures are taken in the considered system to produce easily various chaotic tori/spheres with different input functions. Moreover, controlling the shape of these chaotic attractors can be realized by adjusting some parameters with the help of Fourier series. The underlying chaos-generation mechanism is also explored briefly.     

A multi-directional controllable multi-scroll conservative chaos generator:Modelling,analysis,and FPGA implementation

Combing with the generalized Hamiltonian system theory,by introducing a special form of sinusoidal function,a class of n-dimensional(n=1,2,3)controllable multi-scroll conservative chaos with complicated dynamics is constructed.The dynamics characteristics including bifurcation behavior and coexistence of the system are analyzed in detail,the latter reveals abundant coexisting flows.Furthermore,the proposed system passes the NIST tests and has been implemented physically by FPGA.Compared to the multi-scroll dissipative chaos,the experimental portraits of the proposed system show better ergodicity,which have potential application value in secure communication and image encryption.     

On the weak-noise limit of Fokker-Planck models

The weak-noise limit of Fokker-Planck models leads to a set of nonlinear Hamiltonian canonical equations. We show that the existence of a nonequilibrium potential in the weak-noise limit requires the existence of whiskered tori in the Hamiltonian system and, therefore, the complete integrability of the latter. A specific model is considered, where the Hamiltonian system in the weak-noise limit is not integrable. Two different perturbative solutions are constructed: the first solution describes analytically the breakdown of the whiskered tori due to the appearance of wild séparatrices; the second solution allows the analytic construction of an approximate nonequilibrium potential and an asymptotic expression for the probability density in the steady state.On leave from Institute for Theoretical Physics, Eötvös University, Budapest, Hungary.     

Contact flows and integrable systems

We consider Hamiltonian systems restricted to the hypersurfaces of contact type and obtain a partial version of the Arnold–Liouville theorem: the system need not be integrable on the whole phase space, while the invariant hypersurface is foliated on an invariant Lagrangian tori. In the second part of the paper we consider contact systems with constraints. As an example, the Reeb flows on Brieskorn manifolds are considered.     

Dynamics analysis of a 5-dimensional hyperchaotic system with conservative flows under perturbation

Conservative chaotic flows have better ergodicity, therefore researching dynamics and applications of conservative systems has become a hot topic. We introduce a constant-perturbation into a 5-dimensional(5D) conservative model.Consequently, the line equilibria of original model have been changed to non-equilibrium. Plentiful chaos phenomena such as coexisting conservative flows can be observed in this modified system. In addition, by increasing the magnitude of the disturbance, the conservative system can be transformed to a dissipative system. Then, the modified system is realized by an xc7z020clg400 field programmable gate array(FPGA) chip. The designed chaotic oscillator consumes fewer resources and has high iteration speed. Finally, a pseudo random number generator based on this novel digital oscillator is designed.     

混沌信号的压缩感知去噪

对非线性时间序列进行噪声抑制是从中提取有效信息的前提. 混沌信号的去噪算法不仅要使滤波后的信号具有较高的信噪比, 也要具有较好的不确定性. 从压缩感知的角度出发,提出了一种新的噪声抑制方法. 该方法包括估计噪声方差, 以及依据动态的稀疏度将观测值往确定的过完备字典上投影. 仿真实验表明, 该方法比常用的小波阈值法和局部曲线拟合法具有更高的输出信噪比, 而原始信号的混沌特性也能得到较大程度的恢复.     

Energy behavior of Boris algorithm

Boris numerical scheme due to its long-time stability, accuracy and conservative properties has been widely applied in many studies of magnetized plasmas. Such algorithms conserve the phase space volume and hence provide accurate charge particle orbits. However, this algorithm does not conserve the energy in some special electromagnetic configurations,particularly for long simulation times. Here, we empirically analyze the energy behavior of Boris algorithm by applying it to a 2 D autonomous Hamiltonian. The energy behavior of the Boris method is found to be strongly related to the integrability of our Hamiltonian system. We find that if the invariant tori is preserved under Boris discretization, the energy error can be bounded for an exponentially long time, otherwise the said error will show a linear growth. On the contrary,for a non-integrable Hamiltonian system, a random walk pattern has been observed in the energy error.     

