Periodic shadowing

Coomes, Brian A.; Koçak, Hüseyin; Palmer, Kenneth J.

doi:10.1090/conm/172/01801

Cited by 19 publications

(25 citation statements)

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“…Method 1 [3], [8], [14] The multiple shooting method is one of standard methods to compute a periodic orbit of (1). This method divides the periodic orbit with the period T into M + 1 sub-orbits such that…”

Section: The Multiple Shooting Methods For Periodic Orbitsmentioning

confidence: 99%

“…There are various numerical methods to solve periodic orbits of (1) such as the shooting method and the collocation method [3], [8], and some standard bifurcation analysis tools such as AUTO [4] are easily available. On the other hand, it has been reported that these commonly used numerical methods can incorrectly catch stability of periodic orbits under some conditions, even if the periodic orbits were accurately computed [5], [9], [10].…”

Section: Introductionmentioning

confidence: 99%

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Computation of Floquet Multipliers Using an Iterative Method for Variational Equations

Nureki

Murashige

2009

IEICE Trans. Fundamentals

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SUMMARYThis paper proposes a new method to numerically obtain Floquet multipliers which characterize stability of periodic orbits of ordinary differential equations. For sufficiently smooth periodic orbits, we can compute Floquet multipliers using some standard numerical methods with enough accuracy. However, it has been reported that these methods may produce incorrect results under some conditions. In this work, we propose a new iterative method to compute Floquet multipliers using eigenvectors of matrix solutions of the variational equations. Numerical examples show effectiveness of the proposed method.

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Section: The Multiple Shooting Methods For Periodic Orbitsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Computation of Floquet Multipliers Using an Iterative Method for Variational Equations

Nureki

Murashige

2009

IEICE Trans. Fundamentals

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“…[1][2][3][4] This abundance of UPOs is utilized in a variety of theoretical and practical applications and provides a useful way of investigating a chaotic system since many of its properties can be expressed in terms of these orbits. [5][6][7][8][9] Another approach is to rapidly direct trajectories to desired locations in phase space, a process known as targeting. [10][11][12] Several algorithms have since been developed that combine both types of chaos control techniques in order to switch a chaotic system from one stabilized periodic orbit to another.…”

Section: Introductionmentioning

confidence: 99%

“…Only small perturbations are required to generate cupolets, which in turn suggests that they are shadowing true periodic orbits and theorems have been developed that establish conditions under which this holds. [5][6][7]17 II. GENERATING CUPOLETS Our method for producing cupolets is derived from a chaotic communication scheme developed by Hayes, Grebogi, and Ott (HGO).…”

Section: Introductionmentioning

confidence: 99%

Controlled transitions between cupolets of chaotic systems

Morena

Short

Cooke

2014

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We present an efficient control scheme that stabilizes the unstable periodic orbits of a chaotic system. The resulting orbits are known as cupolets and collectively provide an important skeleton for the dynamical system. Cupolets exhibit the interesting property that a given sequence of controls will uniquely identify a cupolet, regardless of the system's initial state. This makes it possible to transition between cupolets, and thus unstable periodic orbits, simply by switching control sequences. We demonstrate that although these transitions require minimal controls, they may also involve significant chaotic transients unless carefully controlled. As a result, we present an effective technique that relies on Dijkstra's shortest path algorithm from algebraic graph theory to minimize the transients and also to induce certainty into the control of nonlinear systems, effectively providing an efficient algorithm for the steering and targeting of chaotic systems. Efficiently directing a dynamical system to a desired state is a goal of many engineering applications. This process is known as "targeting" and is often achievable using small nonlinear controls. For a chaotic system, the goal of many control methods is either to stabilize the system onto one of its many unstable periodic orbits (UPOs) or to induce the system to transition between its UPOs until a targeted orbit is attained. The set of UPOs plays a significant role in determining the dynamics of a chaotic system and is said to form the skeleton of an associated attractor. In this paper, we combine a particularly effective chaos control method with algebraic graph theory and present a new technique allowing for the efficient transitioning between periodic orbits of chaotic systems. This allows one to navigate efficiently around a chaotic attractor and in physical applications conserve energy and power.

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“…[2,4,8]). In [3] Corless and Pilyugin proved that weak shadowing is a C 0 generic property for discrete dynamical systems of a compact smooth manifold M .…”

Section: Introductionmentioning

confidence: 99%