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UDC 517.9
MATHEMATICS
L. P. SHILNIKOV
ON THE EXISTENCE OF A COUNTABLE SET OF PERIODIC MOTIONS IN FOUR-DIMENSIONAL SPACE IN AN EXTENDED NEIGHBORHOOD OF A SADDLE-FOCUS
(Presented by Academician L. S. Pontryagin on 11 III 1966)
Consider a system of 4 differential equations
\[ dz/dt = Z(z), \tag{1} \]
where \(Z(z)\) is an analytic vector-function. Suppose that system (1) has an equilibrium state \(O\), at which the characteristic equation
\[ |\partial Z/\partial z-\lambda E|=0 \]
has two pairs of complex conjugate roots \(\lambda \pm i\omega,\ \gamma \pm i\Omega\), where \(\lambda<0,\ \gamma>0\). Such an equilibrium state \(O\) will be called a saddle-focus. As is known, in this case system (1) has two two-dimensional integral manifolds \(\mathfrak M^{+}\) and \(\mathfrak M^{-}\), on which, respectively, the \(0^{+}\)-curves and \(0^{-}\)-curves lie.
Suppose that there exists a trajectory \(\Gamma_0\), leaving \(O\) and returning to it again as \(t\to+\infty\), i.e. \(\Gamma_0\subset(\mathfrak M^{+}\cap\mathfrak M^{-})\). We shall assume that the indicated intersection is non-rough of first degree. This means that the relation
\[ \dim(\mathcal L\mathfrak M_M^{+}\cap \mathcal L\mathfrak M_M^{-})=1, \tag{2} \]
holds, where by \(\mathcal L\mathfrak M_M^{+}\) and \(\mathcal L\mathfrak M_M^{-}\) are denoted the tangent spaces to \(\mathfrak M_M^{+}\) and \(\mathfrak M_M^{-}\) at the point \(M\in\Gamma_0\).
Let \(U(\Gamma_0)\) be some neighborhood of \(\Gamma_0\). Obviously, \(O\in U(\Gamma_0)\). The domain \(U(\Gamma_0)\) will be called an extended neighborhood of the saddle-focus.
Theorem. If \(-\lambda\ne\gamma\) and condition (2) is fulfilled, then any extended neighborhood of the saddle-focus contains a countable set of periodic motions of saddle type.
Without loss of generality one may assume that \(\lambda+\gamma<0\).
By means of a linear transformation, system (1) can be reduced to the form:
\[ \begin{aligned} dx_1/dt &= \lambda x_1-\omega x_2+P, &\qquad dx_2/dt &= \omega x_1+\lambda x_2+P_2,\\ dy_1/dt &= \gamma y_1-\Omega y_2+Q, &\qquad dy_2/dt &= \Omega y_1+\gamma y_2+Q_2, \end{aligned} \tag{3} \]
where \(O(0,0,0,0)\) is a saddle-focus; the functions \(P_1(x_1,x_2,y_1,y_2),\ldots,Q_2(x_1,x_2,y_1,y_2)\) vanish at the origin together with their first derivatives.
In a sufficiently small neighborhood of \(O\), the equations of \(\mathfrak M^{+}\) will be written in the form
\[ y_1=\varphi_1(x_1,x_2),\qquad y_2=\varphi_2(x_1,x_2), \]
and the equations of \(\mathfrak M^{-}\) in the form
\[ x_1=\psi_1(y_1,y_2),\qquad x_2=\psi_2(y_1,y_2). \]
* The existence of a countable set of periodic motions in a neighborhood of \(\Gamma_0\), leaving the saddle and returning to it as \(t\to+\infty\), was discovered in works \((^3,^4)\) under the assumption that \(\Gamma_0\) as \(t\to-\infty\) enters the saddle, tangent to a certain one-dimensional axis.
