scholarly journals PROBING THE LOCAL DYNAMICS OF PERIODIC ORBITS BY THE GENERALIZED ALIGNMENT INDEX (GALI) METHOD

2012 ◽  
Vol 22 (09) ◽  
pp. 1250218 ◽  
Author(s):  
T. MANOS ◽  
CH. SKOKOS ◽  
CH. ANTONOPOULOS
Keyword(s):  
Periodic Orbits ◽  
Nonlinear Dynamical Systems ◽  
Power Laws ◽  
Oscillatory Behavior ◽  
Unstable Periodic Orbits ◽  
Symplectic Maps ◽  
Local Dynamics ◽  
Hamiltonian Flows ◽  
Homoclinic Tangle ◽  
Stable Periodic

As originally formulated, the Generalized Alignment Index (GALI) method of chaos detection has so far been applied to distinguish quasiperiodic from chaotic motion in conservative nonlinear dynamical systems. In this paper, we extend its realm of applicability by using it to investigate the local dynamics of periodic orbits. We show theoretically and verify numerically that for stable periodic orbits, the GALIs tend to zero following particular power laws for Hamiltonian flows, while they fluctuate around nonzero values for symplectic maps. By comparison, the GALIs of unstable periodic orbits tend exponentially to zero, both for flows and maps. We also apply the GALIs for investigating the dynamics in the neighborhood of periodic orbits, and show that for chaotic solutions influenced by the homoclinic tangle of unstable periodic orbits, the GALIs can exhibit a remarkable oscillatory behavior during which their amplitudes change by many orders of magnitude. Finally, we use the GALI method to elucidate further the connection between the dynamics of Hamiltonian flows and symplectic maps. In particular, we show that, using the components of deviation vectors orthogonal to the direction of motion for the computation of GALIs, the indices of stable periodic orbits behave for flows as they do for maps.

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 THE STRUCTURE AND EVOLUTION OF CONFINED TORI NEAR A HAMILTONIAN HOPF BIFURCATION

2011 ◽  
Vol 21 (08) ◽  
pp. 2321-2330 ◽  
Author(s):  
M. KATSANIKAS ◽  
P. A. PATSIS ◽  
G. CONTOPOULOS
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbits ◽  
Transition Point ◽  
Initial Conditions ◽  
Unstable Periodic Orbits ◽  
Unstable Periodic Orbit ◽  
Surface Of Section ◽  
Autonomous Hamiltonian System ◽  
Hamiltonian Hopf Bifurcation ◽  
Stable Periodic

We study the orbital behavior at the neighborhood of complex unstable periodic orbits in a 3D autonomous Hamiltonian system of galactic type. At a transition of a family of periodic orbits from stability to complex instability (also known as Hamiltonian Hopf Bifurcation) the four eigenvalues of the stable periodic orbits move out of the unit circle. Then the periodic orbits become complex unstable. In this paper, we first integrate initial conditions close to the ones of a complex unstable periodic orbit, which is close to the transition point. Then, we plot the consequents of the corresponding orbit in a 4D surface of section. To visualize this surface of section we use the method of color and rotation [Patsis & Zachilas, 1994]. We find that the consequents are contained in 2D "confined tori". Then, we investigate the structure of the phase space in the neighborhood of complex unstable periodic orbits, which are further away from the transition point. In these cases we observe clouds of points in the 4D surfaces of section. The transition between the two types of orbital behavior is abrupt.

Periodic orbits and chaos around two fixed black holes. II

1991 ◽  
Vol 435 (1895) ◽  
pp. 551-562 ◽  
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Periodic Orbits ◽  
Satellite Orbits ◽  
Unstable Periodic Orbits ◽  
Zero Velocity ◽  
Stable Periodic

We study the orbits of particles (time-like geodesics) around two fixed black holes when the energy is elliptic, i. e. it does not allow the motion to extend to infinity. Most orbits are chaotic, but in many cases there are also ordered motions around stable periodic orbits. The orbits that fall into the first or the second black hole are separated by unstable periodic orbits. These are the satellite periodic orbits around the black holes when they exist. But for certain intervals of parameters there are no satellite orbits around the first or the second black hole. Then the limiting orbits are like arcs of hyperbolae, reaching the curve of zero velocity.

scholarly journals Time Series and the Dynamics of Demand Pacing

2000 ◽  
Vol 39 (02) ◽  
pp. 114-117
Author(s):  
D. T. Kaplan
Keyword(s):  
Time Series ◽  
Periodic Orbits ◽  
Chaotic Systems ◽  
Unstable Periodic Orbits ◽  
Periodic Response ◽  
Stable Periodic

Abstract:Motivated by a common practice in cardiology, we analyze the dynamics of a demand paced system where one seeks to create a stable periodic response. By using techniques originally developed for controlling chaotic systems, one can enhance the information contained in time series regarding hidden, unstable periodic orbits. This makes it possible, for example, to track drifts in a system‘s dynamics.

