*2.1.2. Topological structure of singular attractors*

The problem which can be named as a main problem of chaotic dynamics of nonlinear systems of differential equations, is to find out how the boundary of the separatrix surface of the original singular cycle becomes more complex as the bifurcation parameter increases and how the onset of infinitely many regular and singular attractors of the system settle down on this separatrix manifold in accordance with a certain order (Sharkovskii order, homoclinic or heteroclinic order).

Note, that the simplest performance of a two-dimensional manifold in three-dimensional space on which all cycles in the Sharkovskii order and singular attractors can be placed without self-intersections was found in (Gilmore & Lefranc, 2002) in the form of branching manifold with the use of the Birman-Williams theorem and the principles of symbolic dynamics. However, such manifold must have a gluing, so that one can use it to explain the chaotic structure of semiflows but cannot generalize these results to flows, because this contradicts with the uniqueness theorem for solutions of differential equations. Hence, the representation given in (Gilmore & Lefranc, 2002) cannot be considered satisfactory.

We obtained a representation of the boundary of the separatrix surface of an original singular cycle of an arbitrary nonlinear dissipative system in a form of an infinitely folded two-dimensional heteroclinic separatrix manifold which Poincare section is named as **heteroclinic separatrix zigzag** (Magnitskii, 2010). It spanned by Moebius bands joining various cycles from the Feigenbaum period doubling cascade of bifurcations. From this consideration it becomes clear how and why cycles are arranged on this manifold in subharmonic and homoclinic order in the case of sufficiently strong dissipation, and why this order can be violated in systems with small dissipation and in conservative systems.

Rewrite the system of Eqs. (4) in the form of autonomous 4d-system

136 Nonlinearity, Bifurcation and Chaos – Theory and Applications

and for growth of the parameter

Magnitskii) scenario of transition to dynamical chaos is realized.

cycles of periods four (d) and five (e) in the system of Eqs. (4).

*2.1.2. Topological structure of singular attractors* 

homoclinic or heteroclinic order).

Eqs. (4) in accordance with the Sharkovskii order (3) and then in accordance with the Magnitskii homoclinic order. These cycles of period two and three, one of the singular attractors and homoclinic cycles of periods four and five are presented in Fig.1 (Magnitskii & Sidorov, 2006). Thus, in this system full bifurcation FSM (Feigenbaum-Sharkovskii-

**Figure 1.** Stable cycles of period two (a) and period three (b), singular attractor (c) and homoclinic

The problem which can be named as a main problem of chaotic dynamics of nonlinear systems of differential equations, is to find out how the boundary of the separatrix surface of the original singular cycle becomes more complex as the bifurcation parameter increases and how the onset of infinitely many regular and singular attractors of the system settle down on this separatrix manifold in accordance with a certain order (Sharkovskii order,

Note, that the simplest performance of a two-dimensional manifold in three-dimensional space on which all cycles in the Sharkovskii order and singular attractors can be placed without self-intersections was found in (Gilmore & Lefranc, 2002) in the form of branching manifold with the use of the Birman-Williams theorem and the principles of symbolic dynamics. However, such manifold must have a gluing, so that one can use it to explain the chaotic structure of semiflows but cannot generalize these results to flows, because this contradicts with the uniqueness theorem for solutions of differential equations. Hence, the

representation given in (Gilmore & Lefranc, 2002) cannot be considered satisfactory.

We obtained a representation of the boundary of the separatrix surface of an original singular cycle of an arbitrary nonlinear dissipative system in a form of an infinitely folded two-dimensional heteroclinic separatrix manifold which Poincare section is named as **heteroclinic separatrix zigzag** (Magnitskii, 2010). It spanned by Moebius bands joining various cycles from the Feigenbaum period doubling cascade of bifurcations. From this consideration it becomes clear how and why cycles are arranged on this manifold in subharmonic and homoclinic order in the case of sufficiently strong dissipation, and why this order can be violated in systems with small dissipation and in conservative systems.

stable cycles are generated in the system of

For 4 

$$\begin{aligned} \dot{u} &= (2(\mu - b) + 2bp)u + (2bq - \alpha/2)v - v^2, & \dot{p} &= -\alpha q \\ \dot{v} &= (2bq + \alpha/2)u + (2(\mu - b) - 2bp)v, & \dot{q} &= \alpha p \end{aligned} \tag{5}$$

with 2 *b uv* 1, and with the cycle 2 2 *p q* <sup>1</sup> . The parameter in system of Eqs. (5) is a bifurcation parameter, and the parameter *b* is responsible for dissipation. For small *b* and small the system is weakly dissipative, for large *b* and small it is strongly dissipative. Besides at *b* the system of Eqs. (5) is conservative.