Note that \(\varphi_1,\ldots,\psi_2\) are analytic functions, vanishing at the origin together with their first derivatives. After the substitution
\[ \begin{aligned} \xi_1&=x_1-\psi_1(y_1,y_2),&\qquad \xi_2&=x_2-\psi_2(y_1,y_2),\\ \eta_1&=y_1-\varphi_1(x_1,x_2),&\qquad \eta_2&=y_2-\varphi(x_1,x_2), \end{aligned} \]
in some neighborhood \(O\) the system takes the form
\[ \begin{aligned} d\xi_1/dt&=\lambda \xi_1-\omega \xi_2+P_{11}\xi_1+P_{12}\xi_2,\\ d\xi_2/dt&=\omega \xi_1+\lambda \xi_2+P_{21}\xi_1+P_{22}\xi_2,\\ d\eta_1/dt&=\gamma \eta_1-\Omega \eta_2+Q_{11}\eta_1+Q_{12}\eta_2,\\ d\eta_2/dt&=\Omega \eta_1+\gamma \eta_2+Q_{21}\eta_1+Q_{22}\eta_2, \end{aligned} \tag{4} \]
where \(P_{ij}(0,\ldots,0)=Q_{ij}(0,\ldots,0)=0\). In the new variables the equations of \(\mathfrak{M}^{+}\) will be \(\eta_1=0,\eta_2=0\), and those of \(\mathfrak{M}^{-}\) will be \(\xi_1=0,\xi_2=0\).
Denote by \(S_0\) the surface \(\xi_1^2+\xi_2^2=r_2,\ \eta_1^2+\eta_2^2\le r^2\), and by \(S_1\) the surface \(\eta_1^2+\eta_2^2=r^2,\ \xi_1^2+\xi_2^2\le r^2\). From the form of equations (4) it readily follows that
Lemma 1. For all sufficiently small \(r>0\), the surfaces \(S_0\) and \(S_1\) are surfaces without contact for the trajectories of system (4).
Let
\[ \xi_i(t)=\xi_i(t,\xi_1^0,\xi_2^0,\eta_1^0,\eta_2^0), \qquad \eta_i(t)=\eta_i(t,\xi_1^0,\xi_2^0,\eta_1^0,\eta_2^0) \]
be the equation of a trajectory \(l\) of system (4) passing through the point
\(M_0(\xi_1^0,\xi_2^0,\eta_1^0,\eta_2^0)\in S_0\) at \(t=0\), where
\(\eta_1^{0\,2}+\eta_2^{0\,2}\ne0\). From Lemma 1 we obtain that \(l\), for some \(t_0\), intersects \(S_1\) at the point
\(M_1(\xi_1^1,\xi_2^1,\eta_1^1,\eta_2^1)\), where \(\xi_1^{1\,2}+\xi_2^{1\,2}<r^2\). We denote this mapping of \(S_0\) into \(S_1\) along trajectories by \(T_0\). Evidently, it will be written in the form:
\[ \xi_i^1=\xi_i(t_0,\xi_1^0,\xi_2^0,\eta_1^0,\eta_2^0), \qquad \eta_i^1=\eta_i(t_0,\xi_1^0,\xi_2^0,\eta_1^0,\eta_2^0), \tag{5} \]
where the transition time \(t_0\) of the phase point from \(S_0\) to \(S_1\) is found from the equation
\[ \eta_1^2(t_0,\xi_1^0,\xi_2^0,\eta_1^0,\eta_2^0) + \eta_2^2(t_0,\xi_1^0,\xi_2^0,\eta_1^0,\eta_2^0) =r^2. \]
Along with this form of writing the mapping \(T\), which was constructed in an analogous way in works \((1\text{--}4)\), we shall here construct another representation of the mapping \(T_0\), which we shall call parametric.
Consider the system of integral equations
\[ \begin{aligned} \xi_1(t)&=e^{\lambda t}\,[\,\xi_1^0\cos\omega t-\xi_2^0\sin\omega t\,]+\\ &\quad+\int_0^t e^{\lambda(t-\tau)} [\,\overline{P}_1\cos\omega(t-\tau)-\overline{P}_2\sin\omega(t-\tau)\,]\,d\tau,\\ \xi_2(t)&=e^{\lambda t}\,[\,\xi_1^0\sin\omega t+\xi_2^0\cos\omega t\,]+\\ &\quad+\int_0^t e^{\lambda(t-\tau)} [\,\overline{P}_1\sin\omega(t-\tau)+\overline{P}\cos\omega(t-\tau)\,]\,d\tau,\\ \eta_1(t)&=e^{\gamma(t-t_0)}[\,\eta_1^1\cos\Omega(t-t_0)-\eta_2^1\sin\Omega(t-t_0)\,]+\\ &\quad+\int_{t_0}^t e^{\gamma(t-\tau)} [\,\overline{Q}_1\cos\Omega(t-\tau)-\overline{Q}_2\sin\Omega(t-\tau)\,]\,d\tau,\\ \eta_2(t)&=e^{\gamma(t-t_0)}[\,\eta_1^1\sin\Omega(t-t_0)+\eta_2^1\cos\Omega(t-t_0)\,]+\\ &\quad+\int_{t_0}^t e^{\gamma(t-\tau)} [\,\overline{Q}_1\sin\Omega(t-\tau)+\overline{Q}_2\cos\Omega(t-\tau)\,]\,d\tau, \end{aligned} \tag{6} \]
where \(\overline{P}_1\) denotes the nonlinear terms of the first equation of system (4), \(\overline{P}_2\) those of the second, etc.
