Hyperchaotic Oscillations Research Articles

Offset-Dominated Uncountably Many Hyperchaotic Oscillations

Offset-Dominated Uncountably Many Hyperchaotic Oscillations

Nonlinear feedback synthesis and control of periodic, quasiperiodic, chaotic and hyper-chaotic oscillations in mechanical systems

Nonlinear feedback synthesis and control of periodic, quasiperiodic, chaotic and hyper-chaotic oscillations in mechanical systems

A probe into the fatigue crack growth in mechanical systems with hyperchaotic/chaotic dynamics

This study presents experimental and analytical fatigue frameworks for the dynamic fatigue life prediction of nonlinear mechanical systems under hyperchaotic oscillations. Fatigue in aluminium notched beams under hyperchaotic/chaotic vibrations is studied using a time-based analytical model. The fatigue lives obtained by the analytical model provided satisfactory correlations with those acquired by the dynamic fatigue experiments conducted on cracked aluminium beams. The model is then used to study the fatigue under Duffing chaotic, Duffing hyperchaotic, and periodic dynamic forcing. Results indicate that the classic linear cumulative damage rule delivers inaccurate high cycle fatigue (HCF) predictions under chaotic and hyperchaotic loading. Proper prediction of the HCF in metallic mechanical components experiencing chaotic and hyperchaotic dynamics requires time-based fracture-mechanics analytical models, such as that proposed in the current study. Under similar elastic energy inputs, hyperchaotic loading brings higher fatigue damage than chaotic and periodic loading. The fatigue life sensitivity to initial conditions is high under hyperchaotic, moderate under chaotic and next-zero under periodic loading.

Switching of behavior: From hyperchaotic to controlled magnetoconvection model

The switching of behavior, from the hyperchaotic to controlled magnetoconvection model, is studied by a feedback control technique. The magnetoconvection model shows hyperchaotic oscillations for different values of parameters: Rayleigh number r, Chandrasekhar number Q, and diffusivity ratio l. Chaotic responses of the magnetoconvection model are considered through boundedness and Lyapunov exponents to specify the place where the controller needs to be applied. The controller for the magnetoconvection model is calculated by using the concept of the Lie derivative, which is the most significant facet of control analytical techniques. Speed and dislocated feedback techniques are also utilized with the consideration of stability analysis through feedback gains. To show the advantages of the feedback control technique, we give a comparison with other control techniques such as speed and dislocated feedback techniques. Simulation results indicate that the analytical strategy for controlling the oscillation is effective and controlled within a small duration of time.

Hyperchaos and multistability in the model of two interacting microbubble contrast agents.

We study nonlinear dynamics of two coupled contrast agents that are micrometer size gas bubbles encapsulated into a viscoelastic shell. Such bubbles are used for enhancing ultrasound visualization of blood flow and have other promising applications like targeted drug delivery and noninvasive therapy. Here, we consider a model of two such bubbles interacting via the Bjerknes force and exposed to an external ultrasound field. We demonstrate that in this five-dimensional nonlinear dynamical system, various types of complex dynamics can occur, namely, we observe periodic, quasiperiodic, chaotic, and hypechaotic oscillations of bubbles. We study the bifurcation scenarios leading to the onset of both chaotic and hyperchaotic oscillations. We show that chaotic attractors in the considered system can appear via either the Feigenbaum cascade of period-doubling bifurcations or the Afraimovich-Shilnikov scenario of torus destruction. For the onset of hyperchaotic dynamics, we propose a new bifurcation scenario, which is based on the appearance of a homoclinic chaotic attractor containing a saddle-focus periodic orbit with its two-dimensional unstable manifold. Finally, we demonstrate that the dynamics of two bubbles can be essentially multistable, i.e., various combinations of the coexistence of the above mentioned attractors are possible in this model. These cases include the coexistence of a hyperchaotic regime with an attractor of any other remaining type. Thus, the model of two coupled gas bubbles provides a new example of physically relevant system with multistable hyperchaos.

Exciting Chaotic and Quasi-Periodic Oscillations in a Multicircuit Oscillator with a Common Control Scheme

We propose a multicircuit oscillator with a common control scheme driving self-sustained oscillations in each circuit, which can exhibit quasi-periodic and chaotic oscillations. The proposed oscillator was investigated by experimental and numerical methods that confirmed the possibility of exciting multifrequency quasi-periodic, chaotic, and hyperchaotic oscillations with several positive Lyapunov exponents.

