Visualizing Attractors of the Three-Dimensional Generalized Hénon Map

The two-dimensional Hénon mapHénon (1976) is the quadratic diffeomorphism of the plane: every such diffeomorphism is conjugate to a map in the two-parameter Hénon family. Prominent rigorous results for this diffeomorphism include that of Devaney and Nitecki, Devaney and Nitecki (1979) who showed that the dynamics of its bounded orbits are conjugate to a Smale horseshoe in some parameter regimes, and that of Benedicks and Carleson,Benedicks and Carleson (1991) who showed that when the Jacobian is sufficiently small there are cases for which the map has a transitive attractor with a positive Lyapunov exponent. It was also shown that the horseshoe in this map can be thought of as arising from an anti-integrable limit, where the dynamics becomes non-deterministic.Sterling and Meiss (1998)

Some of these results have been generalized to higher-dimensional maps. For the three-dimensional (3D) case it was shown in Ref. [Lomelí and Meiss, 1998] that every quadratic diffeomorphism with a quadratic inverse is conjugate to the map $L:{\mathbb{R}}^{3}\to{\mathbb{R}}^{3}$

This analysis was later generalized to include 3D quadratic diffeomorphisms that have quartic inverses, Elhadj and Sprott (2009) and to higher dimensions. Lenz, Lomelí, and Meiss (1999) The diffeomorphism (1) arises as a normal from in the volume preserving ($\delta=1$) case near a fixed point with three unit multipliers.Dullin and Meiss (2008, 2009) It has also been shown to be a normal form near a map with a saddle-focus fixed point that has a quadratic homoclinic tangency. Gonchenko et al. (2005); Gonchenko, Meiss, and Ovsyannikov (2006) The theory of anti-integrability also applies to these maps when $\alpha\to-\infty$, see e.g., Ref. [Hampton and Meiss, 2022].

The map $L$ has seven parameters, six in the quadratic polynomial $G$ and $\delta=\det{DL}$, the Jacobian determinant. This parameter set can be reduced, as noted in Ref. [Lomelí and Meiss, 1998]: whenever $a+b+c\neq 0$ and $2a+b\neq 0$,^***If one of these is violated, other scaling transformations can be found to eliminate two of the parameters. an affine coordinate transformation allows one to set

Using this simplification, (1) depends only on $(\alpha,\sigma,a,c)$ and the Jacobian $\delta$.

The case $(a,b,c)=(1,0,0)$ has been called “the 3D Hénon map” since it only has one nonlinear term, similar to the classical Hénon map.Hénon (1976) In a series of papers starting with Refs. [Gonchenko et al., 2005; Gonchenko, Meiss, and Ovsyannikov, 2006], Gonchenko and collaborators focus on the formation of chaotic attractors for this map.^††† In their notation $M_{1}=-\alpha,B=\delta,M_{2}=-\sigma$. These attractors are shown to be “wild hyperbolic”^‡‡‡Nearby maps have Newhouse tangencies between its stable and unstable manifolds. Turaev and Shilnikov (1998) near the parameters $(\alpha,\sigma,\delta)=(\tfrac{1}{4},-1,1)$, where a pair of fixed points are born with multipliers $(-1,-1,1)$. These attractors are also “pseudo-hyperbolic,”^§§§Its tangent space splits into a a strong stable subspace and a complementary subspace that exponentially expands volume. like the Lorenz attractor, implying that every orbit in the attractor has at least one positive and one negative Lyapunov exponent. According to Ref. [Gonchenko and Gonchenko, 2016], there are five types of such attractors formed from the unstable manifold of a fixed point, including “Lorenz-like” and several “figure-eight” attractors.Gonchenko, Simo, and Vieiro (2013) Moreover, there are infinite cascades in parameter space of nearby systems with Lorenz-like attractors.Gonchenko and I. (2017) Chaotic attractors for the non-orientable case have also been studied.Gonchenko et al. (2021a)

Following Gonchenko et al., we will primarily study the 3D Hénon case using the scaling (2) and taking $0<\delta<1$, so that the map is volume contracting and orientation preserving. We focus on two examples:

1. (SC)