As a rule, all known dissipative systems of nonlinear differential equations are strongly dissipative, which has for many decades prevented one from studying the structure of their irregular attractors even with the use of most advanced computers. Last circumstance stimulated the development of numerous definitions of irregular attractors, ostensibly distinguished in their topological structure (strange, chaotic, stochastic, etc.). We illustrate this circumstance by the example of system of Eqs. (5) with strong dissipation for *b* 1, 4 , that is for the system of Eqs. (4). In this case, as the parameter 0 increases, system of Eqs. (5) has not only a complete subharmonic cascade of bifurcations in accordance with the Sharkovskii order, but also it has complete homoclinic cascade of bifurcations of cycles converging to the rotor homoclinic loop. The cause is clarified in Fig. 2a in which the Poincare section ( 0, 0) *q p* of the singular attractor of system of Eqs. (5) for 0.12 lying between cycles of period 5 and 3 in the Sharkovskii order is shown. The graph of the section almost coincides with the graph of onedimensional unimodal mapping of a segment into itself, which has the above-listed cascades of bifurcations (Feigenbaum, 1978; Sharkovskii, 1964; Magnitskii & Sidorov, 2006). The projection of the manifold of the singular attractor onto the plane (,) *p u* corresponding to the section is shown in Fig. 2b.

**Figure 2.** The projection of the Poincare section ( 0, 0) *q p* of solution of system of Eqs. (5) for *b* 1, 4, 0.12 (a) and the projection of the manifold of the singular attractor onto the plane (,) *p u* corresponding to the section (b).

It seems that this is a two-dimensional strip whose lower part rotates around the original cycle, goes into its upper part in a revolution around it without twisting, and, in turn, the upper part goes into the lower part with twisting by 180 degrees in the next revolution. But in this case, to avoid contradiction with uniqueness theorem for solutions of systems of differential equations, two branches of the upper part should go into two branches of the lower part, which can be detected even under tenfold magnification (Fig.2a). Therefore, the upper part of the graph of the section in Fig. 2a should also consist of two branches, which makes its lower part to consist of four branches and so on. Consequently, the invariant manifold of the singular attractor shown in Fig. 2 should be a two-dimensional infinitelysheeted folded surface. However, strong dissipation of the system in this case prevents correct understanding a topological structure of separatrix manifold of original singular cycle.

Universality of Transition to Chaos in All Kinds of Nonlinear Differential Equations 139

(a) and the projection of the manifold of the singular attractor onto the plane

has simultaneously two stable cycles of periods one and three (for

trajectory contains singular unstable cycles from the Feigenbaum cascade. This nonperiodic almost stable trajectory is the Feigenbaum attractor, which is the first and the simplest

**Figure 3.** Projection of the Poincare section ( 0, 0) *q p* of solution of system of Eqs. (5) for

Along with separatrix branches connecting unstable singular cycles of the Feigenbaum cascade, the Feigenbaum separatrix tree contains stable one-sided separatrix branches only entering cycles. These branches begin to close each other and form heteroclinic separatrix folds passing through various stable and unstable cycles from the Sharkovskii subharmonic cascade of bifurcations, which are generated during saddle-node bifurcations. New Feigenbaum separatrix trees are generated on the separatrices of the newly generated singular cycles, and the tops of these trees contain more complicated singular attractors. An infinitely folded separatrix two-dimensional manifold, which Poincare section is referred to as a **heteroclinic separatrix zigzag** is thereby generated. It is shown in Fig. 3a. In the case of weak dissipation Poincare section of solutions of the system (a heteroclinic separatrix zigzag) is already not close to the graph of the one-dimensional unimodal mapping, which leads to the violation of the Sharkovskii order in its right-hand side, i.e. cycles of periods 7, 5 and 3 may not exist in the system but may also be stable either simultaneously with cascades of bifurcations of some other cycle or without them. For example, system of Eqs. (5)