Chaotic itinerancy generated by coupling of Milnor attractors

We report the existence of chaotic itinerancy in a coupled Milnor attractor system. The attractor ruins consist of tori or local chaos generated from the original Milnor attractors. The chaotic behavior exhibited by a single orbit can be considered a "nonstationary" state, due to the extremely slow convergence of the Lyapunov exponents, but the behavior averaged over randomly chosen initial conditions is consistent with the limit theorem. We present as a possibly new indication of chaotic itinerancy the presence of slow decay of large fluctuations of the largest Lyapunov exponent.     

Quasi-Chaplygin Systems and Nonholonimic Rigid Body Dynamics

We show that the Suslov nonholonomic rigid body problem studied in by Fedorov and Kozlov (Am. Math. Soc. Transl. Ser. 2 168:141–171, 1995), Jovanović (Reg. Chaot. Dyn. 8(1):125–132, 2005), and Zenkov and Bloch (J. Geom. Phys. 34(2):121–136, 2000) can be regarded almost everywhere as a generalized Chaplygin system. Furthermore, this provides a new example of a multidimensional nonholonomic system which can be reduced to a Hamiltonian form by means of Chaplygin reducing multiplier. Since we deal with Chaplygin systems in the local sense, the invariant manifolds of the integrable examples are not necessary tori     

The dynamical feature of transition of a Hamiltonian system to a dissipative system

The mechanism of generation and annihilation of attractors during transition from a Hamiltonian system to a dissipative system is studied numerically using the dissipative standard map. The transient process related to the formation of attracting basins of periodic attractors is studied by discussing the evolution of the KAM tori of the standard map. The result shows that as damping increases, attractors are mainly generated from elliptic orbits of the Hamiltonian system and annihilated by colliding with unstable periodic orbits originating from the corresponding hyperbolic orbits of the Hamiltonian system. The transient process also exhibits the general feature of bifurcation.     

Novel routes to chaos through torus breakdown in non-invertible maps

The paper describes a number of new scenarios for the transition to chaos through the formation and destruction of multilayered tori in non-invertible maps. By means of detailed, numerically calculated phase portraits we first describe how three- and five-layered tori arise through period-doubling and/or pitchfork bifurcations of the saddle cycle on an ordinary resonance torus. We then describe several different mechanisms for the destruction of five-layered tori in a system of two linearly coupled logistic maps. One of these scenarios involves the destruction of the two intermediate layers of the five-layered torus through the transformation of two unstable node cycles into unstable focus cycles, followed by a saddle-node bifurcation that destroys the middle layer and a pair of simultaneous homoclinic bifurcations that produce two invariant closed curves with quasiperiodic dynamics along the sides of the chaotic set. Other scenarios involve different combinations of local and global bifurcations, including bifurcations that lead to various forms of homoclinic and heteroclinic tangles. We finally demonstrate that essentially the same scenarios can be observed both for a system of nonlinearly coupled logistic maps and for a couple of two-dimensional non-invertible maps that have previously been used to study the properties of invariant sets.     

Detecting chaos,determining the dimensions of tori and predicting slow diffusion in Fermi–Pasta–Ulam lattices by the Generalized Alignment Index method

The recently introduced GALI method is used for rapidly detecting chaos, determining the dimensionality of regular motion and predicting slow diffusion in multi-dimensional Hamiltonian systems. We propose an efficient computation of the GALI_k indices, which represent volume elements of k randomly chosen deviation vectors from a given orbit, based on the Singular Value Decomposition (SVD) algorithm. We obtain theoretically and verify numerically asymptotic estimates of GALIs long-time behavior in the case of regular orbits lying on low-dimensional tori. The GALI_k indices are applied to rapidly detect chaotic oscillations, identify low-dimensional tori of Fermi–Pasta–Ulam (FPU) lattices at low energies and predict weak diffusion away from quasiperiodic motion, long before it is actually observed in the oscillations.     

Generalized Joseph estimates of stability of plane shear flows with scalar nonlinearity

Possible statements of eigenvalue problems generalizing the classical Orr-Sommerfeld problem are given for incompressible stationary flows of non-Newtonian fluids; these problems are interpreted within the mechanics of a continuum as problems of the shear stability of such flows. A schematic diagram of the integral relation method as applied to these flows is described. In the case of an unperturbed Couette flow, Joseph estimates are generalized for any type of rheological curve and domains of guaranteed stability are constructed on a plane with the axes showing Reynolds numbers corresponding to the tangent and secant dynamic viscosity at one point.