## Institute of Radio-Engineering and Electronics of RAS, (2001)

### BibTeX

@MISC{Isaeva01instituteof,

author = {O. B. Isaeva and S. P. Kuznetsov},

title = {Institute of Radio-Engineering and Electronics of RAS,},

year = {2001}

}

### OpenURL

### Abstract

Scaling properties in dynamics of non-analytic complex maps near the accumulation point of the period-tripling cascade

### Citations

414 |
An Introduction to Chaotic Dynamical Systems
- Devaney
(Show Context)
Citation Context ...ge to observe the parameter space scaling. PACS numbers: 05.45.-a, 05.10.Cc, 05.45.Df.1 Introduction One of the most popular illustrations in the nonlinear science is a picture of the Mandelbrot set =-=[1, 2]-=- (Fig. 1). On the complex parameter plane it is defined as the set of points λ, at which the iterations of the complex quadratic map z ′ = λ − z 2 , λ, z ∈ C (1) launched from the origin z = 0 never d... |

165 | Quantitative universality for a class of nonlinear transformations
- Feigenbaum
- 1978
(Show Context)
Citation Context ...verges to the fixed point function g(z) = lim k→∞ fk(z), which obeys the equation g(z) = αg(g(g(z/α))), α = lim k→∞ αk = 1/g(g(g(0))). (7) It is a generalization of the Feigenbaum-Cvitanović equation =-=[6, 7]-=- for the case of period-tripling. This functional equation may be solved numerically by means of approximating the function g(z) via a finite polynomial expansion and computing the coefficients with a... |

80 | The universal metric properties of nonlinear transformations
- Feigenbaum
- 1979
(Show Context)
Citation Context ...on is disposed on the real axis at λF = 1.401155..., and it is called the Feigenbaum critical point. The scaling properties follow from the renormalization group (RG) analysis developed by Feigenbaum =-=[6, 7]-=-. Behaviour of the solution of the RG equation in complex domain was discussed e.g. by Nauenberg [8] and by Wells and Overill [9]. One can find many other critical points in the parameter plane of the... |

42 |
The Beauty of Fractals: Images of Complex Dynamical Systems
- Peitgen, Richter
- 1986
(Show Context)
Citation Context ...ge to observe the parameter space scaling. PACS numbers: 05.45.-a, 05.10.Cc, 05.45.Df.1 Introduction One of the most popular illustrations in the nonlinear science is a picture of the Mandelbrot set =-=[1, 2]-=- (Fig. 1). On the complex parameter plane it is defined as the set of points λ, at which the iterations of the complex quadratic map z ′ = λ − z 2 , λ, z ∈ C (1) launched from the origin z = 0 never d... |

18 |
Self-similarity and hairiness in the Mandelbrot set
- Milnor
- 1989
(Show Context)
Citation Context ...ross the sequence of leaves corresponding to periods 3, 9, 27..., 3k ..., and arrive at the period-tripling accumulation point λ = 0.0236411685377 + 0.7836606508052i. (2) 1In fact, as shown by Milnor =-=[4]-=-, the subtle structure (”hairiness”) of the Mandelbrot set does not reproduce itself on deep levels of resolution in small-scales. Speaking more accurately, one should relate the property of self-simi... |

7 |
Complex universality
- Cvitanović, Myrheim
- 1989
(Show Context)
Citation Context ...not reproduce itself on deep levels of resolution in small-scales. Speaking more accurately, one should relate the property of self-similarity rather to a definite subset called the Mandelbrot cactus =-=[5]-=-. It consists of a domain of existence of a stable fixed point and of domains associated with attractive periodic orbits, which originate from the fixed point via all possible bifurcation sequences (s... |

5 |
Universality for Period nTuplings in Complex Mappings
- Cvitanović
- 1983
(Show Context)
Citation Context ...functional equation may be solved numerically by means of approximating the function g(z) via a finite polynomial expansion and computing the coefficients with a use of multidimensional Newton method =-=[10, 11]-=-. We have reproduced these calculations, and present the results in Table I. Note that only terms of even power are nonzero. As follows from the computations, the scaling constant is α = −2.09691989 +... |

