^{1,a)}and James A. Yorke

^{2,b)}

### Abstract

A period-doubling cascade is often seen in numerical studies of those smooth (one-parameter families of) maps for which as the parameter is varied, the map transitions from one without chaos to one with chaos. Our emphasis in this paper is on establishing the existence of such a cascade for many maps with phase space dimension 2. We use continuation methods to show the following: under certain general assumptions, if at one parameter there are only finitely many periodic orbits, and at another parameter value there is chaos, then between those two parameter values there must be a cascade. We investigate only families that are generic in the sense that all periodic orbit bifurcations are generic. Our method of proof in showing there is one cascade is to show there must be infinitely many cascades. We discuss in detail two-dimensional families like those which arise as a time- maps for the Duffing equation and the forced damped pendulum equation.

In a series of papers in 1958–1963, Pekka Myrberg was the first to discover that as a parameter is varied in the one-dimensional quadratic map (cf. Figure 1), periodic orbits of periods

*k*, 2

*k*, 4

*k*, 8

*k*, … occur for a variety of

*k*values (cf. Refs. 45 and 46 and references therein). This now well known phenomenon of period-doubling cascades has been seen in a large variety of parameter-dependent dynamical systems. Though it is not always possible to detect cascades numerically, as they may not be stable, they have been observed numerically in the contexts of maps, ordinary differential equations, partial differential equations, delay differential equations, and even in physical experiments. See Refs. 6–9, 21, 24, 26–28, 31, 38, 41, 54, 55, 60, and 62 and the double-well Duffing equation in Figure 2. It has often been observed that cascades occur in a dynamical system varying with a parameter during the transition from a system without chaos to one with chaos. In the current paper, we rigorously show the existence of a cascade (in fact of infinitely many) for certain two-dimensional systems, which include a transition to chaos.

E.S. was partially supported by National Science Foundation Grants DMS-0639300 and DMS-0907818. A portion of this work was performed while ES was a visitor at the Institute for Mathematics and Its Applications. The paper has substantially improved due to the input of the referees. We would like to thank both the anonymous referees and Charles Tresser for their helpful and thorough comments and corrections. We also thank Madhura Joglekar for her comments.

I. INTRODUCTION

A. Components, cascades, and continuation

B. A comparison to previous results

II. CASCADES FROM BOUNDARIES

III. APPLICATIONS AND OFF-ON-OFF CHAOS

### Key Topics

- Bifurcations
- 34.0
- Chaos
- 26.0
- Attractors
- 12.0
- Eigenvalues
- 4.0
- Implicit function theorems
- 4.0

## Figures

The parameterized map . For each fixed μ value, the attracting set in [0, 1] is shown. There are infinitely many cascades. For this map, each cascade has precisely one saddle-node bifurcation. The branch of unstable orbits ending at the saddle-node bifurcation is shown for period-1 and period-3. Each regular periodic orbit within the map's horseshoe at is uniquely connected to one of the (infinitely many) cascades. The regular period-one and period-three orbits are shown here, depicted respectively by one and three dots at . This is the bifurcation diagram most frequently displayed to illustrate the phenomenon of period-doubling cascades. However, cascades occur for much more complex dynamical systems that are completely unrelated to quadratic maps, as shown for example in Fig. 2 .

The parameterized map . For each fixed μ value, the attracting set in [0, 1] is shown. There are infinitely many cascades. For this map, each cascade has precisely one saddle-node bifurcation. The branch of unstable orbits ending at the saddle-node bifurcation is shown for period-1 and period-3. Each regular periodic orbit within the map's horseshoe at is uniquely connected to one of the (infinitely many) cascades. The regular period-one and period-three orbits are shown here, depicted respectively by one and three dots at . This is the bifurcation diagram most frequently displayed to illustrate the phenomenon of period-doubling cascades. However, cascades occur for much more complex dynamical systems that are completely unrelated to quadratic maps, as shown for example in Fig. 2 .

The attracting set for the double-well Duffing equation: This equation is periodically forced with period . Therefore, the F = time- map is a diffeomorphism on parameterized by ω. Depicted here is the attracting set of F, projected to the -plane. The constant 0.01 has been added to destroy symmetry in order to avoid non-generic symmetry-breaking bifurcations.

The attracting set for the double-well Duffing equation: This equation is periodically forced with period . Therefore, the F = time- map is a diffeomorphism on parameterized by ω. Depicted here is the attracting set of F, projected to the -plane. The constant 0.01 has been added to destroy symmetry in order to avoid non-generic symmetry-breaking bifurcations.