From the method of successive approximations it follows easily
Lemma 2. There exists a sufficiently small neighborhood \(\Sigma\) of the origin of the coordinates such that, under the condition \(M_0(\xi_1^0,\xi_2^0,0,0)\in\Sigma\), \(M_1(0,0,\eta_1^1,\eta_2^1)\in\Sigma\), the solution of system (6) for all \(0\leq t\leq t_0\) exists and is unique.
By verification we are convinced that the solution found,
\[ \begin{aligned} \xi_i(t)&=\xi_i^{\Pi}(t,t_0,\xi_1^0,\xi_2^0,\eta_1^0,\eta_2^0),\\ \eta_i(t)&=\eta_i^{\Pi}(t,t_0,\xi_1^0,\xi_2^0,\eta_1^0,\eta_2^0),\quad i=1,2, \end{aligned} \tag{7} \]
is a solution of system (4), satisfying the conditions
\[ \xi_i(0)=\xi_i^0,\qquad \eta_i(t_0)=\eta_i^1,\qquad i=1,2. \]
Lemma 3. The solution (7) is representable in the form:
\[ \begin{aligned} \xi_1(t)&=e^{\lambda t}\bigl[\xi_1^0(1+\alpha_{11}^0+\beta_{11}^0)\cos\omega t -\xi_2^0(1+\alpha_{12}^0+\beta_{12}^0)\sin\omega t\bigr],\\ \xi_2(t)&=e^{\lambda t}\bigl[\xi_1^0(1+\alpha_{21}^0+\beta_{21}^0)\sin\omega t +\xi_2^0(1+\alpha_{22}^0+\beta_{22}^0)\cos\omega t\bigr],\\ \eta_1(t)&=e^{\gamma(t-t_0)}\bigl[\eta_1^1(1+\alpha_{11}^1+\beta_{11}^1)\cos\Omega(t-t_0)-\\ &\qquad\qquad\qquad -\eta_2^1(1+\alpha_{12}^1+\beta_{12}^1)\sin\Omega(t-t_0)\bigr],\\ \eta_2(t)&=e^{\gamma(t-t_0)}\bigl[\eta_1^1(1+\alpha_{21}^1+\beta_{21}^1)\sin\Omega(t-t_0)+\\ &\qquad\qquad\qquad +\eta_2^1(1+\alpha_{22}^1+\beta_{22}^1)\cos\Omega(t-t_0)\bigr], \end{aligned} \tag{8} \]
where \(\alpha_{ij}^0(t-t_0,\xi_1^0,\xi_2^0,\eta_1^1,\eta_2^1)\), \(\alpha_{ij}^1(t,\xi_1^0,\xi_2^0,\eta_1^1,\eta_2^1)\), \(\beta_{ij}^k(t,t_0,\xi_1^0,\ldots,\eta_2^1)\) are analytic functions satisfying the conditions: \(\alpha_{ij}^k(0,\xi_1^0,\ldots,\eta_2^1)\) vanish for \(\xi_1^0=\xi_2^0=\eta_1^1=\eta_2^1=0\), while \(\beta_{ij}^0(t_0,t_0,\xi_1^0,\xi_2^0,\eta_1^1,\eta_2^1)\), \(\beta_{ij}^1(0,t_0,\xi_1^0,\xi_2^0,\eta_1^1,\eta_2^1)\) tend to zero as \(t_0\to+\infty\) together with their first derivatives.