RECURRENCE OF ORDER IN CHAOS

2005 ◽  
Vol 15 (09) ◽  
pp. 2865-2882 ◽  
Author(s):  
G. CONTOPOULOS ◽  
M. HARSOULA ◽  
R. DVORAK ◽  
F. FREISTETTER
Keyword(s):  
Periodic Orbits ◽  
Period Doubling ◽  
Power Laws ◽  
Numerical Calculations ◽  
Bifurcation Ratio ◽  
Group Iii ◽  
Unperturbed Problem ◽  
Stable Periodic ◽  
Theoretical Results ◽  
Period Doubling Bifurcations

The standard map x′ = x + y′, y′ = y + (K/2π) sin (2πx), where both x and y are given modulo 1, becomes mostly chaotic for K ≥ 8, but important islands of stability appear in a recurrent way for values of K near K = 2nπ (groups of islands I and II), and K = (2n + 1)π (group III), where n ≥ 1. The maximum areas of the islands and the intervals ΔK, where the islands appear, follow power laws. The changes of the areas of the islands around a maximum follow universal patterns. All islands surround stable periodic orbits. Most of the orbits are irregular, i.e. unrelated to the orbits of the unperturbed problem K = 0. The main periodic orbits of periods 1, 2 and 4 and their stability are derived analytically. As K increases these orbits become unstable and they are followed by infinite period-doubling bifurcations with a bifurcation ratio δ = 8.72. We find theoretically the connections between the various families and the extent of their stability. Numerical calculations verify the theoretical results.

Lobe dynamics and the escape from a potential well

1991 ◽  
Vol 435 (1895) ◽  
pp. 659-672 ◽  
Keyword(s):  
Periodic Orbits ◽  
Invariant Manifolds ◽  
Numerical Technique ◽  
Basin Of Attraction ◽  
Invariant Sets ◽  
Potential Well ◽  
Unstable Periodic Orbits ◽  
Homoclinic Tangle ◽  
Lobe Dynamics ◽  
Basin Erosion

Periodically-forced nonlinear oscillators that permit escape from a potential well frequently possess unstable periodic orbits whose invariant manifolds are homoclinically tangled. The trellises formed by such overlapping manifolds separate the Poincaré plane into regions (called lobes ) whose fates at successive time periods are amenable to analysis. The lobe configurations observed in simple escape systems are discussed, together with their interrelation with the location of periodic orbits and other invariant sets. Three applications of this procedure are illustrated: (i) a numerical technique for localizing the non-wandering set; (ii) a demonstration that the existence of a saddle possessing a bounded homoclinically tangled unstable manifold branch implies the existence of safe open regions in the global basin of attraction; (iii) the proposal of definitions of global integrity and escape times which may be used to relate safe basin erosion to the evolution of a homoclinic tangle.

Calculation of stable and unstable periodic orbits in a chopper-fed DC drive

2020 ◽  
Vol 8 (1) ◽  
pp. 43-57
Author(s):  
O. O. Kuznyetsov ◽  
Keyword(s):  
Periodic Orbits ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Nonlinear Phenomena ◽  
Unstable Periodic Orbits ◽  
Border Collision Bifurcations ◽  
Stable Periodic ◽  
Main Period ◽  
Dc Drive ◽  
The Stability

It is well known that electric drives demonstrate various nonlinear phenomena. In particular, a chopper-fed analog DC drive system is characterized by the route to chaotic behavior though period-doubling cascade. Besides, the considered system demonstrates coexistence of several stable periodic modes within the stability boundaries of the main period-1 orbit. We discover the evolution of several periodic orbits utilizing the semi-analytical method based on the Filippov theory for the stability analysis of periodic orbits. We analyze, in particular, stable and unstable period-1, 2, 3 and 4 orbits, as well as independent on stability they are significant for the organization of phase space. We demonstrate, in particular, that the unstable periodic orbits undergo border collision bifurcations; those occur according to several scenarios related to the interaction of different orbits of the same period, including persistence border collision, when a periodic orbit is changed by a different orbit of the same period, and birth or disappearance of a couple of orbits of the same period characterized by different topology.