4 |
Universal Properties for Sequences of Bifurcations of Period 3
- Golberg, Sinai, et al.
- 1983
(Show Context)
Citation Context ...its, which originate from the fixed point via all possible bifurcation sequences (see the grey colored part of the picture in Fig. 1).As this point was first discovered by Golberg, Sinai, and Khanin =-=[10]-=-, we call it the GSK critical point. Independently, it was found and studied also by Cvitanović and Myrheim [5, 11]. As in the case of period-doubling, the RG analysis has been applied, and self-simil... |

4 |
Continuation from the Complex Quadratic Family: Fixed-Point Bifurcation Sets
- Real
(Show Context)
Citation Context ...ity of surviving for the universality intrinsic to the period-tripling. Recently Peckham and Montaldi have presented extensive bifurcation analysis of a complex quadratic map with a non-analytic term =-=[16, 17]-=-. However, the question what happens with the bifurcation cascades of period-tripling remains not clear. 3 Sure enough, this matter deserves accurate quantitative analysis in terms of the RG approach,... |

4 |
ThreeParameter Scaling for One-Dimensional Maps
- Kuznetsov, Sataev
- 1994
(Show Context)
Citation Context ...e of the term εz ∗ , these equalities are violated. Let us consider trace S = a11 + a22 (34) 4 See other examples of nonlinear parameter change for observation of the multi-parameter scaling in Refs. =-=[22, 23, 24]-=-.and determinant J = a11a22 − a12a21 (35) of the matrix J at the cycle. Surely, these values depend on the parameters of the map: S = S(λ, ε) = S(a, b, u, v) and J = J(λ, ε) = J(a, b, u, v). Precisel... |

4 |
A Variety of the Period-Doubling Universality Classes in Multi-Parameter Analysis of Transition to Chaos
- Kuznetsov, Sataev
- 1997
(Show Context)
Citation Context ...e of the term εz ∗ , these equalities are violated. Let us consider trace S = a11 + a22 (34) 4 See other examples of nonlinear parameter change for observation of the multi-parameter scaling in Refs. =-=[22, 23, 24]-=-.and determinant J = a11a22 − a12a21 (35) of the matrix J at the cycle. Surely, these values depend on the parameters of the map: S = S(λ, ε) = S(a, b, u, v) and J = J(λ, ε) = J(a, b, u, v). Precisel... |

4 |
Critical Point of Tori Collision in Quasiperiodically Forced Systems
- Kuznetsov, Pikovsky
- 1995
(Show Context)
Citation Context ...e of the term εz ∗ , these equalities are violated. Let us consider trace S = a11 + a22 (34) 4 See other examples of nonlinear parameter change for observation of the multi-parameter scaling in Refs. =-=[22, 23, 24]-=-.and determinant J = a11a22 − a12a21 (35) of the matrix J at the cycle. Surely, these values depend on the parameters of the map: S = S(λ, ε) = S(a, b, u, v) and J = J(λ, ε) = J(a, b, u, v). Precisel... |

3 |
Physical Meaning for Mandelbrot and Julia Set. Physica D125
- Beck
- 1999
(Show Context)
Citation Context ...ould be interesting to discuss a possibility of observation of phenomena associated with complex analytic dynamics and the Mandelbrot set in physical systems. Recently this question was posed by Beck =-=[12]-=-. He has considered motion of a particle in a double-well potential in a time-depended magnetic field. As he has stated, under certain assumptions dynamics of the particle may be described by the comp... |

3 |
On Perturbations Modulated Non-Linear Processes and a Novel Mechanism to Induce Chaos
- Rössler, Hess
- 1989
(Show Context)
Citation Context ...nsional cross-sections of the parameter space and look at the ”topography charts” for dynamical regimes on these surfaces. For graphical presentation we will use the technique of Lyapunov charts (see =-=[19, 20, 21]-=- for previous applications of this method). As we have chosen the parameter-space cross-section to be studied, we compute the senior Lyapunov exponent for the model map at each pixel of the two-dimens... |

