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We used a method based on calculating distributions for the angles between expanding, contracting, and neutral manifolds for trajectories on an attractor. Three specific examples are outlined.

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For two of them, the previously assumed hyperbolicity was confirmed. Farmer, Physica D,4 3] corresponds to Keysight DSOX4034A Oscilloscope chaos. Techniques that are useful for recognizing the types of complex dynamical behavior and allow one to distinguish a strange nonchaotic attractor from quasiperiodic and chaotic oscillations are developed and tested. A scheme was developed and a laboratory prototype of a rough hyperbolic chaos generator based on two alternately excited self-oscillating elements on field-effect transistors for a frequency range of MHz was compiled.

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The scheme of the "electronic rotator" - a system similar in the equation form to the mathematical pendulum on a basis of the available components for the laboratory design, was implemented, and circuit simulation in the Multisim environment was povided. In the participant of the project V. Kruglov defended the thesis "Finite-dimensional and distributed systems of ring structure that generate rough chaos" Keysight DSOX4034A Oscilloscope the scientific degree of candidate of physical and mathematical sciences supervisor S. Main results A research was performed within the framework of numerical and circuit modeling of the electronic scheme proposed at the previous stage of the project, where the Anosov dynamics on the surface of negative curvature is reproduced on the attractor of the system.

As elements of the electronic circuit, the phase-locked loop chains are used, which are analogs of rotators in mechanics, in which the states are characterized by a variable defined modulo 2pi and its derivative, which is a phase of the voltage-controlled generators with respect to the reference signal at fixed frequency.

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  • DSOX4034A Oscilloscope: 350 MHz, 4 Analog Channels

A comparative analysis of the dynamic behavior of models delivering different degrees of accuracy of the description is carried out. Schemes of alternately excited oscillating elements based on field-effect transistors lambda-diode and on tunnel diodes, for which, as expected, it will be possible to select parameters providing hyperbolic dynamics, are developed. A new simple scheme of chaos generator is proposed, realizing Keysight DSOX4034A Oscilloscope pseudo-hyperbolic attractor of the Lorentz type, and its functioning was demonstrated through modeling in the Multisim environment. The scheme is based on an equivalent representation of the Lorentz equations in the form of a nonlinear oscillator with the help of the variable change of Yudovich Kuznetsov S.

In the current version, the circuit uses an operational amplifier and Keysight DSOX4034A Oscilloscope analog multipliers, but they can apparently be replaced by elements on transistors reproducing their properties.

On the basis of the criterion of angles, for the scheme Keysight DSOX4034A Oscilloscope the Lorenz attractor, the property of pseudohyperbolicity is checked, which ensures robustness of chaos and consists in the fact that one invariant subspace is compressive and the other is characterized by an exponential extension of the volume for perturbation vectors of trajectories on the attractor. A bibliographic list was compiled on electronic chaos oscillators.

In the Multisim environment, the circuits were reproduced, and chaotic dynamics were simulated providing phase portraits and spectra for a number of chaos generators described in the literature. These results are supposed to be used as a encouragement for construction of modified circuits that provide generation of rough chaos. We have considerd ensembles of three and four oscillatory elements with a nonlinearity of sine, which can be realized, in particular, on the basis of non-identical Keysight DSOX4034A Oscilloscope junctions.

Numerical modeling of the dynamics is carried out; areas of various behaviors in the parameter space are revealed, including quasiperiodic dynamics on tori of different dimensions and chaos. As Keysight DSOX4034A Oscilloscope promising for realization of complex dynamics, systems composed of van der Pol oscillators are analyzed, namely, a system of five ring-connected oscillators and a spatially developed network of star-like type.

For the first system, variants with different types of connection were studied: The bifurcations Keysight DSOX4034A Oscilloscope during the transition from the Keysight DSOX4034A Oscilloscope torus to the four-frequency torus are considered. A possibility of a quasi-periodic Hopf bifurcation is discovered. For the second system, non-trivial scenarios for emergence and development of the cluster synchronization are identified and described when parameters are varied.

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On the basis of a method of checking transversality of the intersection of manifolds of trajectories on Keysight DSOX4034A Oscilloscope chaotic attractor the criterion of anglesthe hyperbolic nature of chaotic dynamics for several time-delay systems implementable as electronic devices has been confirmed. Applied Nonlinear Dynamics, 22,No 2, Dynamical characteristics of signasl produced by microwave oscillators of chaos in the form of gyroklistron and gyro-TWT with delayed feedback were considered.


A model of a multi-loop generator with a common control scheme has been developed. Numerical simulation and laboratory experiments demonstrated a possibility of formation of chaos and hyperchaos as a Keysight DSOX4034A Oscilloscope of destruction of the multifrequency quasiperiodic dynamics. MHz; 4 analog channels; Easily view signals on the large inch capacitive touch screen; Isolate signals in seconds with exclusive Zone touch triggering. With the Keysight DSOXA see more signal detail with the 1, wfms/s update rate. Expand your measurement capabilities with full upgradability: Add.

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