We study nonsmooth bifurcations of four types of families of one-dimensional quasiperiodically forced maps of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$F_i(x,\theta ) = (f_i(x,\theta ), \theta +\omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>F</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi>θ</mi> <mo>+</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$i=1,\dots ,4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, where x is real, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\theta \in \mathbb {T}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>∈</mo> <mi mathvariant="double-struck">T</mi> </mrow> </math></EquationSource> </InlineEquation> is an angle, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation> is an irrational frequency, and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$f_i(x,\theta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is a real piecewise linear map with respect to x. The first two types of families <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$f_i$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>f</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation> have a symmetry with respect to x, and the other two could be viewed as quasiperiodically forced piecewise-linear versions of saddle-node and period-doubling bifurcations. The four types of families depend on two real parameters, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$a\in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$b\in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>. Under certain assumptions for a, we prove the existence of a continuous map <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$b^(a)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>b</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> where for <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$b=b^(a)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>=</mo> <msup> <mi>b</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> there exists a nonsmooth bifurcation for these types of systems. In particular we prove that for <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$b=b^*(a)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>=</mo> <msup> <mi>b</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> we have a strange nonchaotic attractor. It is worth to mention that the four families are piecewise-linear versions of smooth families which seem to have nonsmooth bifurcations. Moreover, as far as we know, we give the first example of a family with a nonsmooth period-doubling bifurcation.

Nonsmooth Bifurcations in Families of One-Dimensional Piecewise-Linear Quasiperiodically Forced Maps

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2026年1月9日

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Rafael Martinez-Vergara,
Joan Carles Tatjer

摘要

We study nonsmooth bifurcations of four types of families of one-dimensional quasiperiodically forced maps of the form $F_i(x,\theta ) = (f_i(x,\theta ), \theta +\omega )$ $F_{i} (x, θ) = (f_{i} (x, θ), θ + ω)$ for $i=1,\dots ,4$ $i = 1, \dots, 4$ , where x is real, $\theta \in \mathbb {T}$ $θ \in T$ is an angle, $\omega $ $ω$ is an irrational frequency, and $f_i(x,\theta )$ $f_{i} (x, θ)$ is a real piecewise linear map with respect to x. The first two types of families $f_i$ $f_{i}$ have a symmetry with respect to x, and the other two could be viewed as quasiperiodically forced piecewise-linear versions of saddle-node and period-doubling bifurcations. The four types of families depend on two real parameters, $a\in \mathbb {R}$ $a \in R$ and $b\in \mathbb {R}$ $b \in R$ . Under certain assumptions for a, we prove the existence of a continuous map $b^*(a)$ $b^{*} (a)$ where for $b=b^*(a)$ $b = b^{*} (a)$ there exists a nonsmooth bifurcation for these types of systems. In particular we prove that for $b=b^*(a)$ $b = b^{*} (a)$ we have a strange nonchaotic attractor. It is worth to mention that the four families are piecewise-linear versions of smooth families which seem to have nonsmooth bifurcations. Moreover, as far as we know, we give the first example of a family with a nonsmooth period-doubling bifurcation.