Bifurcations of Periodic Orbits of a One-Dimensional Pre-Compressed Granular Array

2018 ◽  
Author(s):  
Gizem Dilber Acar ◽  
Balakumar Balachandran
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Bifurcation Point ◽  
Granular Media ◽  
Period Doubling ◽  
Initial Step ◽  
Hertzian Contact ◽  
One Dimensional ◽  
The Stability ◽  
External Forces

Bifurcations of periodic orbits of a one-dimensional granular array are numerically investigated in this study. A conservative two-bead system is considered without any damping or external forces. By using the Hertzian contact model, and confining the system’s total energy to a certain level, changes in in-phase periodic orbit are studied for various pre-compression levels. At a certain pre-compression level, symmetry breaking and period doubling occur, and an asymmetric period-two orbit emerges from the in-phase periodic orbit. Floquet analysis is conducted to study the stability of the in-phase periodic solution, and to detect the bifurcation location. Although the trajectory of period-two orbit is close to the in-phase orbit at the bifurcation point, the asymmetry of the period-two orbit becomes more pronounced as one moves away from the bifurcation point. This work is meant to serve as an initial step towards understanding how pre-compression may introduce qualitative changes in system dynamics of granular media.

Bifurcations of one- and two-dimensional maps

1984 ◽  
Vol 311 (1515) ◽  
pp. 43-102 ◽  
Keyword(s):  
Periodic Orbits ◽  
Period Doubling ◽  
Jacobian Determinant ◽  
Two Dimensional ◽  
Hénon Map ◽  
Stable And Unstable Manifolds ◽  
Henon Map ◽  
One Dimensional ◽  
The Family ◽  
Area Preserving

We study the qualitative dynamics of two-parameter families of planar maps of the form F^e(x, y) = (y, -ex+f(y)), where f :R -> R is a C 3 map with a single critical point and negative Schwarzian derivative. The prototype of such maps is the family f(y) = u —y 2 or (in different coordinates) f(y) = Ay(1 —y), in which case F^ e is the Henon map. The maps F e have constant Jacobian determinant e and, as e -> 0, collapse to the family f^. The behaviour of such one-dimensional families is quite well understood, and we are able to use their bifurcation structures and information on their non-wandering sets to obtain results on both local and global bifurcations of F/ ue , for small e . Moreover, we are able to extend these results to the area preserving family F/u. 1 , thereby obtaining (partial) bifurcation sets in the (/u, e)-plane. Among our conclusions we find that the bifurcation sequence for periodic orbits, which is restricted by Sarkovskii’s theorem and the kneading theory for one-dimensional maps, is quite different for two-dimensional families. In particular, certain periodic orbits that appear at the end of the one-dimensional sequence appear at the beginning of the area preserving sequence, and infinitely many families of saddle node and period doubling bifurcation curves cross each other in the ( /u, e ) -parameter plane between e = 0 and e = 1. We obtain these results from a study of the homoclinic bifurcations (tangencies of stable and unstable manifolds) of F /u.e and of the associated sequences of periodic bifurcations that accumulate on them. We illustrate our results with some numerical computations for the orientation-preserving Henon map.

Period Doubling Bifurcation Point Detection Strategy with Nested Layer Particle Swarm Optimization

2017 ◽  
Vol 27 (07) ◽  
pp. 1750101 ◽  
Author(s):  
Haruna Matsushita ◽  
Yusho Tomimura ◽  
Hiroaki Kurokawa ◽  
Takuji Kousaka
Keyword(s):  
Particle Swarm Optimization ◽  
Bifurcation Point ◽  
Particle Swarm ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Swarm Optimization ◽  
Detection Strategy ◽  
The Stability ◽  
Point Detection ◽  
Discrete Time Dynamical Systems

This paper proposes a bifurcation point detection strategy based on nested layer particle swarm optimization (NLPSO). The NLPSO is performed by two particle swarm optimization (PSO) algorithms with a nesting structure. The proposed method is tested in detection experiments of period doubling bifurcation points in discrete-time dynamical systems. The proposed method directly detects the parameters of period doubling bifurcation regardless of the stability of the periodic point, but require no careful initialization, exact calculation or Lyapunov exponents. Moreover, the proposed method is an effective detection technique in terms of accuracy, robustness usability, and convergence speed.

