scholarly journals On the existence of periodic orbits and KAM tori in the Sprott A system: a special case of the Nosé–Hoover oscillator

2018 ◽  
Vol 92 (3) ◽  
pp. 1287-1297 ◽  
Author(s):  
Marcelo Messias ◽  
Alisson C. Reinol
Keyword(s):  
Periodic Orbits ◽  
Kam Tori ◽  
System A ◽  
Sprott A System ◽  
Special Case

scholarly journals Symbolic Encoding of Periodic Orbits and Chaos in the Rucklidge System

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Chengwei Dong ◽  
Lian Jia ◽  
Qi Jie ◽  
Hantao Li
Keyword(s):  
Periodic Orbits ◽  
Nonlinear Dynamical Systems ◽  
Unstable Periodic Orbits ◽  
Nonlinear Dynamical ◽  
System A ◽  
Return Maps ◽  
Phase Portrait Analysis ◽  
Encoding Method ◽  
First Return Maps ◽  
Parameter Values

To describe and analyze the unstable periodic orbits of the Rucklidge system, a so-called symbolic encoding method is introduced, which has been proven to be an efficient tool to explore the topological properties concealed in these periodic orbits. In this work, the unstable periodic orbits up to a certain topological length in the Rucklidge system are systematically investigated via a proposed variational method. The dynamics in the Rucklidge system are explored by using phase portrait analysis, Lyapunov exponents, and Poincaré first return maps. Symbolic encodings of the periodic orbits with two and four letters based on the trajectory topology in the phase space are implemented under two sets of parameter values. Meanwhile, the bifurcations of the periodic orbits are explored, significantly improving the understanding of the dynamics of the Rucklidge system. The multiple-letter symbolic encoding method could also be applicable to other nonlinear dynamical systems.

scholarly journals Stability of Periodic Planetary-Type Orbits of the General Planar N-Body Problem

1979 ◽  
Vol 81 ◽  
pp. 23-28
Author(s):  
John D. Hadjidemetriou
Keyword(s):  
Periodic Orbits ◽  
Body Problem ◽  
Frame Of Reference ◽  
Rotating Frame ◽  
N Body Problem ◽  
Families Of Periodic Orbits ◽  
Special Case

It is known that families of periodic orbits in the general N-body problem (N≥3) exist, in a rotating frame of reference (Hadjidemetriou 1975, 1977). A special case of the above families of periodic orbits are the periodic orbits of the planetary type. In this latter case only one body, which we shall call sun, is the more massive one and the rest N-1 bodies, which we shall call planets, have small but not negligible masses. The aim of this paper is to study the properties of the families of periodic planetary-type orbits, with particular attention to stability. To make the presentation clearer, we shall start first with the case N=3 and we shall extend the results to N>3. We shall discuss planar orbits only.

scholarly journals Control of Chaos in the Rössler System

1997 ◽  
Vol 50 (2) ◽  
pp. 263 ◽  
Author(s):  
Stuart Corney
Keyword(s):  
Periodic Orbits ◽  
Time Series Data ◽  
Control Method ◽  
Three Dimensional ◽  
Dynamical Variable ◽  
Series Data ◽  
External Parameter ◽  
Unstable Periodic Orbits ◽  
System A ◽  
Linear Differential

The control method of Ott, Grebogi and Yorke (1990) as applied to the Rössler system, a set of three-dimensional non-linear differential equations, is examined. Using numerical time series data for a single dynamical variable the method was successfully employed to control several of the unstable periodic orbits in a three-dimensional embedding of the data. The method also failed for a number of unstable periodic orbits due to difficulties in linearising about the orbit or the tangential coincidence of the stable manifold and the motion of the orbit with external parameter.

TIME-DELAYED IMPULSIVE CONTROL OF CHAOTIC HYBRID SYSTEMS

2004 ◽  
Vol 14 (03) ◽  
pp. 1091-1104 ◽  
Author(s):  
YU-PING TIAN ◽  
XINGHUO YU ◽  
LEON O. CHUA
Keyword(s):  
Hybrid Systems ◽  
Periodic Orbits ◽  
Special Class ◽  
Manufacturing Systems ◽  
Impulsive Control ◽  
Prototype Model ◽  
Piecewise Affine Systems ◽  
Piecewise Affine ◽  
System A ◽  
Feedback Method

This paper presents a time-delayed impulsive feedback approach to the problem of stabilization of periodic orbits in chaotic hybrid systems. The rigorous stability analysis of the proposed method is given. Using the time-delayed impulsive feedback method, we analyze the problem of detecting various periodic orbits in a special class of hybrid system, a switched arrival system, which is a prototype model of many manufacturing systems and computer systems where a large amount of work is processed in a unit time. We also consider the problem of stabilization of periodic orbits of chaotic piecewise affine systems, especially Chua's circuit, which is another important special class of hybrid systems.