Thus, any unstable cycle of the system is unstable singular cycle joining neiboring separatrices of a heteroclinic zigzag. Any simple singular attractor is almost stable nonperiodic trajectory passing through vertices of some infinite Feigenbaum tree. Any trajectory of system from the attraction domain of the separatrix zigzag is first attracted to it along the nearest stable Moebius band, then approaches unstable sheets, goes along them, and tends either to a stable cycle or to a singular attractor depending on value of bifurcation

attractor in the infinite family of singular attractors.

*b* 0.05, 0.8, 0.02 

for *b* 0.05, 1.5 

0.0355 ).

parameter.

(,) *u v* corresponding to the section (b).

So, let us analyze the behavior of attractors of the system of Eqs. (5) with weak dissipation for *b* 0.05, 0.8 . A stable cycle of the double period, which is the boundary of the unstable Moebius band (an unstable two-dimensional manifold) of the original unit singular cycle 2 2 *p q* 1 , is generated in system of Eqs. (5) for small 0 . It is an ordinary simple cycle of the period 4 / in the projection onto the two-dimensional subspace (u,v) . Initially this cycle has two multipliers lying on the positive part of the real axis inside the unit circle and moving towards each other as the parameter grows. Then multipliers meet, become complex conjugated and continue to move on positive and negative half-circles inside an unit circle towards the negative part of the real axis. In this case, the unstable Moebius band of the original singular cycle becomes a complex roll around the stable cycle of the double period. The frequency of rotation of a trajectory on the roll around the stable cycle of the double period is specified by the frequency and also by imaginary parts of complex conjugated multipliers. Therefore, the approach of the multipliers to the negative part of the real axis leads to the flattening of the roll in one direction and to its degeneration into a stable Moebius band around the stable cycle of the double period.

Further multipliers of the cycle begin to move along the negative part of the real axis in opposite directions, which leads to appearance of two stable two-dimensional manifolds in the form of two transversal Moebius bands for the cycle of double period. Therefore, the cycle of double period becomes a singular stable cycle. Next, at the moment of intersection of the unit circle by one of the multiplies at the point -1, the cycle of double period becomes an unstable singular cycle, whose stable and unstable manifolds are two transversal Moebius bands. The boundary of its unstable manifold is a stable cycle of quadruple period. Thus, we came to an original situation, but for a singular cycle of double period.

The cascade of Feigenbaum period doubling bifurcations continues, up to infinity, the process of construction of a two-dimensional heteroclinic separatrix manifold, which consists of Moebius bands, joining the unstable singular cycles of the cascade. The selfsimilar separatrix figure obtained in the Poincare section is referred to as **Feigenbaum separatrix tree**. A nonperiodic stable trajectory passes through the top of the Feigenbaum tree and through the endpoints of all of its branches, and each neighborhood of that trajectory contains singular unstable cycles from the Feigenbaum cascade. This nonperiodic almost stable trajectory is the Feigenbaum attractor, which is the first and the simplest attractor in the infinite family of singular attractors.

138 Nonlinearity, Bifurcation and Chaos – Theory and Applications

cycle.

for *b* 0.05, 0.8 

cycle of the period 4 /

It seems that this is a two-dimensional strip whose lower part rotates around the original cycle, goes into its upper part in a revolution around it without twisting, and, in turn, the upper part goes into the lower part with twisting by 180 degrees in the next revolution. But in this case, to avoid contradiction with uniqueness theorem for solutions of systems of differential equations, two branches of the upper part should go into two branches of the lower part, which can be detected even under tenfold magnification (Fig.2a). Therefore, the upper part of the graph of the section in Fig. 2a should also consist of two branches, which makes its lower part to consist of four branches and so on. Consequently, the invariant manifold of the singular attractor shown in Fig. 2 should be a two-dimensional infinitelysheeted folded surface. However, strong dissipation of the system in this case prevents correct understanding a topological structure of separatrix manifold of original singular