   Strongly Contracting: $(a,c,\delta)=(1,0,0.05)$,

2. (MC)

   Moderately Contacting: $(a,c,\delta)=(1,0,0.7)$,

allowing $(\alpha,\sigma)$ to vary. In §II, we recall basic bifurcation behavior of periodic orbits for a 3D map using the trace and second trace of the Jacobian. These bifurcation conditions are transformed to conditions on $(\alpha,\sigma)$ for the fixed points of (1) in §III. Similar bifurcation criterion were obtained for the 3D Hénon map in Ref. [Zhao et al., 2017] for the case that $\sigma=\delta$, as well as in Ref. [Richter, 2002] when $\sigma=0$ for three and more dimensions. In §IV, we prove that all bounded orbits of the 3D Hénon map lie within a finite cube about origin, following the proof of a related theorem in Ref. [Lomelí and Meiss, 1998].

The simplest bounded orbits are stable and periodic. Parameter regions in the $(\alpha,\sigma)$-plane containing such attractors, analogous to Arnold tongues, are computed in §V, where we also compute heteroclinic manifolds between stable and unstable orbits. In §VI, we study aperiodic attractors, computing the maximal Lyapunov exponent to distinguish between regular and chaotic cases. Regular aperiodic attractors come in the form of invariant circles, which we study in §VI.1 extending the volume-preserving case studied in Ref. [Dullin and Meiss, 2009]. In §VI.2, we find Hénon-like attractors for case (SC) and discrete, Lorenz-like attractorsGonchenko et al. (2021b) for case (MC). Additionally, we observe chaotic attractors arising from bifurcations of invariant circles that are unlike those in Refs. [Gonchenko et al., 2014; Gonchenko, Gonchenko, and Turaev, 2021]; these also do not seem to be related to the generalizations of Smale’s horseshoe to 3D found by Ref. [Zhang, 2016].

For a fixed point $\xi^{*}=f(\xi^{*})$ of a 3D map $f$, the eigenvalues of the Jacobian $A=Df(\xi^{*})$ are given by the characteristic polynomial

We refer to these as the multipliers of the fixed point. Here $t=\operatorname{\mbox{tr}}(A)$ is the trace and ${d}=\det(A)$. An expression for the “second trace”, $s$, can be obtained from the Cayley-Hamilton theorem: a matrix satisfies its own characteristic polynomial, $A^{3}-tA^{2}+sA-{d}I=0$. Multiplying this by $A^{-3}$ implies that $s={d}\operatorname{\mbox{tr}}(A^{-1})$. Finally, multiplying the Cayley polynomial by $A^{-1}$ and taking the trace gives

Of course, the multipliers can be related to the coefficients of (3) by the symmetric polynomials,

More generally, for an orbit, $\xi_{t}=f(\xi_{t-1})$, that has period $n$, $\xi_{0}=f^{n}(\xi_{0})$, the Jacobian becomes

Thus, using the same process as above, we can find a general expression for the trace and second trace of a period-$n$ orbit:

Following Ref. [Kuznetsov and Meijer, 2019], the simplest, local codimension-one bifurcations—saddle-node, period-doubling and Neimark-Sacker—occur when at least one multiplier has unit modulus. Each of these occurs on a surface in $(t,s,{d})$, given in Table 1. Sections through these surfaces for four values of ${d}$ are shown in Fig. 1; similar figures can be found in Ref. [Lomelí and Meiss, 1998] for the case ${d}=1$ and in Ref. [Gonchenko and Gonchenko, 2016], where they are referred to as “saddle-charts.”

Saddle-node (SN) bifurcations require a unit multiplier, so that (3) gives $p_{A}(1)=1-t+s-{d}=0$. This corresponds to a line in the $(t,s)$-plane, shown in blue in Fig. 1. A period-doubling (PD) bifurcation occurs when there is a $-1$ multiplier, or $p_{A}(-1)=-1-t-s-{d}=0$; shown in red in Fig. 1. The third codimension-one bifurcation, the Neimark-Sacker (NS), occurs when there is a pair of complex multipliers with unit modulus; it generically results in the creation of an invariant circle or pair of periodic orbits with differing stabilities. Since $|\lambda_{1,2}|=1$, (5) gives $\lambda_{3}={d}$, so that $p_{A}({d})={d}({d}^{2}-t{d}+s-1)=0$. The resulting line, $s={d}(t-{d})+1$, is shown in black in Fig. 1. Solving for the complex multipliers gives

or that $t-d=2\cos(2\pi\omega)$, for rotation number $\omega$.