3 |
Lyapunov Exponents of the Logistic Map with Periodic Forcing
- Markus, Hess
- 1989
(Show Context)
Citation Context ...nsional cross-sections of the parameter space and look at the ”topography charts” for dynamical regimes on these surfaces. For graphical presentation we will use the technique of Lyapunov charts (see =-=[19, 20, 21]-=- for previous applications of this method). As we have chosen the parameter-space cross-section to be studied, we compute the senior Lyapunov exponent for the model map at each pixel of the two-dimens... |

2 |
Feigenvalues for Mandelsets
- Briggs, Quispel
- 1991
(Show Context)
Citation Context ... has subtle and complicated structure, which is a subject of extensive researches. Objects analogous to the Mandelbrot set are present in parameter planes of other complex analytic iterative maps too =-=[2, 3]-=-. The Mandelbrot set is regarded as a classic example of fractal; this suggests a sort of self-similarity, or scaling. 1 One particular manifestation of scaling is the Feigenbaum perioddoubling cascad... |

2 |
Fractal boundary of domain of analyticity of the Feigenbaum function and relation to the Mandelbrot set
- Nauenberg
- 1987
(Show Context)
Citation Context ...e scaling properties follow from the renormalization group (RG) analysis developed by Feigenbaum [6, 7]. Behaviour of the solution of the RG equation in complex domain was discussed e.g. by Nauenberg =-=[8]-=- and by Wells and Overill [9]. One can find many other critical points in the parameter plane of the map (1), at which the Mandelbrot cactus displays other scaling regularities. In particular, we can ... |

2 |
The extension of the Feigenbaum – Cvitanović function to the complex plane
- Wells, Overill
- 1994
(Show Context)
Citation Context ...rom the renormalization group (RG) analysis developed by Feigenbaum [6, 7]. Behaviour of the solution of the RG equation in complex domain was discussed e.g. by Nauenberg [8] and by Wells and Overill =-=[9]-=-. One can find many other critical points in the parameter plane of the map (1), at which the Mandelbrot cactus displays other scaling regularities. In particular, we can select a path on the complex ... |

2 |
Perturbation of Complex Analytic Families: Points to Regions
- Real
- 1998
(Show Context)
Citation Context ...ity of surviving for the universality intrinsic to the period-tripling. Recently Peckham and Montaldi have presented extensive bifurcation analysis of a complex quadratic map with a non-analytic term =-=[16, 17]-=-. However, the question what happens with the bifurcation cascades of period-tripling remains not clear. 3 Sure enough, this matter deserves accurate quantitative analysis in terms of the RG approach,... |

2 | E.Karakasidis. On Perturbations of the Maldelbrot Map - Argyris, T |

1 |
Trajectory scaling for period tripling in near conformal mappings. Phys. Rev. A36
- Gunaratne
- 1987
(Show Context)
Citation Context ...hy – Riemann conditions. In the original work [10] the authors claimed that the infinite period-tripling cascade could occur typically in two-parameter families of real two-dimensional maps (see also =-=[13]-=-). In contrast, from studies of Peinke et al. [14], Klein [15], and from qualitative analysis of Cvitanović and Myrheim [5] it follows that in non-analytic maps the parameter space arrangement is chan... |

1 |
Mandelbrot set in a non-analytic map
- Klein
- 1988
(Show Context)
Citation Context ... claimed that the infinite period-tripling cascade could occur typically in two-parameter families of real two-dimensional maps (see also [13]). In contrast, from studies of Peinke et al. [14], Klein =-=[15]-=-, and from qualitative analysis of Cvitanović and Myrheim [5] it follows that in non-analytic maps the parameter space arrangement is changed drastically, and, apparently, there is no opportunity of s... |

1 |
Lyapunov graph for two-parameters map: Application to the circle map
- Figueiredo, Malta
- 1998
(Show Context)
Citation Context ...nsional cross-sections of the parameter space and look at the ”topography charts” for dynamical regimes on these surfaces. For graphical presentation we will use the technique of Lyapunov charts (see =-=[19, 20, 21]-=- for previous applications of this method). As we have chosen the parameter-space cross-section to be studied, we compute the senior Lyapunov exponent for the model map at each pixel of the two-dimens... |