On the piecewise-linear approximation of the polynomial chaotic dynamics

Petržela, Jiří

doi:10.1109/tsp.2011.6043717

Cited by 17 publications

(18 citation statements)

References 8 publications

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“…Currently, knowledge of the basins of attraction is used e.g. in chaos circuit, robotics, engineering systems and neural networks [23][24][25][26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

Variations of Boundary Surface in Chua’s Circuit

Guzan¹

2015

RADIOENGINEERING

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The paper compares the boundary surfaces using cross-sections in three projection planes, for the four changes of Chua's circuit parameters. It is known that due to changing the parameters, the Chua's circuit can be characterized not only by stable limit cycle, but also by one double scroll chaotic attractor, two single scroll chaotic attractors or other two stable limit cycles. Chua's circuit can even start working as a binary memory. It is not known yet, how changes in parameters, and consequently in attractors in the circuit, will affect the morphology of the boundary surface. The boundary surface separates the double scroll chaotic attractor from the stable limit cycle. In a variation of the parameters presented in this paper, the boundary surface will separate even single scroll chaotic attractors from each other. Dividing the state space into regions of attraction for different attractors, however, remains fundamentally the same.

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“…Currently, knowledge of the basins of attraction is used e.g. in chaos circuit, robotics, engineering systems and neural networks [23][24][25][26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

Variations of Boundary Surface in Chua’s Circuit

Guzan¹

2015

RADIOENGINEERING

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“…This single real number can be adopted in routine for optimal approximation of the polynomial chaotic dynamics. Some simple examples can be found in [34].…”

Section: Discussionmentioning

confidence: 99%

New Chaotic Dynamical System with a Conic-Shaped Equilibrium Located on the Plane Structure

Petržela

Götthans

2017

Applied Sciences

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Featured Application: Theoretical foundations provided in this contribution demonstrate that the existence of chaos is not bound to mathematical models and electronic circuits with singular equilibrium points. The practical part of this work proves that a structurally stable hidden chaotic attractor can be both excited and experimentally observed in a lumped electronic circuit; that is, it can be measured by a digital oscilloscope.Abstract: This paper presents a new autonomous deterministic dynamical system with equilibrium degenerated into a plane-oriented hyperbolic geometrical structure. It is demonstrated via numerical analysis and laboratory experiments that the discovered system has both a structurally stable strange attractor and experimentally measurable chaotic behavior. It is shown that the evolution of complex dynamics can be associated with a single parameter of a mathematical model and, due to one-to-one correspondence, to a single circuit parameter. Two-dimensional high resolution plots of the largest Lyapunov exponent and basins of attraction expressed in terms of final state energy are calculated and put into the context of the discovered third-order mathematical model and real chaotic oscillator. Both voltage-and current-mode analog chaotic oscillators are presented and verified by visualization of the typical chaotic attractor in a different fashion.

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“…A successive application of this algorithm is demonstrated via few examples in Refs. [66][67][68]. Proposed algorithm can be used for the detection of chaotic motion in the real physical system; both continuous 69 and discrete.…”

Section: Mathematical Backgroundmentioning

confidence: 99%

Current-Mode Network Structures Dedicated for Simulation of Dynamical Systems with Plane Continuum of Equilibrium

Petržela

Götthans

Guzan

2018

J CIRCUIT SYST COMP

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This review paper describes di®erent lumped circuitry realizations of the chaotic dynamical systems having equilibrium degeneration into a plane object with topological dimension of the equilibrium structure equals one. This property has limited amount (but still increasing, especially recently) of third-order autonomous deterministic dynamical systems. Mathematical models are generalized into classes to design analog networks as universal as possible, capable of modeling the rich scale of associated dynamics including the so-called chaos. Reference state trajectories for the chaotic attractors are generated via numerical analysis. Since used active devices can be precisely approximated by using third-level frequency dependent model, it is believed that computer simulations are close enough to capture real behavior. These simulations are included to demonstrate the existence of chaotic motion.

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On the piecewise-linear approximation of the polynomial chaotic dynamics

Cited by 17 publications

References 8 publications

Variations of Boundary Surface in Chua’s Circuit

Variations of Boundary Surface in Chua’s Circuit

New Chaotic Dynamical System with a Conic-Shaped Equilibrium Located on the Plane Structure

Current-Mode Network Structures Dedicated for Simulation of Dynamical Systems with Plane Continuum of Equilibrium

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