Some codimension-two bifurcations occur along the NS line when $\omega=\tfrac{p}{q}$, and generically result in the birth of a pair of period-$q$ orbits. The endpoints of the NS line occur where it intersects the SN and PD lines at $\omega=0$ and $\omega=\tfrac{1}{2}$, labeled (R1) and (R2), respectively, in Table 1. The table also shows the period-three (R3) and period-four (R4) cases. Finally, the saddle-node flip (SNf) bifurcation point, with multipliers $(-1,1,-{d})$ is at the intersection of the PD and SN lines.

A curve in parameter space along which there is a double multiplier, say $\lambda_{1}=\lambda_{2}=r$, for $r\in{\mathbb{R}}$, corresponds to the transition from real to complex multipliers. Using (5), this occurs when $\lambda_{3}=d/r^{2}$ along the parametric curve

This curve has two branches when ${d}\neq 0$, one of which has a cusp at $r^{3}={d}$, i.e., at $(t,s)=3({d}^{1/3},{d}^{2/3})$. When $r=1$, this curve is tangent to the SN line and crosses the R1 point, and when $r=-1$ the curve is tangent to the PD line and crosses the R2 point. Four segments of these curves, labelled by ranges of the double multiplier, are shown in Fig. 1.

We also include representative configurations of the multipliers in the complex plane relative to the unit circle for each region of the $(t,s)$ plane in Fig. 1. For $0\leq d<1$, there exists a small triangular region with all stable multipliers ($|\lambda_{i}|<1$) that is bounded by the SN, PD and NS lines with vertices R1, R2, and SNf. Otherwise, the SN and PD lines divide the plane into four regions where stability types of the multipliers of the fixed points alternate between having one unstable and two stable multipliers and two unstable and one stable multiplier.

We now apply the results of §II to fixed points of (1) with parameter convention (2). The fixed points have the form $\xi_{\pm}=(x_{\pm},x_{\pm},x_{\pm})$ where

provided $\alpha\leq\alpha_{SN}$. These are born in a saddle-node bifurcation when $\alpha=\alpha_{SN}$ with $x_{\pm}^{2}=x_{SN}^{2}=\alpha_{SN}$. Thinking of $\alpha$ as a bifurcation parameter, the form (8) implies that there is a fixed point at $x^{*}$, say, when $\alpha=\alpha^{*}\leq\alpha_{SN}$ whenever $(x^{*}-x_{SN})^{2}=\alpha_{SN}-\alpha^{*}$, or equivalently when

The stability of the fixed points is determined by the linearization of (1),

This is in companion form, implying that at a fixed point the trace and second trace are

and the Jacobian is ${d}=\delta$.

The fixed points lie on a line in the $(t,s)$-plane that can be found by eliminating $x_{\pm}$ and $b$ from (10),

and are born at the intersection of this line with the SN line from Table 1,

Moreover, (10) gives

Since $x_{-}<x_{SN}<x_{+}$, this implies that $x_{+}$ lies below and $x_{-}$ above the SN line in the $(t,s)$ plane. Therefore when $0\leq\delta<1$, if there exists an attracting fixed point, it must be $\xi_{-}$.

Using the equations for the SN and PD lines in Table 1, the fixed points are born above the PD line when $t_{SN}>-\delta$, or

Provided $a\neq c$, one of the fixed points undergoes a period-doubling bifurcation at

An expression for the period doubling value of $\alpha$ is obtained from (9): $\alpha_{PD}=x_{PD}(2x_{SN}-x_{PD})$. Note that for a fixed point to ‘double,’ it needs to exist, thus we must have $\alpha_{PD}\leq\alpha_{SN}$. In addition, when $x_{PD}<x_{SN}$, the $\xi_{-}$ fixed point doubles, and when $x_{PD}>x_{SN}$, the $\xi_{+}$ point doubles.

