Chaos control in delayed phase space constructed by the Takens embedding theory

Abstract The problem of controlling the chaotic dynamics of an inverted pendulum is addressed. Starting from the optimal excitation obtained in previous works, which gives satisfactory results but only on one side of the phase space, two different implementations have been developed in order to improve the controlled feature of the system. The implementations consist in a shrewd alternation of one-side optimal and of harmonic excitation, and in an alternation of right and left optimal excitations. They are aimed at reducing the scattered nature of the response, which has been synthetized by the number of jumps between the two potential wells. Some numerical simulations have been performed, showing the effectiveness of the proposed procedure. Furthermore, the comparison between the two cases has been discussed on both a local and a global basis.

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Control of Continuous Time Chaotic Systems With Unknown Dynamics and Limitation on State Measurement

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4041968 ◽

2018 ◽

Vol 14 (1) ◽

Cited By ~ 2

Author(s):

Hojjat Kaveh ◽

Hassan Salarieh

Keyword(s):

Chaotic Attractor ◽

Chaos Control ◽

Chaotic Systems ◽

Control Method ◽

Original System ◽

Control Of Chaos ◽

State Measurement ◽

New Chaotic Attractor ◽

Systematic Control ◽

Takens Embedding

This paper has dedicated to study the control of chaos when the system dynamics is unknown and there are some limitations on measuring states. There are many chaotic systems with these features occurring in many biological, economical and mechanical systems. The usual chaos control methods do not have the ability to present a systematic control method for these kinds of systems. To fulfill these strict conditions, we have employed Takens embedding theorem which guarantees the preservation of topological characteristics of the chaotic attractor under an embedding named “Takens transformation.” Takens transformation just needs time series of one of the measurable states. This transformation reconstructs a new chaotic attractor which is topologically similar to the unknown original attractor. After reconstructing a new attractor its governing dynamics has been identified. The measurable state of the original system which is one of the states of the reconstructed system has been controlled by delayed feedback method. Then the controlled measurable state induced a stable response to all of the states of the original system.

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A one-dimensional self-gravitating stellar gas

Symposium - International Astronomical Union ◽

10.1017/s0074180900105261 ◽

1966 ◽

Vol 25 ◽

pp. 46-48 ◽

Cited By ~ 2

Author(s):

M. Lecar

Keyword(s):

Gravitational Field ◽

Phase Space ◽

Motion Picture ◽

Space Distribution ◽

Two Dimensional ◽

One Dimensional ◽

Phase Space Distribution ◽

Dimensional Phase Space ◽

The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

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