Dynamical analysis and multistability in autonomous hyperchaotic oscillator with experimental verification

In this contribution, a modified oscillator of Tamasevicius et al. (Electron Lett 33:542–544, 1997) (referred to as the mTCMNL oscillator hereafter) is introduced with antiparallel diodes as nonlinear elements. The model is described by a continuous time of four-dimensional autonomous system with hyperbolic sine nonlinearity based on Shockley diode model. Various methods for characterizing chaos/hyperchaos including bifurcation diagrams, Lyapunov exponents spectrum, frequency spectra, phase portraits, Poincare sections and two parameter Lyapunov exponents diagrams are exploited to point out the rich dynamical behaviors in the model. Numerical results indicate that the system displays extremely rich dynamical behaviors including periodic windows, torus, chaotic and hyperchaotic oscillations. One of the main findings of this work is the presence of a region in the parameter space in which the mTCMNL experiences hysteretic behaviors. This later singularity/phenomenon is marked by the coexistence of multiple attractors (i.e., coexistence of asymmetric pair of periodic, torus and chaotic attractors with symmetric periodic, torus and chaotic attractors), for the same parameters settings. Basins of attractions of various competing attractors depicts symmetric complex basin boundaries, thus suggesting possible jumps between coexisting solutions (i.e., asymmetric pair of attractors with symmetric one) in experiments. A predominant route to chaos/hyperchaos observed in the system for different system parameters is the Afraimovich–Shilnikov scenario with tiny periodic regions. Experimental results from real-time circuit implementation are in good agreement with numerical analysis.

Bifurcation and Chaos in Integer and Fractional Order Two-Degree-of-Freedom Shape Memory Alloy Oscillators

A two-degree-of-freedom shape memory oscillator derived using polynomial constitutive model is investigated. Periodic, quasiperiodic, chaotic, and hyperchaotic oscillations are shown by the shape memory alloy based oscillator for selected values of the operating temperatures and excitation parameters. Bifurcation plots are derived to investigate the system behavior with change in parameters. A fractional order model of the shape memory oscillator is presented and dynamical behavior of the system with fractional orders and parameters are investigated.

Возбуждение хаотических и квазипериодических колебаний в многоконтурном генераторе с общей схемой управления

AbstractWe propose a multicircuit oscillator with a common control scheme driving self-sustained oscillations in each circuit, which can exhibit quasi-periodic and chaotic oscillations. The proposed oscillator was investigated by experimental and numerical methods that confirmed the possibility of exciting multifrequency quasi-periodic, chaotic, and hyperchaotic oscillations with several positive Lyapunov exponents.

A simple chaotic and hyperchaotic time-delay system: design and electronic circuit implementation

The present paper reports the design, analysis and experimental implementation of a simple first-order nonlinear time-delay dynamical system, which is capable of showing chaotic and hyperchaotic oscillations even at low values of intrinsic time delay. The system consists of a nonlinearity that has a closed-form mathematical function, which enables one to derive exact stability and bifurcation conditions. A detailed stability and bifurcation analyses reveal that a limit cycle is born via a supercritical Hopf bifurcation. Also, we observe both mono-scroll and double-scroll chaotic and hyperchaotic attractors even for a small value of time delay. The complexity of the system is characterized by bifurcation diagram and Lyapunov exponent spectrum. We implement the system in an electronic circuit, and a data acquisition (DAQ) system with LabView environment is used to visualize the experimental bifurcation diagram along with the other characteristic diagrams, namely time series waveform, phase plane plots and frequency spectra. Experimental observations agree well with the analytical and numerical results.

Characterization of the chaos-hyperchaos transition based on return times.

We discuss the problem of the detection of hyperchaotic oscillations in coupled nonlinear systems when the available information about this complex dynamical regime is very limited. We demonstrate the ability of diagnosing the chaos-hyperchaos transition from return times into a Poincaré section and show that an appropriate selection of the secant plane allows a correct estimation of two positive Lyapunov exponents (LEs) from even a single sequence of return times. We propose a generalized approach for extracting dynamics from point processes that allows avoiding spurious identification of the dynamical regime caused by artifacts. The estimated LEs are nearly close to their expected values if the second positive LE is essentially different from the largest one. If both exponents become nearly close, an underestimation of the second LE may be obtained. Nevertheless, distinctions between chaotic and hyperchaotic regimes are clearly possible.