Similarly, using (10) and the expression in Table 1 for the NS bifurcation gives

provided the denominator is nonzero. Of course, this bifurcation only exists if the fixed points intersect the NS line on the interval $\delta-2<t<\delta+2$.Again, (9) can be used to determine the bifurcation value, $\alpha_{NS}$.

When the parameters $(a,c,\delta)$ are fixed, the generic bifurcation diagrams of Fig. 1 can be transformed to corresponding diagrams for a fixed point in the $(\alpha,\sigma)$ plane; details are given in Appendix A, including criteria for codimension-two bifurcations. The results are shown in Fig. 2 for the fixed point $\xi_{-}$—as this is the only fixed point that can be attracting—for cases (SC) and (MC).

In Ref. [Lomelí and Meiss, 1998], it was shown that if the quadratic form $Q(x,y)=ax^{2}+bxy+cy^{2}$ is positive definite and $\delta=1$, then there is a cube that contains all bounded orbits of (1); equivalently, all points outside of this cube escape to infinity. We show in this section that a similar argument can be used for the case $(a,b,c)=(1,0,0)$, where $Q$ is semi-definite, for any $\delta>0$.

To illustrate this result, we compute the volume of bounded orbits, i.e. the volume of the basin of any attractors, for the cases (SC) and (MC), see Fig. 3. For each point on a $500^{2}$ grid in the $(\alpha,\sigma)$-plane for which the fixed points (8) exist ($\alpha\leq\alpha_{SN}$), we choose $50^{3}$ initial conditions on a uniform grid in a cube with bounds (14) that vary with $(\alpha,\sigma)$. An orbit is declared to be unbounded if it leaves the $\kappa$-cube within $200$ iterates.

The selected parameters in Fig. 3 focus on the triangular regions of Fig. 2 where $\xi_{-}$ is an attracting fixed point. The bifurcation curves (dashed) and codimension-two points are also shown (refer to the legend in Fig. 2). Note that near the SN line, the volume of bounded orbits in the $\kappa$-cube is small since most are found near the fixed points, which are close together.

In this section we compute regions in parameter space for which (1) has attracting periodic orbits. If there is a unique, bounded attractor, it can easily be found by choosing an appropriate initial condition within the cube of Lem. 1 and iterating until the orbit limits on the attractor. Once the transient is removed, parameter regions of periodic behavior can be computed by looking for recurrence.

Computed periodic regions are shown in Fig. 4 for the two cases, (SC) and (MC). To compute these, the initial point is set to $\xi_{0}=0$ when neither of the fixed points exist (i.e., to the right of the SN line in the $(\alpha,\sigma)$-plane) or to $\xi_{0}=\xi_{-}+(0.001,0,0)$, near the fixed point. For each $(\alpha,\sigma)$ on a $1000^{2}$ grid, the map (1) is iterated $T=5000$ times to eliminate transients. The orbit is declared to diverge (white region) if $|\xi_{t}|>\kappa_{max}$ for some $t\leq T$, where $\kappa_{max}$ is the maximum of (14) over the studied parameter region ($\kappa_{max}=3.237$ for (SC) and $4.632$ for (MC)). Otherwise the point $\xi_{T}$ is iterated up to $90$ more steps, checking for return time, defined as the first time, $p$, for which the distance

Thus we find approximately periodic orbits up to period $90$; the period $p$ is indicated by the color map in Fig. 4. If an orbit is not periodic by this definition, the point is colored black or gray; we will discuss the dynamics in these regions in §VI.

The number of transient iterations, recurrence tolerance, and grid size were chosen to provide sufficient detail at the pictured resolution but still lessen the computational expense.

Note that since this method uses a single initial condition, it cannot find cases where there are multiple attractors. Additionally, it is possible that the chosen initial condition leads to an unbounded orbit when there is still an attractor somewhere in the $\kappa$-cube. Nevertheless, for almost all $(\alpha,\sigma)$ points that have bounded orbits in Fig. 3, the orbit iterated in Fig. 4 is also bounded; therefore, with a few exceptions, it does not appear that the orbit of $\xi_{0}$ is unbounded when there are other bounded orbits. We plan to discuss such exceptional cases and the case of multiple attractors in future papers.

