*2.1.3. Some examples of classical two-dimensional nonautonomous systems*

Consider three classical nonlinear ordinary differential equations of the second order with periodic coefficients such as Duffing-Holmes equation

$$
\ddot{\mathbf{x}} + k\dot{\mathbf{x}} + a\rho^2 \mathbf{x} + \mu \mathbf{x}^3 = f\_0 \cos \Omega t,\tag{6}
$$

Universality of Transition to Chaos in All Kinds of Nonlinear Differential Equations 141

**Figure 6.** Original cycle (a), cycle of period two (b), Feigenbaum attractor (c), cycle of period three (d) from subharmonic cascade and more complex singular attractor (e) in the Croquette equation (8).

Consider a smooth family of three-dimensional nonlinear dissipative autonomous systems

 

It is shown by the author in (Magnitskii & Sidorov, 2006; Magnitskii, 2008) that if a threedimensional system of Eqs. (9) has a singular cycle of period *T* defined by complex Floquet exponents with equal imaginary parts (i.e. Moebius bands are its stable and unstable invariant manifolds), then by passing to a coordinate system rotating around the cycle, one can reduce such system to a two-dimensional nonautonomous system in coordinates, transversal to the singular cycle with zero rotor-type singular point corresponding to the cycle. So, all arguments listed in the previous section hold completely for autonomous three-

Therefore, three-dimensional autonomous system with singular cycle should have the same FSM scenario of transition to chaos as two-dimensional nonautonomous system with periodic coefficients and zero rotor-type singular point. As an example, consider the

> 22 22 1 2 1 1 2 11 2 2 3 22 22 2 1 2 1 2 11 2 2 3

> >

( ) ( ), ( ),(0,0,1) , 0 0 *<sup>T</sup> Qt x t x t* one can reduce the system of Eqs. (10) to two-dimensional

*x x x x x xx x x x x x x x x xx x x x*

 

> 

[(( 1) )( 1) (2 / 2) ],

 

in the plane of variables 1 2 (,) *x x* . By

(10)

and limit cycle



[(( 1) )( 1) (2 / 2) ],

with 2 /

2 2 3 2 12 1 3

 

 

with period 2 / *T*

changing the variables <sup>0</sup> 1 2 ( , ) , ( , )(0, ( ), ( ))*<sup>T</sup> xt x t Qt u t u t* 

> 

*x x xx x x*

( / 4)( 1) 2( 1 ) .

system of Eqs.(10) has the singular point (0,0, / 8( 1))

.

*М R I R FC* , ,, (9)

**2.2. Three-dimensional autonomous systems** 

 <sup>3</sup> *x Fx x* ( , ), 

depending on a scalar system parameter

dimensional systems with singular cycles.

*2.2.1. FSM – scenario of transition to chaos* 

autonomous three-dimensional system

 

<sup>0</sup> , (cos ,sin ,0)*<sup>T</sup> xt t t*

 

nonautonomous system with 2 /

For 1 

of ordinary differential equations

modified dissipative Mathieu equation

$$
\ddot{\mathbf{x}} + \mu \dot{\mathbf{x}} + (\delta + \mathbf{z} \cos \alpha t)\mathbf{x} + \alpha \mathbf{x}^3 = \mathbf{0},\tag{7}
$$

and Croquette dissipative equation

$$
\ddot{\mathbf{x}} + \mu \dot{\mathbf{x}} + \alpha \sin \mathbf{x} + \beta \sin(\mathbf{x} - \alpha \vartheta) = \mathbf{0}.\tag{8}
$$

All these equations are equivalent to two-dimensional nonlinear dissipative systems of ordinary differential equations with periodic coefficients and all of them have the same universal FSM scenario of transition to dynamical chaos (Magnitskii & Sidorov, 2006). For these equations, some important stable cycles and singular subharmonic attractors in accordance with the Sharkovskii order are presented in Fig. 4 - Fig. 6.

**Figure 4.** Original cycle (a), cycle of period two (b), Feigenbaum attractor (c), cycle of period six (d) from subharmonic cascade and more complex singular attractor (e) in the Duffing-Holmes equation (6).

**Figure 5.** Original cycle (a), cycle of period two (b), Feigenbaum attractor (c), cycle of period three (d) from subharmonic cascade and more complex singular attractor (e) in the Mathieu equation (7).

Note that double period bifurcations were found also in (Awrejcewicz, 1989; Awrejcewicz 1991) for some other nonlinear ordinary differential equations of the second order with periodic coefficients.

**Figure 6.** Original cycle (a), cycle of period two (b), Feigenbaum attractor (c), cycle of period three (d) from subharmonic cascade and more complex singular attractor (e) in the Croquette equation (8).
