##
**Property of period-doubling bifurcations.**
*(English)*
Zbl 1081.34517

This paper considers an \(n\)-dimensional ordinary differential equation in the form of \(n\) relations giving the derivative of the phase vector with respect to the time. This equation depends on a parameter vector and is nonautonomous, i.e., the functions defining the right part of the \(n\) relations depend on the time.

Via an analytical way, the authors’ purpose is to establish a relation between a period \(T\) solution and a period \(2T\) one resulting from a period doubling when the parameter varies. They want to show that the period \(2T\) solution contains the whole information of the period \(T\) solution, and conversely. Relations between these two solutions are given. From these relations, the authors claim that it is possible to obtain properties of the power spectrum of period doubling sequences.

About this paper some remarks can be made. (1) The text has forgotten to specify that the functions defining the right parts of the n relations (defining the equation) must depend periodically on the time. (2) The period doubling phenomenon in ordinary differential equation is wrongly attributed to Feigenbaum (1978), who only considered the one-dimensional unimodal map. Moreover, for such maps this result was established before by Myrberg (1963). For ordinary differential equations, the credit of the period doubling phenomenon (called subharmonic \(1/(2k)\) oscillations) must be given to C. Hayashi (1953). His book [Nonlinear oscillations in physical systems. McGraw-Hill Series in Electrical and Electronic Engineering. Maidenhead, Berksh.: Mc Graw-Hill Publishing Company (1964; Zbl 0192.50605) (Reprint) (1985; Zbl 0604.70043)] describes and gives an analytical study of the phenomenon. From 1967, his disciples Y. Ueda and H. Kawakami published many papers including this topic, in particular using the nonautonomous Duffing’s equation, the example of the paper.

Via an analytical way, the authors’ purpose is to establish a relation between a period \(T\) solution and a period \(2T\) one resulting from a period doubling when the parameter varies. They want to show that the period \(2T\) solution contains the whole information of the period \(T\) solution, and conversely. Relations between these two solutions are given. From these relations, the authors claim that it is possible to obtain properties of the power spectrum of period doubling sequences.

About this paper some remarks can be made. (1) The text has forgotten to specify that the functions defining the right parts of the n relations (defining the equation) must depend periodically on the time. (2) The period doubling phenomenon in ordinary differential equation is wrongly attributed to Feigenbaum (1978), who only considered the one-dimensional unimodal map. Moreover, for such maps this result was established before by Myrberg (1963). For ordinary differential equations, the credit of the period doubling phenomenon (called subharmonic \(1/(2k)\) oscillations) must be given to C. Hayashi (1953). His book [Nonlinear oscillations in physical systems. McGraw-Hill Series in Electrical and Electronic Engineering. Maidenhead, Berksh.: Mc Graw-Hill Publishing Company (1964; Zbl 0192.50605) (Reprint) (1985; Zbl 0604.70043)] describes and gives an analytical study of the phenomenon. From 1967, his disciples Y. Ueda and H. Kawakami published many papers including this topic, in particular using the nonautonomous Duffing’s equation, the example of the paper.

Reviewer: Christian Mira (Quint)

### MSC:

34C23 | Bifurcation theory for ordinary differential equations |

34C25 | Periodic solutions to ordinary differential equations |

### Keywords:

ordinary differential equations; nonlinear dynamics; bifurcation; nonlinear oscillations; period doubling
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\textit{L. Wang} and \textit{M. Xu}, Chaos Solitons Fractals 24, No. 2, 527--532 (2005; Zbl 1081.34517)

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