The largest, blue regions in Fig. 4 correspond to period-one, where $\xi_{-}$ is attracting. This region is bounded by the SN, PD, and NS curves seen in Fig. 2. When the fixed point loses stability at the PD curve an attracting period-two orbit is born; the period-two (vivid orange) region is prominent in Fig. 4(a), though there is also a thin period-two region just below the PD curve in Fig. 4(b). Period-doubling bifurcations leading to period-four (magenta) and eight (red) are also seen in Fig. 4(a). By contrast, in Fig. 4(b), the period-two orbit in (MC) looses stability by a NS bifurcation, so a doubling cascade is not seen in this case.

Resonant “tongues”, analogous to the Arnold tongues found in circle maps, start along the NS curve where $\omega$ is rational; these are prominent in Fig. 4(b). Especially visible are the $\omega=\tfrac{1}{3}$ (yellow), $\tfrac{1}{4}$ (magenta), $\tfrac{2}{5}$ (dark green), and $\tfrac{3}{7}$ (brown) tongues. Bifurcation points along the NS curve for the first two of these were indicated in Fig. 2 as R3 and R4.

Enlargements of case (SC), shown in Fig. 5, and of case (MC), Fig. 7, provide more detail. In the following two subsections, we show a few examples of the corresponding orbits in phase space for the period-doubling cascade and resonant tongues.

We now discuss the black and gray regions seen in Fig. 4, and its enlargements Fig. 5 and Fig. 7. These represent parameters for which there is an attractor that is either aperiodic or periodic with period larger than $90$. Possible aperiodic attractors include invariant circles formed at an NS bifurcation with irrational $\omega$, as well as more complex, possibly chaotic cases. In this section, we will distinguish between regular and chaotic cases by computing the maximal Lyapunov exponent. In a number of studies, multiple Lyapunov exponents were calculated to characterize the dimensionality of the unstable spaces of attractors.Gonchenko et al. (2021b); Gonchenko and Gonchenko (2016) Here we only compute one, as our focus is simply the distinction between chaotic and regular.

Formally, the Lyapunov exponent is

for an initial point $\xi_{0}$, and an initial vector $v_{0}$, which evolves linearly as $v_{t}=DL(\xi_{t-1})v_{t-1}$. Note for a generic initial vector, $v_{0}$, this limits to the maximal exponent (MLE). If there exists an attracting regular orbit, then any initial condition in its basin will have a non-positive MLE, whereas chaotic orbits will have $\mu>0$.

Numerically, we approximate the $\limsup$ of (18) by choosing an increment $\Delta T$, and computing

This approximates (18) for a “large” enough $T$ and $\Delta T$, up to some tolerance. We estimate the MLEs for those bounded orbits that have periods larger than 90. The initial point used for (19) is the point used in the recurrence algorithm described in §V found after $5000+90$ iterates, and the initial vector is set to $v_{0}=(1,1,1)/\sqrt{3}$. To avoid overflow, the length of $v_{n}$ is renormalized every 10 iterates. As is well known, it is computationally expensive to achieve convergence of MLEs. Indeed, since the ostensible error for (19) is ${\cal O}(T^{-1})$, we chose a threshold appropriate for $T\sim 10^{3}-10^{4}$: an orbit is deemed regular if, $\mu_{T}\leq\mu_{o}=3(10)^{-4}$. We selected the interval $\Delta T=100$ as it gave reasonable convergence for a number of trials. The time $T$ in (19) is increased in steps of $\Delta T$ until the error $|\mu_{T}-\mu_{T+\Delta T}|<10^{-4}$, or until $T$ reaches $T_{max}=10^{5}$. In this case, the MLE is set to $\mu_{T_{max}}$. For the results in Fig. 4, $\mu_{T}$ is set to $\mu_{T_{max}}$ for $0.14\%$ of the calculations for case (MC) and $1.2\%$ for case (SC).

