Three-Dimensional State Space and Chaos

2000 ◽  
pp. 117-156 ◽  
Author(s):  
Robert C. Hilborn
Keyword(s):  
State Space ◽  
Three Dimensional ◽  
Dimensional State Space

Efficient photon sorter in a high-dimensional state space

2011 ◽  
Vol 11 (3&4) ◽  
pp. 313-325
Author(s):  
Warner A. Miller
Keyword(s):  
Quantum Information ◽  
State Space ◽  
Quantum Key Distribution ◽  
Quantum Information Processing ◽  
Three Dimensional ◽  
High Dimensional ◽  
Coupled Mode Theory ◽  
Coupled Mode ◽  
Dimensional State Space ◽  
Significant Obstacle

An increase in the dimension of state space for quantum key distribution (QKD) can decrease its fidelity requirements while also increasing its bandwidth. A significant obstacle for QKD with qu$d$its ($d\geq 3$) has been an efficient and practical quantum state sorter for photons whose complex fields are modulated in both amplitude and phase. We propose such a sorter based on a multiplexed thick hologram, constructed e.g. from photo-thermal refractive (PTR) glass. We validate this approach using coupled-mode theory with parameters consistent with PTR glass to simulate a holographic sorter. The model assumes a three-dimensional state space spanned by three tilted planewaves. The utility of such a sorter for broader quantum information processing applications can be substantial.

scholarly journals The shape of higher-dimensional state space: Bloch-ball analog for a qutrit

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 485
Author(s):  
Christopher Eltschka ◽  
Marcus Huber ◽  
Simon Morelli ◽  
Jens Siewert
Keyword(s):  
State Space ◽  
Quantum System ◽  
Dimensional Space ◽  
Three Dimensional ◽  
The State ◽  
Level System ◽  
Three Dimensional Model ◽  
System A ◽  
Dimensional State Space ◽  
Geometric Intuition

Geometric intuition is a crucial tool to obtain deeper insight into many concepts of physics. A paradigmatic example of its power is the Bloch ball, the geometrical representation for the state space of the simplest possible quantum system, a two-level system (or qubit). However, already for a three-level system (qutrit) the state space has eight dimensions, so that its complexity exceeds the grasp of our three-dimensional space of experience. This is unfortunate, given that the geometric object describing the state space of a qutrit has a much richer structure and is in many ways more representative for a general quantum system than a qubit. In this work we demonstrate that, based on the Bloch representation of quantum states, it is possible to construct a three dimensional model for the qutrit state space that captures most of the essential geometric features of the latter. Besides being of indisputable theoretical value, this opens the door to a new type of representation, thus extending our geometric intuition beyond the simplest quantum systems.

Impact of control representations on efficiency of local nonholonomic motion planning

2011 ◽  
Vol 59 (2) ◽  
pp. 213-218 ◽  
Author(s):  
I. Duleba
Keyword(s):  
Motion Planning ◽  
State Space ◽  
Three Dimensional ◽  
Nonholonomic Systems ◽  
State Spaces ◽  
Controlled Systems ◽  
Higher Dimensional ◽  
Dimensional State Space ◽  
Optimal Motion Planning ◽  
Nonholonomic Motion Planning

Impact of control representations on efficiency of local nonholonomic motion planning In this paper various control representations selected from a family of harmonic controls were examined for the task of locally optimal motion planning of nonholonomic systems. To avoid dependence of results either on a particular system or a current point in a state space, considerations were carried out in a sub-space of a formal Lie algebra associated with a family of controlled systems. Analytical and simulation results are presented for two inputs and three dimensional state space and some hints for higher dimensional state spaces were given. Results of the paper are important for designers of motion planning algorithms not only to preserve controllability of the systems but also to optimize their motion.

A Polytopic System Approach for the Hybrid Control of a Diesel Engine Using VGT/EGR

2004 ◽  
Vol 127 (1) ◽  
pp. 13-21 ◽  
Author(s):  
Sorin Bengea ◽  
Ray DeCarlo ◽  
Martin Corless ◽  
Giorgio Rizzoni
Keyword(s):  
Diesel Engine ◽  
State Space ◽  
Hybrid Control ◽  
Three Dimensional ◽  
Control Design ◽  
Error Model ◽  
Operating Point ◽  
Third Order ◽  
Dimensional State Space ◽  
Operating Points

This paper develops a hybrid/gain scheduled controller for moving the state of a diesel engine through a driving profile represented as a sequence of operating points in the seven-dimensional state space of a mean value breathing nonlinear engine state model. The calculations for the control design are based on a third-order (reduced) model of the diesel engine, on whose state space the operating points are projected. About each operating point, we generate a third-order nonlinear error model in polytopic form. Using the polytopic error model at each operating point, a control design is set forth as a system of LMIs. The solution of each system of LMIs produces a norm bounded controller guaranteeing that xi−1d→xid where xid is the ith desired operating point in the three-dimensional state space. The control performance is then evaluated on the seventh order model.

Nonlinear Dynamics and Chaos

2019 ◽  
pp. 111-153
Author(s):  
David D. Nolte
Keyword(s):  
Nonlinear Dynamics ◽  
State Space ◽  
Autonomous Systems ◽  
Three Dimensional ◽  
Fractal Dimensions ◽  
Phase Portraits ◽  
Two Dimensional ◽  
Floquet Multipliers ◽  
Dimensional State Space ◽  
State Space System

The geometric structure of state space is understood through phase portraits and stability analysis that classify the character of fixed points based on the properties of Lyapunov exponents. Limit cycles are an important type of periodic orbit that occurs in many nonlinear systems, and their stability is analyzed according to Floquet multipliers. When time dependence is added to an autonomous two-dimensional state space system to make it non-autonomous, then chaos can emerge, as in the case of the driven damped pendulum. Autonomous systems with three-dimensional chaos include the Lorenz and Rössler models. Dissipative chaos often displays strange attractors with fractal dimensions.

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