An Analysis for Crack Instability in Elastically Confined Yielding Regime

1984 ◽  
Vol 106 (4) ◽  
pp. 488-494 ◽  
Author(s):  
A. Zahoor
Keyword(s):  
Control Mode ◽  
Small Scale ◽  
Small Scale Yielding ◽  
Yield Model ◽  
System A ◽  
Strip Yield Model ◽  
Loading System ◽  
Scale Yielding ◽  
Crack Tip Plasticity ◽  
Special Case

An analysis for crack instability is presented which utilizes a J-integral-based tearing modulus approach. In particular, a plane stress center-cracked panel which experiences elastically confined yielding is analyzed for a displacement controlled loading. The analysis assumes a compliant loading system, a special case of which leads to a load control mode of loading. The effects of the crack tip plasticity are taken into account by using the strip-yield model of Dugdale-Barenblatt. A method of predicting the amount of crack growth at the onset of instability is presented. Numerical results suggest that under conditions of small-scale yielding, crack instability can be achieved in materials having very low tearing modulus values.

scholarly journals Theoretical and experimental evidence of non-symmetric doubly localized rogue waves

2014 ◽  
Vol 470 (2171) ◽  
pp. 20140318 ◽  
Author(s):  
Jingsong He ◽  
Lijuan Guo ◽  
Yongshuai Zhang ◽  
Amin Chabchoub
Keyword(s):  
Deep Water ◽  
Rogue Waves ◽  
Nls Equation ◽  
Third Order ◽  
Weakly Nonlinear ◽  
Wave Flume ◽  
Symmetric Pattern ◽  
System A ◽  
Special Case ◽  
Good Agreement

We present determinant expressions for vector rogue wave (RW) solutions of the Manakov system, a two-component coupled nonlinear Schrödinger (NLS) equation. As a special case, we generate a family of exact and non-symmetric RW solutions of the NLS equation up to third order, localized in both space and time. The derived non-symmetric doubly localized second-order solution is generated experimentally in a water wave flume for deep-water conditions. Experimental results, confirming the characteristic non-symmetric pattern of the solution, are in very good agreement with theory as well as with numerical simulations, based on the modified NLS equation, known to model accurately the dynamics of weakly nonlinear wave packets in deep water.

A Generalization of Poincaré’s Theorem to Periodic Hybrid and Impulsive Dynamical Systems

2001 ◽  
Author(s):  
V. Chellaboina ◽  
S. G. Nersesov ◽  
W. M. Haddad
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Discrete Time ◽  
Discrete Time System ◽  
Time System ◽  
Stability Of Periodic Orbits ◽  
Poincaré Return Map ◽  
The Stability ◽  
Time Periodic ◽  
Special Case

Abstract Poincaré’s method is well known for analyzing the stability of continuous-time periodic dynamical systems by studying the stability properties of a fixed point as an equilibrium point of a discrete-time system. In this paper we generalize Poincaré’s method to dynamical systems possessing left-continuous flows to address the stability of limit cycles and periodic orbits of left-continuous, hybrid, and impulsive dynamical systems. It is shown that resetting manifold (which gives rise to the state discontinuities) provides a natural hyperplane for defining a Poincaré return map. In the special case of impulsive dynamical systems, we show the Poincaré map replaces an nth-order impulsive dynamical system by an (n − 1)th-order discrete-time system for analyzing the stability of periodic orbits.

Periodic Orbits in a Second-Order Discontinuous System with an Elliptic Boundary

2016 ◽  
Vol 26 (13) ◽  
pp. 1650224 ◽  
Author(s):  
Liping Li ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Orbits ◽  
Vector Fields ◽  
Sliding Mode ◽  
Elliptic Boundary ◽  
Second Order ◽  
Discontinuous System ◽  
System A ◽  
Discontinuous Dynamical System

This paper develops the analytical conditions for the onset and disappearance of motion passability and sliding along an elliptic boundary in a second-order discontinuous system. A periodically forced system, described by two different linear subsystems, is considered mainly to demonstrate the methodology. The passable, sliding and grazing conditions of a flow to the elliptic boundary in the discontinuous dynamical system are provided through the analysis of the corresponding vector fields and [Formula: see text]-functions. Moreover, by constructing appropriate generic mappings, periodic orbits in such a discontinuous system are predicted analytically. Finally, three different cases are discussed to illustrate the existence of periodic orbits with passable and/or sliding flows. The results obtained in this paper can be applied to the sliding mode control in discontinuous dynamical systems.

scholarly journals Existence and uniqueness of periodic orbits in a discrete model on Wolbachia infection frequency

2021 ◽  
Vol 11 (1) ◽  
pp. 212-224
Author(s):  
Bo Zheng ◽  
Jianshe Yu
Keyword(s):  
Periodic Solution ◽  
Periodic Orbits ◽  
Discrete Model ◽  
Wolbachia Infection ◽  
Discrete Models ◽  
Infection Frequency ◽  
Globally Asymptotically Stable ◽  
Release Ratio ◽  
Impulsive Release ◽  
Special Case

Abstract In this paper, we study a discrete model on Wolbachia infection frequency. Assume that a periodic and impulsive release strategy is implemented, where infected males are released during the first N generations with the release ratio α, and the release is terminated from (N + 1)-th generation to T-th generation. We find a release ratio threshold denoted by α *(N, T), and prove the existence of a T-periodic solution for the model when α ∈ (0, α *(N, T)). For the special case when N = 1 and T = 2, we prove that the model has a unique T-periodic solution which is unstable when α ∈ (0, α *(N, T)). While α ≥ α *(N, T), no periodic phenomenon occurs and the Wolbachia fixation equilibrium is globally asymptotically stable. Numerical simulations are also provided to illustrate our theoretical results. One main contribution of this work is to offer a new method to determine the exact number of periodic orbits to discrete models.

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