# Belykh map

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The Belykh map is a piecewise linear map possessing a chaotic attractor, called the Belykh attractor. The Belykh attractor contains only hyperbolic (saddle) orbits and belongs to the class of quasi-hyperbolic chaotic attractors.

## The Map

Figure 1: First image of the unit square $$S=S_1 \cup S_2$$ under the map $$f\ ,$$ where $$S_1$$ and $$S_2$$ are separated by the discontinuity line $$L(x,y)=0.$$ One iterate of $$f$$ transforms trapezoids $$S_1$$ (light brown) and $$S_2$$ (light blue) into $$fS_1$$ (dark brown) and $$fS_2$$ (dark blue), respectively.

The two-dimensional Belykh map $$f$$ is defined by $$\tag{1} \begin{cases} \bar{x}= \lambda x\\ \bar{y}= \gamma y \end{cases} \mbox{if} \;\;L(x,y)\le 0$$

and $$\tag{2} \begin{cases} \bar{x}= \lambda (x-1)+1\\ \bar{y}= \gamma (y-1)+1 \end{cases} \mbox{if} \;\;L(x,y)>0,$$

where $$L(x,y)=k(2x-1)+2y-1 \ .$$ The map $$f$$ is defined on a square $$S=\{(x,y): 0\le x\le 1,\;\; 0\le y\le 1\}\ .$$ The square $$S$$ is cut into two parts $$S_1$$ and $$S_2$$ by a line $$L(x,y)=k(2x-1)+2y-1=0$$ (see Figure 1). The dynamics of the map $$f$$ is governed by (1) on the lower part $$S_1 \ ,$$ where $$L(x,y) \le 0 \ ,$$ and by (2) on the upper part $$S_2 \ .$$

The set $$A=\bigcap \limits_{n=1}^{\infty} f^n(S)$$ is called the attractor of $$f\ ,$$ or the Belykh attractor.

The map was first introduced and studied in (Belykh, 1976; Belykh, 1980; Belykh, 1982) as a simple model of digital phase locked loop (PLL). The Belykh map is a special case of a hyperbolic map with singularities. Another examples of such a map are the Lozi map (Lozi, 1978) and a dispersing dynamical billiard (Sinai, 1970; Bunimovich, 1974).

The Belykh map is invertible if $$0<\lambda<1/2\ .$$ The ergodic properties of the Belykh map in the parameter range of invertibility were studied in (Pesin, 1992; Afraimovich et al, 1995; Sataev, 1992; Belykh 1995; Sataev, 1999). It has been proved that under the conditions $$\tag{3} 0<\lambda<1/2, \;\;1<\gamma \le \frac{2}{1+|k|}, \;\;\mbox{and}\;\; |k|<1$$

the map $$f$$ has a hyperbolic attractor in the sense that it contains only hyperbolic (saddle) orbits. However, the attractor might be structurally unstable as its saddle periodic orbits bifurcate or disappear when at least one point on the periodic orbit hits the discontinuity line $$L(x,y)=0.$$ Such structurally unstable attractors containing only saddle orbits are called quasi-hyperbolic chaotic attractors (see, for example, (Afraimovich and Hsu, 2003)). The famous Lorenz attractor (Lorenz, 1963) also belongs to this class.

Figure 2: Chaotic Belykh attractor generated by the map $$f$$ with parameters $$\lambda=0.48,$$ $$\gamma=1.3,$$ and $$k=0.5$$ for which the map is invertible.
Figure 3: Fat Belykh attractor generated by the non-invertible map $$f\ .$$ Parameters are $$\lambda=0.8,$$ $$\gamma=1.3,$$ and $$k=0.5\ .$$

Pesin (1992) and Sataev (1992) proved the existence of the Sinai-Ruelle-Bowen measure for the chaotic map $$f\ .$$

Schmeling and Troubetzkoy (1998) considered the Belykh map in a wider range of parameters ($$1/2<\lambda <1$$) where the map is non-invertible. In analogy with the fat baker's transformation (Alexander and Yorke, 1984), they called the map $$f$$ within the wider range of parameters the fat Belykh map and proved that the fat map is chaotic and has a continuous invariant measure (Schmeling, 1998; Schmeling and Troubetzkoy, 1998; Persson, 2008).

Figure 2 demonstrates the chaotic structure of the Belykh attractor in the parameter range of invertibility (3). The fat Belykh attractor in the parameter range of non-invertibility is depicted in Figure 3.

## Embedding of the Belykh attractor into 3-D phase space

Figure 4: Embedding of the Belykh attractor into the 3-D phase space of the hybrid ODE system (9). The parameters $$\delta=-{\rm ln}\, 0.48=0.73,$$ $$\sigma={\rm ln}\, 1.3=0.262,$$ and $$k=0.5$$ correspond to those of Figure 2.

The Belykh map and its planar attractor can be embedded into the phase space of a system of ordinary differential equations (ODEs). The motivation for finding possible flow embeddings of planar discrete-time chaotic attractors is two-fold. First, it gives specific examples of ODE systems, possessing strange attractors with rigorously proven chaotic properties. While planar attractors of discrete-time maps such as, for example, the baker's map, the Lozi map, and the Belykh map have been analytically shown to exhibit quasi-hyperbolic strange attractors, rigorous proofs of chaoticity in ODE systems are rare and often require computer assistance. In light of this, finding a systematic way of planar attractors embedding has its own value for the theory of dynamical systems. Second, these embeddings are an excellent way of visualizing discrete-time attractors and creating graphically appealing images.

The construction that generates the Belykh attractor is formed by two saddle-focus equilibria of two 3-D linear systems. The details of the construction are given below. The same technique can be used for embedding various piecewise-linear chaotic maps into the 3-D phase space by means of hybrid ODE systems.

