Generalized Attracting Horseshoe in the R\"{o}ssler Attractor

We show that there is a mildly nonlinear three-dimensional system of ordinary differential equations - realizable by a rather simple electronic circuit - capable of producing a generalized attracting horseshoe map. A system specifically designed to have a Poincar\'{e} section yielding the desired map is described, but not pursued due to its complexity, which makes the construction of a circuit realization exceedingly difficult. Instead, the generalized attracting horseshoe and its trapping region is obtained by using a carefully chosen Poincar\'{e} map of the R\"{o}ssler attractor. Novel numerical techniques are employed to iterate the map of the trapping region to approximate the chaotic strange attractor contained in the generalized attracting horseshoe, and an electronic circuit is constructed to produce the map. Several potential applications of the idea of a generalized attracting horseshoe and a physical electronic circuit realization are proposed.


Introduction
The seminal work of Smale [1] showed that the existence of a horseshoe structure in the iterate space of a diffeomorphism is enough to prove it is chaotic. Often these diffeomorphisms arise from certain Poincaré maps of continuous-time chaotic strange attractors (CSA), which in turn are discrete-time CSAs. Some examples of such attractors are the Lorenz strange attractor [2], the Rössler attractor [3], and the double scroll attractor [4]. An example of a Poincaré map of the Lorenz equations is the Hénon map [5], which can be further simplified to the Lozi map [6].
In more recent years Joshi and Blackmore [7] developed an attracting horseshoe (AH) model for CSAs, which has two saddles and a sink. This, however, negates the possibility of the Hénon and Lozi maps, which have two saddles. Fortunately the attracting horseshoe can be modified into a generalized attracting horseshoe (GAH), which can have either one or two saddles while still being an attracting horseshoe [8]. This results in a quadrilateral trapping region. While extensive analysis was done in Joshi et al. [8], a simple concrete example seemed to be illusive.
In this investigation we implement MATLAB codes to find the necessary Poincaré map of the Rössler attractor that would admit a quadrilateral trapping region. The remainder of the paper is organized as follows; in Sec. 2 we give an overview of the algorithm with the MATLAB codes relegated to the Appendix. Once we have the tools for our numerical experiments, we first propose a carefully constructed GAH model in Sec. 3

Poincaré map algorithm
To produce a general Poincaré section of a flow we break up the program into four parts: solving the ODE, computing a Poincaré section perpendicular to either x = 0, y = 0, or z = 0, rotating the Poincaré section, and iterating the Poincaré map. Solving the ODE is standard through ODE45 on MATLAB, which executes a modified Runge-Kutta scheme. Once we have our solution matrix we need to approximate the values of first return maps from the discretized flow. By restricting the first return onto a Poincaré section the iterate space of Poincaré map can be visualized. This is easily done for a section perpendicular to the axes, but in order to locate a highly specialized object such as the GAH we need to be able to rotate the section. Once the desired section is found we can experiment on iterating the points of trapping region candidates.
The first major task is approximating the first return map on a Poincaré section of a flow. Much of the ideas of our initial first return map code came from that of Didier Gonze [9]. Once the discretized flow is found numerically a planar section for a certain value of x, y, or z can be defined, which in general will lie between pairs of simultaneous points. Then we may draw a line between the pair through the planar section and identify the intersecting point, which approximates a point of the first return map. This can also be done with more simultaneous points in order to get higher order approximations.
Once we can approximate a map for a section perpendicular to the axes we need to have the ability to rotate and move the map to any position. This is where our program completely diverges from that of [9]. While the first instinct might be to try to rotate the section, it is equivalent to rotate the flow in the opposite direction to the desired rotation of the section. Once the flow is rotated, the code for the first return map can be readily used. This gives us the ability to analyze the first return map of a general Poincaré section.
Finally, we would like to not only compute a first return map, but also compute the iterates of a Poincaré map of any system; that is, given an initial condition on an arbitrary Poincaré section can we find the subsequent iterates. To accomplish this, we solve the ODE for a given initial condition on the planar section to find the first return. Once we have the first return we record it's location and use that as the new initial condition. This iterates the map for as many returns as desired, thereby filling in a Poincaré map. Now we have the tools needed to run numerical experiments on GAHs.

A constructed GAH system
In this section, we give a brief description of the generalized attracting horseshoe (GAH) map and devise a three-dimensional nonlinear ordinary differential equation with a Poincaré section that produces it.