Orbits with $\mu_{T}>\mu_{o}$ are declared to be “chaotic”, and colored gray, and those with smaller MLEs are declared “regular” and colored black in the figures. As seen in Fig. 4. there is a qualitative difference between the two cases. For case (SC), there are very few regular, aperiodic orbits— these are confined to a narrow strip along the NS curve at the top of the bounded region ($4.3\%$ of the aperiodic orbits). For case (MC), however, there are large regions of regular, aperiodic behavior to the left of the NS curve ($70\%$ of the aperiodic orbits). In the following subsections, we illustrate this difference by looking at the corresponding behavior of orbits in phase space.

Previous research on quadratic 3D maps has focused on the volume-preserving caseLomelí and Meiss (1998); Dullin and Meiss (2009) or on the existence and development of chaotic attractors. Gonchenko et al. (2005); Gonchenko and Gonchenko (2016); Gonchenko et al. (2021b) Here, we have explored a broader range of parameters and studied periodic and aperiodic, regular and chaotic attractors.

The simplest bifurcations of a 3D map were discussed in §II using the trace and second trace of the Jacobian as primary parameters and fixing the Jacobian determinant. These results were applied to the fixed points of the map (1) in §III. We focused on two primary parameters: the more “structural” parameter $\sigma$, that controls the type of bifurcation and, what can be viewed as the “primary unfolding” parameter, $\alpha$. Note that in our previous work on anti-integrability, it was the limit $\alpha\to-\infty$ that corresponded to a non-deterministic limit where the dynamics is conjugate to a shift on a set of symbols.Hampton and Meiss (2022)

For our numerical studies, we chose two cases for the Jacobian $\delta$: a strongly contracting case (SC)—where the map is nearly 2D—and a moderately contracting case (MC).

We showed in §IV that all bounded orbits of the orientation preserving 3D Hénon map lie within a cube about origin as illustrated in Fig. 3. Most of the region of bounded orbits correspond to nonchaotic situations, which could be an attracting fixed point or orbits that arise from this point by doubling or Neimark-Sacker bifurcations. However this figure also shows protruding spikes from the region of stability, that may be related to attractors not born from the fixed point. We hope to study these further in the future.

To classify the behavior of bounded orbits, we computed resonant regions in parameter space in §V; these are analogous to the Arnold tongues of circle maps. The resulting partition of the bounded region, Fig. 4, shows periodic and aperiodic attractors. Using this, we are able to understand the development of attracting periodic orbits and visualize their codimension-one and -two bifurcations. Note that our computations used the attractor arising from a single initial condition. There will be cases with multiple attractors and cases for which the chosen orbit is unbounded even when there might be attractors elsewhere. But given the similarity between Fig. 3 and Fig. 4, the latter possibility is rare. We plan to investigate the more complex, outlying cases in future research.

Aperiodic attractors (defined to be attractors with period greater than $90$), were studied in §VI. We also followed the evolution of invariant circles along several curves in parameter space starting at points on the NS bifurcation curve where the rotation number was irrational. We did not attempt to follow a circle with fixed rotation number, though we expect such a curve exists in a two-dimensional parameter space and plan to do this in future research. Along the parameter curves that we did follow, the invariant circle has a varying rotation number. When this becomes rational, the circle undergoes a resonant bifurcation, generically breaking up into a pair of periodic orbits. As the curve leaves a resonant tongue, an invariant circle can reform. In some cases the destruction of the circle gave rise to chaotic attractors.

Some of the chaotic attractors we studied are well-known: the discrete Lorenz and Hénon attractors found in Refs. [Gonchenko et al., 2021b; Hénon, 1976; Hampton and Meiss, 2022]. More unusual are the chaotic attractors with a paraboloid structure that arise from the destruction of the invariant circles, recall Fig. 13. These are not classified in Ref. [Gonchenko and Gonchenko, 2016] and are not obviously related to the 3D generalized horseshoes studied by Ref. [Zhang, 2016]. We believe that further study of such cases could lead to a broader understanding of attractors that lie within higher-dimensional generalizations of the Smale horseshoe. We plan to analyze these further in future research.