Consider the following linear system of three ODEs that has a saddle-focus equilibrium at the origin$\tag{4} \left \{\begin{array}{l} \dot{x}=-\delta x\\ \dot{y}=\sigma y+ 2 \pi z,\\ \dot{z}=-2\pi y+ \sigma z \end{array} \right.$

where $$\delta$$ and $$\sigma$$ are positive parameters to be defined.


System (4) has a solution $$\tag{5} x=x_0 e^{-\delta t},\;\;y=y_0 e^{\sigma t}\cos 2\pi t,\;\;z=-y_0 e^{\sigma t}\sin 2\pi t$$

for the initial conditions $$x(0)=x_0 \ge 0,$$ $$y(0)=y_0 \ge 0,$$ and $$z(0)=0.$$ Therefore, the system (4) generates a Poincaré map of the quarter plane $$P^{+}=\{(x,y): x\ge 0,\;\; y\ge 0,\;\;z=0\}$$ into itself as the solution (5) returns to $$P^{+}$$ at every instant $$t=i\in \mathbb{Z}.$$ The Poincaré map on the cross-section $$P^{+}$$ reads $$\tag{6} \bar{x}=e^{-\delta}x,\;\;\bar{y}=e^{\sigma}y.$$

Setting $$e^{-\delta}=\lambda$$ and $$e^{\sigma}=\gamma,$$ where $$\lambda$$ and $$\gamma$$ are the original parameters of the Belykh map (1)-(2), one transforms the Poincare map (6) into the first part of the Belykh map (1) defined for the region $$S_1$$ where $$L(x,y)\le 0.$$

Changing the variables $$(x,y)\rightarrow (1-x,1-y)$$ in the system (4) yields another linear system with the saddle-focus equilibrium shifted to $$(1,1,0)\ .$$ The new system reads $$\tag{7} \left \{\begin{array}{l} \dot{x}=-\delta (x-1)\\ \dot{y}=\sigma (y-1)- 2 \pi z\\ \dot{z}=2\pi (y-1)+ \sigma z. \end{array} \right.$$

The corresponding Poincaré map on the cross-section $$P^{-}=\{(x,y): x\le 1,\;\; y\le 1,\;\;z=0\}$$ takes the form $$\tag{8} \bar{x}=e^{-\delta}(x-1)+1,\;\;\bar{y}=e^{\sigma}(y-1)+1$$

which coincides with the second part of the Belykh map (2) for $$L(x,y)>0.$$

Therefore, two ODE systems (4) and (7) act on a common cross-section $$P=P^{+} \cap P^{-}=\{(x,y):0 \le x\le 1,\;\; 0 \le y\le 1,\;\;z=0\}$$ according to the following rule $$\tag{9} \left \{\begin{array}{l} \dot{x}=-\delta [x-H(L(x_i,y_i))]\\ \dot{y}=\sigma [y-H(L(x_i,y_i))]+2\pi(-1)^{H(L(x_i,y_i))}z\\ \dot{z}=-2\pi (-1)^{H(L(x_i,y_i))}[y-H(L(x_i,y_i))]+ \sigma z, \end{array} \right.$$

where $$H \left (L(x_i,y_i) \right )$$ is the Heaviside step function with $$H(0)=0$$ and $$L(x_i,y_i)=k(2x_i-1)+2y_i-1.$$ The system (9) must be integrated over the time intervals $$t\in [i,i+1],$$ $$i=0,1,2,...$$ The Heaviside step function $$H(L(x_i,y_i))$$ can only switch its value at time $$t=i$$ when the trajectory leaves the cross-section $$P$$ to come back at time $$t=i+1\ .$$ Note that the system (9) becomes (4) when $$H(L(x_i,y_i))=0\ .$$ This happens when the trajectory hits the cross-section $$P$$ at a point for which $$L(x_i,y_i) \le 0.$$ At the same time, $$H(L(x_i,y_i))=1$$ yields the system (7).

 Figure 5: Strange attractor of the ODE system (9) generating the Belykh map on the Poincaré cross-section $$P=\{(x,y):0 \le x\le 1,\;\; 0 \le y\le 1,\;\;z=0\}$$ (bright yellow). The parameters are the same as in Figure 4. Figure 6: Corresponding $$xy$$-projection of the ODE attractor. Points of intersections with the Poincaré cross-section $$P$$ yield the Belykh attractor (cf. Figure 7). Figure 7: Belykh attractor on the Poincaré cross-section $$P$$ is identical to the one of Figure 2. The parameters $$\delta=-{\rm ln}\, 0.48=0.73,$$ $$\sigma={\rm ln}\, 1.3=0.262,$$ and $$k=0.5$$ correspond to those of Figure 2.

In short, the hybrid ODE system (9) is a way of gluing trajectories of two linear systems (4) and (7) periodically in time such that the trajectory of one system is continued by the other system if the trajectory returns to the cross-section $$P$$ on the other side of the line $$L(x_i,y_i).$$

To ensure the one-to-one correspondence between the original Belykh map (1)-(2) and the Poincaré map (6)-(8) on the cross-section $$P\ ,$$ the parameters of the system (9) must be recalculated via $$\lambda$$ and $$\gamma$$ as follows$\delta=-{\rm ln} \lambda$ and $$\sigma={\rm ln} \gamma\ .$$ The parameter $$k,$$ present in $$L(x_i,y_i),$$ remains the same.

Under these conditions, the hybrid ODE system (9) with initial conditions $$\{0 \le x(0) \le 1,\;\; 0 \le y(0) \le 1,\;\;z(0)=0\}$$ acts as the Belykh map on the cross-section $$P$$ ( Figure 5-Figure 6) and generates a quasi-hyperbolic Belykh attractor ( Figure 7).

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