The GAH map
The GAH is a modification of the AH that can be represented as a geometric paradigm with either just one or two fixed points, both of which are saddles. Figure 1 shows a rendering of a C 1 GAH with two saddle points, which can be constructed as follows: The rectangle is first contracted vertically by a factor 0 < λ v < 1/2, then expanded horizontally by a factor 1 < λ h < 2 and then folded back into the usual horseshoe shape in such a manner that the total height and width of the horseshoe do not exceed the height and width, respectively of the trapping rectangle Q. Then the horseshoe is translated horizontally so that it is completely contained in Q. Obviously, the map f defined by this construction is a smooth diffeomorphism. Clearly, there are also many other ways to obtain this geometrical configuration. For example, the map f as described above is orientation-preserving, and an orientation-reversing variant can be obtained by composing it with a reflection in the horizontal axis of symmetry of the rectangle, or by composing it with a reflection in the vertical axis of symmetry followed by a composition with a half-turn. Another construction method is to use the standard Smale horseshoe that starts with a rectangle, followed by a horizontal composition with just the right scale factor or factors to move the image of Q into Q, while preserving the expansion and contraction of the horseshoe along its length and width, respectively.
It is important to note that subrectangle S with its left vertical edge through p, which contains the arch of the horseshoe and the keystone region K, plays a key role in the dynamics of the iterates of f . In particular, we require that the map satisfy the following additional property, which is illustrated in Fig. 2 : ( ) f maps the keystone region K (containing a portion of the arch of the horseshoe) to the left of the fixed point p and the portion of its corresponding stable manifold W s (p) containing p and contained in f (Q).
The definition above and ( ) can be shown to lead to the conclusion that where W u (p) is the unstable manifold of p, is a global chaotic strange attractor (CSA).
The map above can be considered to be the paradigm for a GAH, but there are many analogs. In fact, let F :Q →Q be any smooth diffeomorphism of a quadrilateral trapping regionQ possessing a horseshoe-like image with a keystone regionK containing a portion of Figure 1: A planar GAH with two saddle points the arch of F (Q) analogous to that shown in Fig. 1. Suppose that the map is expanding by a scale factor uniformly greater than one along the length of the horseshoe and contracting transverse to it by a scale factor uniformly less than one-half in the complement of a subset ofQ containingK. Then if F satisfies an additional property analogous to ( ), it mapsK into an open subset ofQ to the left of the saddle pointp, and is a global CSA.

A GAH producing system
We now construct an ODE in R 3 with a Poincaré section that is a GAH. The transversal we use is the following square in the xz-plane defined in Cartesian and polar coordinates The trick is to find a relatively simple (necessarily nonlinear) C 1 ODE having Q 0 as a transversal with an induced Poincaré first-return map P : We chose the ODE based upon a rotation about the z-axis so that the square evolves into the GAH as Q 0 makes a full rotation. The first half of the metamorphosis takes care of the vertical squeezing and horizontal stretching, while the second half produces the folding. It is not difficult to show that the system (in cylindrical coordinates)ṙ flows Q 0 to which is the original square in the radial half-plane plane corresponding to θ = 0 stretched by a factor of 1.2 along the x-axis and squeezed by a factor of 1/5 with respect to z = −0.2 along the z-axis in the radial half plane corresponding to θ = π. Consequently, (2) produces the first half of the desired result comprising the stretching and squeezing for 0 ≤ θ ≤ π.
Finally, it is not difficult to show that the Poincaré section of the transversal (and trapping region) Q 0 under the system (12) is a GAH with image that is simply a 180-degree rotation of the horseshoe in Fig. 1. However, it appears that the construction of an electronic circuit simulating (12) would be a rather formidable undertaking, so we selected a simpler system; namely, the Rössler attractor model, which is a mildly nonlinear three-dimensional ODE that has a straightforward circuit realization.

Poincaré maps and circuit realization of the Rössler attractor
We consider the Rössler attractorẋ = y − ż y = x + aẏ where we use the parameters a = 0.2, b = 0.1, and c = 10. This produces the chaotic strange attractor in Fig. 3 and it can also be realized by a rather simple electronic circuit.