A path in a component. There is one component that contains fixed points (as well as orbits of period 2 n for . The path h(s) shown in red passes through each orbit in this component exactly once. As shown, at each period-doubling bifurcation, h(s) passes through the upper branch, but this choice is arbitrary. If the path were chosen so it sometimes went through the lower branch of periodic points, it would still pass through the same orbits. Both branches of periodic points describe the same orbits. Note that the lower branch extends infinitely far to the right and infinitely far down. The limiting end point is not a periodic point and is not a point of the cascade. The path can be thought of as the image of a function for s in an open interval J, and the path h(s) passes through each orbit exactly once. Furthermore, for an appropriate change of variables, we can choose J = (–1, +1).

A path in a component. There is one component that contains fixed points (as well as orbits of period 2 n for . The path h(s) shown in red passes through each orbit in this component exactly once. As shown, at each period-doubling bifurcation, h(s) passes through the upper branch, but this choice is arbitrary. If the path were chosen so it sometimes went through the lower branch of periodic points, it would still pass through the same orbits. Both branches of periodic points describe the same orbits. Note that the lower branch extends infinitely far to the right and infinitely far down. The limiting end point is not a periodic point and is not a point of the cascade. The path can be thought of as the image of a function for s in an open interval J, and the path h(s) passes through each orbit exactly once. Furthermore, for an appropriate change of variables, we can choose J = (–1, +1).

Paths on families of orbits. Each point in the curves of this figure represents an entire orbit. This figure depicts three different paths of periodic orbits. It outlines the situation in Theorem 1 and Lemma 1. Path segments between bifurcations consist of regular saddles (denoted S), attracting orbits, or repelling orbits (both denoted AR). The bifurcation orbits for a generic family are all saddle-node (SN), period-doubling, or period-halving (both denoted PD) bifurcations. If while following a path, the path reverses direction ( is increasing on one branch and decreasing on the other), as with p 3 and p 4 at SN, then one branch is Type S while the other is type AR. The path starting at p 1 reverses direction twice, both times at a SN point. If a path connects orbits at two ends of the parameter interval, as with p 1 and p 5, then they are the same type. We show that there are infinitely many Type S segments starting at that do not return to , and only a finite number of these may connect to orbits at , and these must be Type S. Hence, there are infinitely many paths like the one stemming from p 2, remaining forever in a bounded region between and and containing cascades.

Paths on families of orbits. Each point in the curves of this figure represents an entire orbit. This figure depicts three different paths of periodic orbits. It outlines the situation in Theorem 1 and Lemma 1. Path segments between bifurcations consist of regular saddles (denoted S), attracting orbits, or repelling orbits (both denoted AR). The bifurcation orbits for a generic family are all saddle-node (SN), period-doubling, or period-halving (both denoted PD) bifurcations. If while following a path, the path reverses direction ( is increasing on one branch and decreasing on the other), as with p 3 and p 4 at SN, then one branch is Type S while the other is type AR. The path starting at p 1 reverses direction twice, both times at a SN point. If a path connects orbits at two ends of the parameter interval, as with p 1 and p 5, then they are the same type. We show that there are infinitely many Type S segments starting at that do not return to , and only a finite number of these may connect to orbits at , and these must be Type S. Hence, there are infinitely many paths like the one stemming from p 2, remaining forever in a bounded region between and and containing cascades.

Paired cascades. This figure depicts two sets of period-seven cascade components for the Hénon map , each containing paired cascades. Only one point of each of the period-7 orbits of the Hénon map are shown, so that it is clearer how each pair is connected by its component. The leftmost and rightmost cascades are paired; that is, they lie in the same component of orbits. They are connected by a path of unstable regular periodic orbits (the orbit connecting the lefthand and righthand sides of the figure). Likewise, the two middle cascades are paired. They lie on (and are connected by) a path of attracting period-seven orbits. Paired cascades are not robust to macroscopic changes in the map in that both can be simultaneously destroyed by a large enough local perturbation.

Paired cascades. This figure depicts two sets of period-seven cascade components for the Hénon map , each containing paired cascades. Only one point of each of the period-7 orbits of the Hénon map are shown, so that it is clearer how each pair is connected by its component. The leftmost and rightmost cascades are paired; that is, they lie in the same component of orbits. They are connected by a path of unstable regular periodic orbits (the orbit connecting the lefthand and righthand sides of the figure). Likewise, the two middle cascades are paired. They lie on (and are connected by) a path of attracting period-seven orbits. Paired cascades are not robust to macroscopic changes in the map in that both can be simultaneously destroyed by a large enough local perturbation.

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