Effects of noise on periodic orbits of the logistic map

Open Physics ◽  
2008 ◽  
Vol 6 (3) ◽  
Author(s):  
Feng-guo Li
Keyword(s):  
Periodic Orbit ◽  
White Noise ◽  
Probability Density ◽  
Periodic Orbits ◽  
Gaussian Noise ◽  
Dynamical Behavior ◽  
Gaussian White Noise ◽  
Logistic Map ◽  
Period Doubling ◽  
Colored Gaussian Noise

AbstractNoise can induce an inverse period-doubling transition and chaos. The effects of noise on each periodic orbit of three different period sequences are investigated for the logistic map. It is found that the dynamical behavior of each orbit, induced by an uncorrelated Gaussian white noise, is different in the mergence transition. For an orbit of the period-six sequence, the maximum of the probability density in the presence of noise is greater than that in the absence of noise. It is also found that, under the same intensity of noise, the effects of uncorrelated Gaussian white noise and exponentially correlated colored (Gaussian) noise on the period-four sequence are different.

scholarly journals CONNECTING PERIOD-DOUBLING CASCADES TO CHAOS

2012 ◽  
Vol 22 (02) ◽  
pp. 1250022 ◽  
Author(s):  
EVELYN SANDER ◽  
JAMES A. YORKE
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Duffing Equation ◽  
Period Doubling Bifurcation ◽  
Bifurcation Diagrams ◽  
The Third ◽  
Infinite Collection

The appearance of infinitely-many period-doubling cascades is one of the most prominent features observed in the study of maps depending on a parameter. They are associated with chaotic behavior, since bifurcation diagrams of a map with a parameter often reveal a complicated intermingling of period-doubling cascades and chaos. Period doubling can be studied at three levels of complexity. The first is an individual period-doubling bifurcation. The second is an infinite collection of period doublings that are connected together by periodic orbits in a pattern called a cascade. It was first described by Myrberg and later in more detail by Feigenbaum. The third involves infinitely many cascades and a parameter value μ2 of the map at which there is chaos. We show that often virtually all (i.e. all but finitely many) "regular" periodic orbits at μ2 are each connected to exactly one cascade by a path of regular periodic orbits; and virtually all cascades are either paired — connected to exactly one other cascade, or solitary — connected to exactly one regular periodic orbit at μ2. The solitary cascades are robust to large perturbations. Hence the investigation of infinitely many cascades is essentially reduced to studying the regular periodic orbits of F(μ2, ⋅). Examples discussed include the forced-damped pendulum and the double-well Duffing equation.

A Surfeit of Cycles

2009 ◽  
Vol 20 (6) ◽  
pp. 985-996
Author(s):  
W. M. Schaffer
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Cyclic Behavior ◽  
Intrinsic Variability ◽  
Rotation Numbers ◽  
Wide Range ◽  
Exogenous Forcing ◽  
The Stability ◽  
Parametric Adjustment ◽  
Long Term Prediction

Chaotic sets are organized about “skeletons” of periodic orbits in the sense that every point on a chaotic set is arbitrarily close to such an orbit. The orbits have the stability property of saddles: attracting in some directions; repelling in others. This topology has implications for changing climates that evidence pronounced variability on time scales ranging from decades to tens of thousands of years. Among these implications are the following: 1. A wide range of periodicities should be (and are) observed. 2. Periodicities should (and do) shift – often abruptly – as the evolving climatic trajectory sequentially shadows first one periodic orbit and then another. 3. Models that have been “tuned” (parametric adjustment) to fit trajectorial evolution in the vicinity of one periodic orbit are likely to fail when the real system moves to another region of the phase space. 4. In response to secular forcing, chaotic sets simplify via the elimination of periodic orbits. If one accepts the reality of anthropogenic warming, the long-term prediction is loss of intrinsic variability. 5. In response to periodic forcing, nonlinear systems can manifest subharmonic resonance i.e., “cyclic” behavior with periods and rotation numbers rationally related to the period of the forcing. Such cycling has been implicated in millennial and stadial variations in paleoclimatic time series. 6. Generically, the dynamics of system observables, such as climate sensitivity, are qualitatively equivalent to those of the whole. If the climate is chaotic, so too is sensitivity. These considerations receive minimal attention in consensus views of climate change that emphasize essentially one-to-one correspondence between global temperatures and exogenous forcing. Caveat emptor.

The Dynamics of Symmetric Bimodal Maps

1997 ◽  
Vol 07 (12) ◽  
pp. 2735-2744 ◽  
Author(s):  
Thomas Lofaro
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Period Doubling ◽  
General Setting ◽  
The Family ◽  
Pitchfork Bifurcations ◽  
Parameter Values ◽  
Kneading Theory ◽  
Period Doubling Bifurcations

The dynamics and bifurcations of a family of odd, symmetric, bimodal maps, fα are discussed. We show that for a large class of parameter values the dynamics of fα can be described via an identification with a unimodal map uα. In this parameter regime, a periodic orbit of period 2n + 1 of uα corresponds to a periodic orbit of period 4n + 2 for fα. A periodic orbit of period 2n of uα corresponds to a pair of distinct periodic orbits also of period 2n for fα. In a more general setting we describe the genealogy of periodic orbits in the family fα using symbolic dynamics and kneading theory. We identify which periodic orbits of even periods are born in period-doubling bifurcations and which are born in pitchfork bifurcations and provide a method of describing the "ancestors" and "descendants" of these orbits. We also show that certain periodic orbits of odd periods are born in saddle-node bifurcations.