The Poincaré map
One can use the algorithm in Sec. 2 to compute any Poincaré section of the attractor, however what we are particularly interested in is identifying a trapping region for a generalized attracting horseshoe. Assuming the system contains a GAH, we first look for a Poincaré section with a horseshoe-like structure as shown in Fig. 4. Now, if we can find a trapping region around this horseshoe, we will have shown evidence for the existence of a GAH. First we identify vertices of a quadrilateral that fully encompasses the horseshoe-like structure. Then using a recursive algorithm (described in Sec. 2) we compute the first return map of those vertices on that particular Poincaré section; i.e., the first iteration of the Poincaré map of those points. If the iterates are contained within that quadrilateral, the points on the quadrilateral itself can be tested. In Fig. 5 four thousand points on the quadrilateral are iterated and it is illustrated that this first return is completely contained in the quadrilateral. While this is not a proof, the grid spacing on the quadrilateral provides compelling evidence that this may be a trapping region for the GAH.
In order to provide more compelling evidence, we compute higher order iterations of the Poincaré map in Fig. 6.

Circuit realization of the Rössler system
It happens that there are several known examples of electronic circuits realizing the Rössler attractor system. We chose the one, obtained from [10], shown in Fig. 7 with a list of components in Table 1.  . While the quadrilateral edges look "continuous", it should be noted that it is in fact discretized using four thousand points, which are then mapped back to the Poincaré section (r = 5, θ = 2π/5). Plot is shown in the rotated frame withx andŷ denoting rotated axes.
The physical realization of the Rossler attractor circuit was constructed using summing amplifiers, integrators, and a multipliers. Due to the nature of this system, the operational amplifier must operate within ±15 volts in order to avoid clipping of the Rossler Attractor output waveform. In this circuit, resistors were used to represent constant values for parameters a and b in (14). A potentiometer was used to vary the parameter value of b in order to observe the bifurcations of the physical system. We first test the circuit on Multisim and observe the aforementioned bifurcations in Fig. 8.
Next we built the circuit and observed oscilloscope outputs as shown in Fig. 9. The Poincaré section that we chose was a particular vertical plane through the top arch of the output shown (see also Fig. 3). The acceptable planes were obtained by trial and error via varying the system parameters and rotation of the plane about a vertical axis through the apex of the arc.

Potential applications
One can imagine several practical applications of devices containing electronic circuit realizations of a GAH. Two, which are related to communications and intelligence gathering, immediately come to mind: First, the circuit could be embedded in a communication receiving device, and tuned to certain "static" frequencies different from those in the expected incoming messages. The strong global attracting characteristics of the circuit would separate the static from the incoming messages, thereby enhancing the receiving capabilities of First five iterations of the Poincaré map (blue markers) of the quadrilateral trapping region (red markers) with vertices located at (x,ŷ) = (−3.55, −27), (11.91, −6.6), (12, 0), (−8.5, 3.5). While the quadrilateral edges look "continuous", it should be noted that it is in fact discretized using four thousand points, which are then mapped back to the Poincaré section (r = 5, θ = 2π/5). Plot is shown in the rotated frame withx andŷ denoting rotated axes.   Figure 9: Oscilloscope output from Rössler attractor circuit system. In effect, the GAH circuit would filter out the static. Secondly, a stationary or compact mobile device incorporating the GAH circuit could be used to penetrate and analyze various communication systems. Either be connecting remotely in the case of a stationary device or directly for a mobile version, the global attracting properties could be employed to extract crucial characteristics of the system to which it is connected. Moreover, the same attracting features of the GAH circuit device could be used to absorb various parts of sent messages that would render them useless, false or somply misleading.
The two rather basic applications mentioned provide just a glimpse of the possible applications of GAH circuits, most of which would probably be related to information systems, data collection and filtering. Moreover, there are more applications that could exploit the chaotic strange attractor associated with a GAH circuit. For, example a GAH circuit device could be used either to control chaos, introduce chaos or adjust the fractal dimension of outputs of a variety of applicable processes based on dynamical systems.

Conclusions
We constructed a rather complicated nonlinear three-dimensional ordinary differential equation (ODE) having a Poincaré section that is a GAH map, but is not particularly amenable to electronic circuit realization, which was a goal of the investigation.
So,instead of the initial ODE, we selected the Rössler attractor; a mildly nonlinear threedimensional ODE that has a reasonably simple circuit realization and can actually produce GAH maps for carefully chosen Poincaré sections. We constructed the corresponding GAH circuit and used a novel iteration procedure to generate good approximations of the chaotic strange attractors associated to the GAH maps.
Finally, in addition to the experimental and analytic aspects of our investigation, we discussed a number of potential practical applications of the GAH circuit. Most of the envisioned applications were in the realms of communication and information gathering.