So, let us analyze the behavior of attractors of the system of Eqs. (5) with weak dissipation

unstable Moebius band (an unstable two-dimensional manifold) of the original unit singular

this cycle has two multipliers lying on the positive part of the real axis inside the unit circle

complex conjugated and continue to move on positive and negative half-circles inside an unit circle towards the negative part of the real axis. In this case, the unstable Moebius band of the original singular cycle becomes a complex roll around the stable cycle of the double period. The frequency of rotation of a trajectory on the roll around the stable cycle of the

conjugated multipliers. Therefore, the approach of the multipliers to the negative part of the real axis leads to the flattening of the roll in one direction and to its degeneration into a

Further multipliers of the cycle begin to move along the negative part of the real axis in opposite directions, which leads to appearance of two stable two-dimensional manifolds in the form of two transversal Moebius bands for the cycle of double period. Therefore, the cycle of double period becomes a singular stable cycle. Next, at the moment of intersection of the unit circle by one of the multiplies at the point -1, the cycle of double period becomes an unstable singular cycle, whose stable and unstable manifolds are two transversal Moebius bands. The boundary of its unstable manifold is a stable cycle of quadruple period.

The cascade of Feigenbaum period doubling bifurcations continues, up to infinity, the process of construction of a two-dimensional heteroclinic separatrix manifold, which consists of Moebius bands, joining the unstable singular cycles of the cascade. The selfsimilar separatrix figure obtained in the Poincare section is referred to as **Feigenbaum separatrix tree**. A nonperiodic stable trajectory passes through the top of the Feigenbaum tree and through the endpoints of all of its branches, and each neighborhood of that

Thus, we came to an original situation, but for a singular cycle of double period.

cycle 2 2 *p q* 1 , is generated in system of Eqs. (5) for small 0

stable Moebius band around the stable cycle of the double period.

double period is specified by the frequency

and moving towards each other as the parameter

. A stable cycle of the double period, which is the boundary of the

in the projection onto the two-dimensional subspace (u,v) . Initially

. It is an ordinary simple

grows. Then multipliers meet, become

and also by imaginary parts of complex

**Figure 3.** Projection of the Poincare section ( 0, 0) *q p* of solution of system of Eqs. (5) for *b* 0.05, 0.8, 0.02 (a) and the projection of the manifold of the singular attractor onto the plane (,) *u v* corresponding to the section (b).

Along with separatrix branches connecting unstable singular cycles of the Feigenbaum cascade, the Feigenbaum separatrix tree contains stable one-sided separatrix branches only entering cycles. These branches begin to close each other and form heteroclinic separatrix folds passing through various stable and unstable cycles from the Sharkovskii subharmonic cascade of bifurcations, which are generated during saddle-node bifurcations. New Feigenbaum separatrix trees are generated on the separatrices of the newly generated singular cycles, and the tops of these trees contain more complicated singular attractors. An infinitely folded separatrix two-dimensional manifold, which Poincare section is referred to as a **heteroclinic separatrix zigzag** is thereby generated. It is shown in Fig. 3a. In the case of weak dissipation Poincare section of solutions of the system (a heteroclinic separatrix zigzag) is already not close to the graph of the one-dimensional unimodal mapping, which leads to the violation of the Sharkovskii order in its right-hand side, i.e. cycles of periods 7, 5 and 3 may not exist in the system but may also be stable either simultaneously with cascades of bifurcations of some other cycle or without them. For example, system of Eqs. (5) for *b* 0.05, 1.5 has simultaneously two stable cycles of periods one and three (for 0.0355 ).

Thus, any unstable cycle of the system is unstable singular cycle joining neiboring separatrices of a heteroclinic zigzag. Any simple singular attractor is almost stable nonperiodic trajectory passing through vertices of some infinite Feigenbaum tree. Any trajectory of system from the attraction domain of the separatrix zigzag is first attracted to it along the nearest stable Moebius band, then approaches unstable sheets, goes along them, and tends either to a stable cycle or to a singular attractor depending on value of bifurcation parameter.