Introduce the functions
\[ \xi_i^1=\xi_i^{\Pi}(t_0,t_0,\xi_1^0,\xi_2^0,\eta_1^1,\eta_2^1),\qquad \eta_i^0=\eta_i^{\Pi}(0,t_0,\xi_1^0,\xi_2^0,\eta_1^1,\eta_2^1). \tag{9} \]
From lemma (3) there will follow the estimates
\[ |\xi_1^1|+|\xi_2^1|<Ke^{\lambda t_0},\qquad |\eta_1^0|+|\eta_2^0|<Ke^{-\gamma t_0}, \tag{10} \]
\[ D\xi_1^1+D\xi_2^1<Ke^{\lambda\cdot 2\pi t_0},\qquad D\eta_1^0+D\eta_1^1<Ke^{-\gamma\cdot 2\pi t_0}, \tag{11} \]
where \(D=|\partial/\partial t_0|+\Sigma|\partial/\partial\xi_i^0|+\Sigma|\partial/\partial\eta_i^1|\), and \(K\) is a certain constant. Obviously, when \(\xi_1^{0\,2}+\xi_2^{0\,2}=r^2\), \(\eta_1^{1\,2}+\eta_2^{1\,2}=r^2\), where \(r\) is sufficiently small, formulas (9) give a new form for writing the map \(T_0\).
Without loss of generality one may assume \(r\) to be such that \(\Gamma_0\) intersects \(S_0\) and \(S_1\) at the points \(M_0^+(r,0,0,0)\) and \(M_1^-(0,0,r,0)\). From the theorems on continuous dependence on initial conditions it follows that the trajectory of system (4) intersecting \(S_1\) at points close to the point \(M_1^-\) will also intersect \(S_0\) at points close to the point \(M_1^-\). In the variables under consideration this correspondence, which we denote by \(T_1\), can be written in a sufficiently small neighborhood of \(M_1^-\) in the form:
\[ \bar{\eta}_i^0=f(\xi_1^1,\xi_2^1,\eta_2^1) =A_1\xi_1^1+A_2\xi_2^1+B\eta_2^1+\ldots, \]
\[ \bar{\xi}_i^0=g_i(\xi_1^1,\xi_2^1,\eta_2^1) =A_{i1}\xi_1^1+A_{i2}\xi_2^1+B_i\eta_2^1+\ldots,\qquad i=1,2. \tag{12} \]
In the linear approximation, the trace of the intersection of \(\mathfrak{M}^-\) with \(S_0\) has the equations
\[ \xi_2^0=B\eta_2^1,\qquad \eta_1^0=B_1\eta_2^1,\qquad \eta_2^0=B_2\eta_2^1. \tag{13} \]
It follows from condition (2) that \(B_1\) and \(B_2\) are not simultaneously equal to zero.
Since, for sufficiently small \(\bar\eta_1^0\) and \(\bar\eta_2^0\), the trajectory passing through the point
\(\overline{M}_0(\bar\xi_1,\bar\xi_2^0,\bar\eta_1^0,\bar\eta_2^1)\) intersects \(S_1\) after time \(\bar t_0\) at the point
\(\overline{M}_1(\bar\xi_1,\bar\xi_2^1,\bar\eta_1^1,\bar\eta_2^1)\), then, by virtue of what was indicated above, the relation between the last two coordinates of the points \(\overline{M}_0\) and \(\overline{M}_1=T\overline{M}_0\) can also be written in the form:
\[ \bar\eta_i^0=\eta_i^{\mathrm{II}}(0,\bar t_0,\bar\xi_1^0,\bar\xi_2^0,\bar\eta_1^1,\bar\eta_2^1), \]
where \(\bar\xi_1^0=\sqrt{r^2-(\bar\xi_2^0)^2}\), \(\bar\eta_1^1=\sqrt{r^2-(\bar\eta_2^1)^2}\). Consider the mapping \(T=T_1T_0\)
\[ \begin{gathered} \xi_2^0=f(\xi_1^{\mathrm{II}}(t_0,t_0,\xi_1^0,\ldots,\eta_2^1), \xi_2^{\mathrm{II}}(t_0,t_0,\xi_1^0,\ldots,\eta_2^1),\eta_2^1), \\ \eta_i^{\mathrm{II}}(0,t_0,\xi_1^0,\ldots,\eta_2^1) =g_i(\xi_1^{\mathrm{II}}(t_0,t_0,\xi_1^0,\ldots,\eta_2^1), \xi_2^{\mathrm{II}}(t_0,t_0,\xi_1^0,\ldots,\eta_2^1)\eta_2^1), . \end{gathered} \tag{14} \]