BIFURCATION IN A PIECEWISE LINEAR CIRCUIT WITH SWITCHING BOUNDARIES

2012 ◽  
Vol 22 (02) ◽  
pp. 1250034 ◽  
Author(s):  
ZHENGDI ZHANG ◽  
QINSHENG BI
Keyword(s):  
Periodic Orbits ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Dynamical Evolution ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Generalized Jacobian ◽  
Linear Circuit ◽  
The Stability

By introducing time-dependent power source, a periodically excited piecewise linear circuit with double-scroll is established. In the absence of the excitation, all possible equilibrium points as well as the stability conditions are presented. Analyzing the corresponding characteristic equations with perturbation method, Hopf bifurcation conditions associated with the equilibria are derived, which can be demonstrated by the numerical simulations. The Hopf bifurcations of the two symmetric equilibrium points may cause two symmetric periodic orbits, which lead to single-scroll chaotic attractors via sequences of period-doubling bifurcations with the variation of the parameters. The two chaotic attractors expand to interact with each other to form an enlarged chaotic attractor with double-scroll. The behaviors on the switching boundaries are investigated by the generalized Jacobian matrix. When periodic excitation is applied to work on the circuit, three periodic orbits with the frequency of the excitation may exist, which can be called generalized equilibrium points (GEPs) with the same characteristic polynomials as those of the corresponding equilibrium points for the autonomous case. It is shown that when the trajectories do not pass across the switching boundaries, the solutions are the same as the GEPs. However, when the trajectories pass across the switching boundaries, complicated behaviors will take place. Three forms of chaotic attractors via different bifurcations can be observed and the influence of the switching boundaries on the phase portraits is discussed to explore the mechanism of the dynamical evolution.

Can a Pseudo Periodic Orbit Avoid a Catastrophic Transition?

2015 ◽  
Vol 25 (13) ◽  
pp. 1550185 ◽  
Author(s):  
Tetsushi Ueta ◽  
Daisuke Ito ◽  
Kazuyuki Aihara
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Stable Limit Cycle ◽  
Period Doubling ◽  
Transient Behavior ◽  
Control Input ◽  
Stable Limit ◽  
Unstable Periodic Orbits ◽  
Saddle Node Bifurcation ◽  
Saddle Node

We propose a resilient control scheme to avoid catastrophic transitions associated with saddle-node bifurcations of periodic solutions. The conventional feedback control schemes related to controlling chaos can stabilize unstable periodic orbits embedded in strange attractors or suppress bifurcations such as period-doubling and Neimark–Sacker bifurcations whose periodic orbits continue to exist through the bifurcation processes. However, it is impossible to apply these methods directly to a saddle-node bifurcation since the corresponding periodic orbit disappears after such a bifurcation. In this paper, we define a pseudo periodic orbit which can be obtained using transient behavior right after the saddle-node bifurcation, and utilize it as reference data to compose a control input. We consider a pseudo periodic orbit at a saddle-node bifurcation in the Duffing equations as an example, and show its temporary attraction. Then we demonstrate the suppression control of this bifurcation, and show robustness of the control. As a laboratory experiment, a saddle-node bifurcation of limit cycles in the BVP oscillator is explored. A control input generated by a pseudo periodic orbit can restore a stable limit cycle which disappeared after the saddle-node bifurcation.

Period-multiplying cascades for diffeomorphisms of the disc

1994 ◽  
Vol 116 (2) ◽  
pp. 359-374 ◽  
Author(s):  
Jean-Marc Gambaudo ◽  
John Guaschi ◽  
Toby Hall
Keyword(s):  
Fixed Point ◽  
Periodic Orbits ◽  
Analogous Result ◽  
Period Doubling ◽  
Positive Entropy ◽  
One Dimensional ◽  
Continuous Map ◽  
Period 2 ◽  
Zero Entropy ◽  
Period Doubling Bifurcations

It is a well-known result in one-dimensional dynamics that if a continuous map of the interval has positive topological entropy, then it has a periodic orbit of period 2i for each integer i ≥ 0 [15] (see also [12]). In fact, one can say rather more: such a map has a sequence of periodic orbits (P)i ≥ 0 with per (Pi) = 2i which form a period-doubling cascade (that is, whose points are ordered and permuted in the way which would occur had the orbits been created in a sequence of period-doubling bifurcations starting from a single fixed point). This result reflects the central role played by period-doubling in transitions to positive entropy in a one-dimensional setting. In this paper we prove an analogous result for positive-entropy orientation-preserving diffeomorphisms of the disc. Using the notion [9] of a two-dimensional cascade, we shall show that such diffeomorphisms always have infinitely many ‘zero-entropy’ cascades of periodic orbits (including a period-doubling cascade, though this need not begin from a fixed point).

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.

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