Design of Chaotic and Hyperchaotic Time-Delayed Electronic Circuit

The present paper reports a first order nonlinear retarded type time-delayed chaotic and hyperchaotic electronic circuit. The proposed circuit has three distinct advantages over the existing time-delayed circuits. First, it has a nonlinearity that is expressed by closed form mathematical functions, which makes the analysis and design of the circuit easier. Second, the time-delay part of the proposed circuit is realized with an active All-Pass Filter (APF), in which no inductor is used, and the variation of delay is obtained simply by the variation of a resistor, which is more advantageous than to vary the inductor in LCL delay blocks that is widely used in all the time-delayed circuits existing in the literature. Third, the circuit shows hyperchaos even for a moderate time-delay. We describe the systematic design procedure of the circuit, and whenever necessary, the experimental results are corroborated by the numerical computations. We show that the circuit shows limit cycle oscillation, bifurcation scenario, chaotic and hyperchaotic oscillations.

Complete and generalized synchronization of chaos and hyperchaos in a coupled first-order time-delayed system

This paper explores the synchronization scenario of coupled chaotic and hyperchaotic time delay systems that are coupled through linear, dissipative and unidirectional coupling. For the present study, we choose a prototype first-order nonlinear time delay system, which is recently reported in Banerjee et al. (Nonlinlinear. Dyn., doi:10.1007/s11071-012-0490-3, 2012); the system shows well-characterized chaotic and hyperchaotic oscillations even for a small time delay, and also, experimental implementation of the system is easy. We show that, keeping all the system design parameters the same for the two systems, if the time delays associated with the two systems are equal, then complete synchronization occurs beyond a threshold coupling strength. On the contrary, above a certain coupling strength, generalized synchronization between two identical coupled systems occurs for the unequal time delays. We derive an estimate of the coupling strength and sufficient stability conditions for all the synchronization processes using Krasovskii–Lyapunov theory. We simulate the coupled system numerically to support the analytical results. Also, we implement the coupled system in an electronic circuit to verify all the synchronization phenomena. It is shown that the experimental results agree well with our analytical and numerical results.

HYPERCHAOS IN A CHUA'S CIRCUIT WITH TWO NEW ADDED BRANCHES

In this work a fourth-order Chua's circuit, capable of generating hyperchaotic oscillations in a wide range of parameters, is presented. The circuit is obtained by adding two new branches to the original topology of the Chua's double scroll circuit. One of the added branches is a linear inductor-resistor series connection, and the other one is a nonlinear voltage-controlled current source. A theoretical analysis of the circuit equations is presented, along with numerical and experimental results.

FAST-SCALE HYPERCHAOS ON TOP OF SLOW-SCALE PERIODICITY IN DELAYED DYNAMICAL SYSTEMS

We show the coexistence of a fast-scale hyperchaos with a slow-scale periodic behavior in an integro-differential delayed dynamical system. This behavior arises after a sequence of Hopf bifurcations. This new phenomenology is experimentally evidenced with a Mach-Zehnder modulator optically fed by a semiconductor laser, and subjected to a delayed nonlinear electro-optical feedback. A “breathing” dynamics is therefore observed, consisting of ns-timescale hyperchaotic oscillations with a periodic modulation at a μs-timescale. A theoretical understanding of the phenomenon is also provided.

Complex dynamical behaviours in two non-linearly coupled Chua’s circuits

The complex behaviour in the dynamics of a sixth-order circuit oscillator is presented. The oscillator is made up of two smooth Chua’s circuits, which have been properly bi-directionally and non-linearly coupled. The coupled system exhibits synchronization, antisynchronization, hyperchaos and on–off intermittency. The occurrence of hyperchaotic oscillations is marked by the loss of transverse stability of a chaotic attractor belonging to an invariant subspace. The transverse stability condition of such an attractor has been numerically analyzed by referring to both the properties of the flow transverse to the attractor and its linearization. The presence of a blowout bifurcation is confirmed by the calculus of Lyapunov exponents and by the presence of on–off intermittency. Moreover the occurrence of antisynchronization under the hypothesis of weak coupling and of synchronization under the hypothesis of tight coupling are analytically demonstrated and numerically verified through Lyapunov exponents.

Stabilization of an unstable steady state in a Mackey-Glass system

We present theoretical and experimental results of stabilizing an unstable steady state in a Mackey-Glass system and its electronic analog driven into regions of hyperchaotic oscillations. Three stabilization methods based on conventional feedback, tracking filter, and delayed feedback are considered.