where \(\xi_1^0=\sqrt{r^2-(\xi_2^0)^2}\), \(\eta_1^1=\sqrt{r^2-(\eta_2^1)^2}\). Obviously, \(T\) is defined for sufficiently small \(\xi_2^0,\eta_2^1\) and sufficiently large \(t_0\), and maps the “point” \((\xi_2^0,\eta_2^1,t_0)\) into the point \((\bar\xi_2^0,\bar\eta_2^1,\bar t_0)\). We shall look for fixed points of \(T\). The coordinates of the fixed points \((\xi_2^*,\eta_2^*,t^*)\) satisfy the equations
\[ \xi_2^*-f(\xi_1^{\mathrm{II}},\xi_2^{\mathrm{II}},\eta_2^*)=0,\qquad \eta_i^{\mathrm{II}}-g_i(\xi_1^{\mathrm{II}},\xi_2^{\mathrm{II}},\eta_2^*)=0,\qquad i=1,2. \tag{15} \]
Let, for definiteness, \(B_1\ne0\). It is easily proved that the first two equations of system (15) are solvable with respect to \(\xi_2^*\) and \(\eta_2^*\) for all \(t^*>\bar t^*\). Substituting their expressions into the last equation, we obtain an equation with respect to \(t^*\)
\[ \begin{aligned} &B_1(1-\varphi_{21}^1(0,r,0,r,0)+\ldots)\sin\Omega t^*+{}\\ &\quad+B_2(1+\varphi_{11}^1(0,r,0,r,0)+\ldots)\cos\Omega t^* -{}\\ &\quad-e^{(\lambda+\gamma)t^*}\bigl[\Delta_1(1+\varphi_{11}^0(0,r,0,r,0)+\ldots)\cos\omega t^*+{}\\ &\quad+\Delta_2(1+\varphi_{21}^1(0,r,0,r,0)+\ldots)\sin\omega t^*\bigr]=0, \end{aligned} \tag{16} \]
where \(\Delta_1=A_{11}B_2-A_{21}B_1\), \(\Delta_2=A_{12}B_2-A_{22}B_1\). Since \(\lambda+\gamma<0\) and \(1+\varphi_{11}^1>0\), \(1+\varphi_{21}^1>0\) for sufficiently small \(r\), we obtain that equation (16) has a countable number of roots \(t_n^*\), whose asymptotics as \(n\to\infty\) is determined by the roots of the equation
\[ B_1(1+\varphi_{21}^1)\sin\Omega t^*+B_2(1+\varphi_{11}^1)\cos\Omega t^* =\overline{B}\sin(\Omega t^*+\theta)=0. \]
The characteristic equation of the linearized point mapping (14) at a fixed point is easy to find and can be written in the form
\[ \alpha_1(n)z^3+[\overline{B}\Omega\delta+\alpha_2(n)]z^2+\alpha_3(n)z+\alpha_4(n)=0, \]
where \(\alpha_i(n)\to0\) as \(n\to\infty\), and \(\delta\) is equal either to \(+1\) or to \(-1\). As \(n\to+\infty\), this equation has one root tending to infinity and two tending to zero.
Thus, we have proved that the mapping \(T\) has a countable number of fixed points. Consequently, system (3) has a countable set of periodic solutions of saddle type, which pass through the points
\[ M_n^*(\xi_{1n}^*,\xi_{2n}^*,\eta_1^{\mathrm{II}}(0,t^*,\xi_{1n}^*,\xi_{2n}^*,\eta_{1n}^*,\eta_{2n}^*),\eta_2^{\mathrm{II}})\in S_0. \]
From the construction of the mappings \(T_0\) and \(T_1\) it follows that any extended neighborhood of the saddle-focus contains a countable set of periodic solutions of the indicated type.
Scientific Research Institute
of Applied Mathematics and Cybernetics
at Gorky State University
named after N. I. Lobachevsky
Received
10 III 1966
CITED LITERATURE
- L. P. Shilnikov, DAN, 143, No. 2, 289 (1962).
- L. P. Shilnikov, Matem. sborn., 61 (104), No. 4, 443 (1963).
- L. P. Shilnikov, DAN, 160, No. 3, 558 (1965).
- Yu. I. Neimark, L. P. Shilnikov, DAN, 160, No. 6, 1261 (1965).