Noise destroys the coexistence of periodic orbits of a piecewise linear map

2013 ◽  
Vol 22 (3) ◽  
pp. 030502 ◽  
Author(s):  
Can-Jun Wang ◽  
Ke-Li Yang ◽  
Shi-Xian Qu
Author(s):  
S.J Hogan ◽  
L Higham ◽  
T.C.L Griffin

In this paper, we consider periodic solutions of discontinuous non-smooth maps. We show how the fixed points of a general piecewise linear map with a discontinuity (‘a map with a gap’) behave under parameter variation. We show in detail all the possible behaviours of period 1 and period 2 solutions. For positive gaps, we find that period 2 solutions can exist independently of period 1 solutions. Conversely, for negative gaps, period 1 and period 2 solutions can coexist. Higher periodic orbits can also exist and be stable and we give several examples of how these solutions behave under parameter variation. Finally, we compare our results with those of Jain & Banerjee (Jain & Banerjee 2003 Int. J. Bifurcat. Chaos 13 , 3341–3351) and Banerjee et al . (Banerjee et al . 2004 IEEE Trans. Circ. Syst. II 51 , 649–654) and explain their numerical simulations.


Author(s):  
O. Jenkinson ◽  
M. Pollicott ◽  
P. Vytnova

AbstractIommi and Kiwi (J Stat Phys 135:535–546, 2009) showed that the Lyapunov spectrum of an expanding map need not be concave, and posed various problems concerning the possible number of inflection points. In this paper we answer a conjecture in Iommi and Kiwi (2009) by proving that the Lyapunov spectrum of a two branch piecewise linear map has at most two points of inflection. We then answer a question in Iommi and Kiwi (2009) by proving that there exist finite branch piecewise linear maps whose Lyapunov spectra have arbitrarily many points of inflection. This approach is used to exhibit a countable branch piecewise linear map whose Lyapunov spectrum has infinitely many points of inflection.


2015 ◽  
Vol 25 (13) ◽  
pp. 1550184 ◽  
Author(s):  
Carlos Lopesino ◽  
Francisco Balibrea-Iniesta ◽  
Stephen Wiggins ◽  
Ana M. Mancho

In this paper, we prove the existence of a chaotic saddle for a piecewise-linear map of the plane, referred to as the Lozi map. We study the Lozi map in its orientation and area preserving version. First, we consider the autonomous version of the Lozi map to which we apply the Conley–Moser conditions to obtain the proof of a chaotic saddle. Then we generalize the Lozi map on a nonautonomous version and we prove that the first and the third Conley–Moser conditions are satisfied, which imply the existence of a chaotic saddle. Finally, we numerically demonstrate how the structure of this nonautonomous chaotic saddle varies as parameters are varied.


2010 ◽  
Vol 20 (05) ◽  
pp. 1365-1378 ◽  
Author(s):  
GÁBOR CSERNÁK ◽  
GÁBOR STÉPÁN

In the present paper, we introduce and analyze a mechanical system, in which the digital implementation of a linear control loop may lead to chaotic behavior. The amplitude of such oscillations is usually very small, this is why these are called micro-chaotic vibrations. As a consequence of the digital effects, i.e. the sampling, the processing delay and the round-off error, the behavior of the system can be described by a piecewise linear map, the micro-chaos map. We examine a 2D version of the micro-chaos map and prove that the map is chaotic.


1997 ◽  
Vol 07 (02) ◽  
pp. 351-372 ◽  
Author(s):  
D. Aharonov ◽  
R. L. Devaney ◽  
U. Elias

The paper describes the dynamics of a piecewise linear area preserving map of the plane, F: (x, y) → (1 - y - |x|, x), as well as that portion of the dynamics that persists when the map is approximated by the real analytic map Fε: (x, y) → (1 - y - fε(x), x), where fε(x) is real analytic and close to |x| for small values of ε. Our goal in this paper is to describe in detail the island structure and the chaotic behavior of the piecewise linear map F. Then we will show that these islands do indeed persist and contain infinitely many invariant curves for Fε, provided that ε is small.


2012 ◽  
Vol 241-244 ◽  
pp. 2728-2731
Author(s):  
Yong Zhang

Some chaos-based image encryption schemes using plain-images independent secret code streams have weak encryption security and are vulnerable to chosen plaintext and chosen cipher-text attacks. This paper proposed a two-level secret key image encryption method, where the first-level secret key is the private symmetric secret key, and the second-level secret key is derived from both the first-level secret key and the plain image by iterating piecewise linear map and Logistic map. Even though the first-level key is identical, the different plain images will produce different second-level secret keys and different secret code streams. The results show that the proposed has high encryption speed, and also can effectively resist chosen/known plaintext attacks.


2015 ◽  
Vol 25 (03) ◽  
pp. 1530006 ◽  
Author(s):  
Anastasiia Panchuk ◽  
Iryna Sushko ◽  
Viktor Avrutin

In this work, we investigate the bifurcation structure of the parameter space of a generic 1D continuous piecewise linear bimodal map focusing on the regions associated with chaotic attractors (cyclic chaotic intervals). The boundaries of these regions corresponding to chaotic attractors with different number of intervals are identified. The results are obtained analytically using the skew tent map and the map replacement technique.


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