scholarly journals The Lyapunov exponents and the neighbourhood of periodic orbits

2020 ◽  
Vol 495 (2) ◽  
pp. 1608-1612
Author(s):  
D D Carpintero ◽  
J C Muzzio
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Periodic Orbits ◽  
Monodromy Matrix ◽  
Unstable Periodic Orbits ◽  
Centre Manifold ◽  
Chaotic Orbits ◽  
Galactic Potential ◽  
Stable Periodic ◽  
Centre Manifold Theorem

ABSTRACT We show that the Lyapunov exponents of a periodic orbit can be easily obtained from the eigenvalues of the monodromy matrix. It turns out that the Lyapunov exponents of simply stable periodic orbits are all zero, simply unstable periodic orbits have only one positive Lyapunov exponent, doubly unstable periodic orbits have two different positive Lyapunov exponents, and the two positive Lyapunov exponents of complex unstable periodic orbits are equal. We present a numerical example for periodic orbits in a realistic galactic potential. Moreover, the centre manifold theorem allowed us to show that stable, simply unstable, and doubly unstable periodic orbits are the mothers of families of, respectively, regular, partially, and fully chaotic orbits in their neighbourhood.

Periodic Orbits and Chaos in Nonsmooth Delay Differential Equations

2019 ◽  
Vol 29 (10) ◽  
pp. 1950137
Author(s):  
Andrea Bel ◽  
Romina Cobiaga ◽  
Walter Reartes
Keyword(s):  
Differential Equations ◽  
Periodic Orbits ◽  
Delay Differential Equations ◽  
Order Differential Equation ◽  
Stable Periodic Solution ◽  
Unstable Periodic Orbits ◽  
First Order ◽  
Stable Periodic ◽  
Delay Differential ◽  
Discrete Map

In this paper, we present a method to find periodic solutions for certain types of nonsmooth differential equations or nonsmooth delay differential equations. We apply the method to three examples, the first is a second-order differential equation with a nonsmooth term, in this case the method allows us to find periodic orbits in a nonlinear center. The two remaining examples are first-order nonsmooth delay differential equations. In the first one, there is a stable periodic solution and in the second, the presence of a chaotic attractor was detected. In the latter, the method allows us to obtain unstable periodic orbits within the attractor. For large values of the delay, both examples can be seen as singularly perturbed delay differential equations. For them, an analysis is performed with an associated discrete map which is obtained in the limit of large delays.

Periodic orbits and chaos around two black holes

1990 ◽  
Vol 431 (1881) ◽  
pp. 183-202 ◽  
Keyword(s):  
Black Holes ◽  
Periodic Orbits ◽  
Initial Conditions ◽  
Classical Problem ◽  
Period Doubling ◽  
Satellite Orbits ◽  
Unstable Periodic Orbits ◽  
Newtonian Case ◽  
Stable Periodic ◽  
Escaping Orbits

We calculate orbits of photons and particles in the relativistic problem of two extreme Reissner-Nordström black holes (fixed). In the case of photons there are three types of non-periodic orbits, namely orbits falling into the black holes M 1 and M 2 (types I and II), and orbits escaping to infinity (III). The various types of orbits are separated by orbits asymptotic to the three main types of (unstable) periodic orbits: ( a ) around M 1 , ( b ) around M 2 and ( c ) around both M 1 and M 2 . Between two non-periodic orbits of two different types there are orbits of the third type. The initial conditions of the three types of orbits form three Cantor sets, and this fact is a manifestation of chaos. The role of higher-order periodic orbits is explained. In the case of particles the situation is similar in the parabolic and hyperbolic cases. However, in the elliptic case the orbits are contained inside a curve of zero velocity, and there are no escaping orbits. Instead we have orbits trapped around stable periodic orbits (IV) and stochastic orbits not falling on the black holes M 1 and M 2 (V). The stable periodic orbits become unstable by period doubling, and infinite period doublings lead to chaos. The newtonian limit is an integrable problem (essentially the same as the classical problem of two fixed centres). The periodic orbits of type ( c ) are topologically similar to the corresponding relativistic orbits, but they differ considerably numerically. We prove that in the newtonian case there are no satellite orbits around M 1 or M 2 (of type ( a ) or ( b )). The post-newtonian case is non-integrable. In this case there are in general orbits of all three types ( a ), ( b ) and ( c ).

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