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Chaos 18, 043107 (2008); http://dx.doi.org/10.1063/1.2985853 (9 pages)

Multistability and arithmetically period-adding bifurcations in piecewise smooth dynamical systems

Younghae Do1 and Ying-Cheng Lai2

1Department of Mathematics, Kyungpook National University, Daegu 702-701, South Korea
2Department of Electrical Engineering and Department of Physics, Arizona State University, Tempe, Arizona 85287, USA

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(Received 3 June 2008; accepted 27 August 2008; published online 15 October 2008)

Multistability has been a phenomenon of continuous interest in nonlinear dynamics. Most existing works so far have focused on smooth dynamical systems. Motivated by the fact that nonsmooth dynamical systems can arise commonly in realistic physical and engineering applications such as impact oscillators and switching electronic circuits, we investigate multistability in such systems. In particular, we consider a generic class of piecewise smooth dynamical systems expressed in normal form but representative of nonsmooth systems in realistic situations, and focus on the weakly dissipative regime and the Hamiltonian limit. We find that, as the Hamiltonian limit is approached, periodic attractors can be generated through a series of saddle-node bifurcations. A striking phenomenon is that the periods of the newly created attractors follow an arithmetic sequence. This has no counterpart in smooth dynamical systems. We provide physical analyses, numerical computations, and rigorous mathematical arguments to substantiate the finding.

© 2008 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. MODEL DESCRIPTION AND NUMERICAL EVIDENCE FOR MULTISTABILITY
  3. GLOBAL DYNAMICS IN THE HAMILTONIAN LIMIT
    1. Invariant property
    2. Chaotic orbits and elliptic islands
  4. SYMBOLIC DYNAMICS
  5. PROOF OF EXISTENCE OF PERIODIC ORBITS
    1. Fixed points
    2. Period-2 attractors
    3. Period-5 attractors
    4. Periodic attractor of period 8
    5. Periodic attractor of period 11
    6. Periodic orbits of higher periods
    7. Scaling of the number of attractors
  6. CONCLUSIONS

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KEYWORDS and PACS

Keywords

bifurcation, Jacobian matrices, nonlinear dynamical systems, stability

PACS

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.2985853

PUBLICATION DATA

ISSN

1054-1500 (print)  
1089-7682 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

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Figures (6) Tables (1)

Figures (click on thumbnails to view enlargements)

FIG.1
Bifurcation diagram of Eq. ( 1 ) for a = −2 showing the occurrence of multiple coexisting periodic attractors. At each bifurcation point bi, a new periodic attractor of period i appears. In fact, the period of any newly appeared attractor is increased arithmetically. The precise values of various bi are given in Table 1.

FIG.1 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.2
For a = −2 and b = −0.95 in Eq. ( 1 ), basins of attraction of three distinct periodic attractors and an additional attractor at infinity. Blank regions indicate the initial conditions that lead to trajectories approaching infinity. The blue, yellow, and red regions denote the basins of the periodic attractors of period 2, 5, and 8, respectively.

FIG.2 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.3
(a) For μ<0, partition of B into three regions: R1 (blue), R2 (red), and R3 (yellow). Black filled dots indicate the points v1, v2, and v3, respectively. (b) First iterations of the respective regions in (a) under the map F.

FIG.3 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.4
Transition graph characterizing the dynamics on the invariant set Bμ under Eq. ( 1 ).

FIG.4 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.5
For Eq. ( 1 ) in the Hamiltonian limit (μ = −1), chaotic sea, elliptic periodic orbits, and KAM island chains. Blue lines indicate the boundary of the invariant set and markers indicate elliptic periodic orbits in the KAM islands: red crosses for an unstable period-3 orbit, red filled circles for a period-2 orbit, blue filled circles for a period-5 orbit, red filled diamond for a period-8 orbit, red circles for a period-11 orbit, blue filled diamond for a period-14 orbit, blue filled rectangles for a period-17 orbit, green filled diamond for a period-20 orbit, red filled triangles for a period-23 orbit, and blue filled triangles for a period-26 orbit.

FIG.5 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.6
The number of attractors vs ln(∣b+1∣) as the Hamiltonian limit is approached.

FIG.6 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

Tables

Table I. Existence and stability of periodic orbits, and critical bifurcating point of attracting periodic orbits.

View Table


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