Dynamics and Periodic Solutions in Cubic Polynomial Hamiltonian Systems

Abstract

We consider the Hamiltonian function defined by the cubic polynomial $H= \frac{1}{2}(y_1^2+ y_2^2)+ V(x_1, x_2)$ where the potential $V(x)= \delta V_2(x_1, x_2)+ V_3(x_1, x_2)$, with $V_2(x_1, x_2)=\frac{1}{2}(x_1^2+x_2^2)$ and $V_3(x_1, x_2)= \frac{1}{3} x_1^3+ f x_1 x_2^2+ g x_2^3$, with f and g are real parameters such that $f \ne 0$ and $\delta$ is 0 or 1. Our objective is to study the number and bifurcations of the equilibria and its type of stability. Moreover, we obtain the existence of periodic solutions close to some equilibrium points and an isolated symmetric periodic solution distant of the equilibria for some convenient region of the parameters. We point out the role of the parameters and the difference between the homogeneous potential case ($\delta = 0$) and the general case ($\delta = 1$).

Appendix

1.1 The Lyapunov Center Theorem and Moser’s Theorem

Theorem 5

Assume that the system

$$\begin{aligned} \dot{x}= f(x), \end{aligned}$$

(5.1)

admits a nondegenerate integral and has an equilibrium point with exponents $\pm i \omega$, $\lambda _3, \ldots , \lambda _n$, where $i \omega \ne 0$ is pure imaginary. If $\frac{\lambda _j}{i \omega }$ is never an integer for $j+3, \ldots , n$, then there exists a one-parameter family of periodic orbits emanating from the equilibrium point. Moreover, when approaching the equilibrium point along the family, the periods tend to $2 \pi / \omega$ and the nontrivial multipliers tend to $e^{2 \pi \lambda _j/ \omega }$, $j=3, \ldots , m$.

The proof of this theorem can be found in [9].

Remark 1

When the system (5.1) is an Hamiltonian system then it has always a nondegenerate integral.

Now, we will enunciate Moser’s Theorem (see details in [10]) which is a generalization of Lyapunov’s Theorem.

Theorem 6

Consider a Hamiltonian system

$$\begin{aligned} \dot{z}= J \nabla H(z), \end{aligned}$$

(5.2)

such that $\nabla H(0)=0$. Assume that ${\mathbb {R}}^ {2n}= E + F$ where E, F are invariant subspaces of A where $A= J Hess H(0)$ (the linear part around the equilibrium 0), such that all solutions of this linear system in E have the same period $T>0$ while no nontrivial solution in F has this period. Moreover, assume that the Hessian HessH(0) restricted to E is positive definite. Then, for sufficiently small $\epsilon$, on each energy surface $H(z)= H(0)+ \epsilon ^2$ the number of periodic orbits of this system is at least $1/2\, dim(E)$.

1.2 Isolated Periodic Solution

Here we will only enunciate Theorem 1.1 of [1], the details of the notation can be found there.

It is considered the mechanical systems

$$\begin{aligned} {\ddot{x}}= -\nabla W(x), \, x \in {\mathbb {R}}^2. \end{aligned}$$

(5.3)

Both W and the region will have an axis of symmetry ${\mathcal {A}}$ which will contain the only critical point P of W in this region. The symmetry allows us to restrict our consideration to the ”upper” half-plane defined by ${\mathcal {A}}$. An energy h will be chosen so that the level curve $W(x)= h$ lies above and does not intersect ${\mathcal {A}}$. Note that by symmetry ${\mathcal {A}}$ is a gradient line (homothetic solution). We construct a closed, simply connected region R between $W = h$ and ${\mathcal {A}}$ by first specifying two such solutions that intersect ${\mathcal {A}}$ at points $Q_1$ and $Q_2$, respectively, where $Q_1$ is strictly to the left of P and $Q_2$ is to the right of P ($Q= P$ is allowed). We then obtain R by intersecting this region by a vertical strip whose right-hand boundary is a line perpendicular to ${\mathcal {A}}$ at P.

Theorem 7

(a) the orbit segment $x_q(t)$ in R for points on $W=h$ foliates R; (b) for $q< p$ ($p< q$) any orbit $x_q(t)$ that intersects A does so in an acute (obtuse) angle.

Then there is a periodic orbit $\Pi$, unstable and it is the only solution of ${\ddot{x}}=- W_x$ with energy h that remains in $R \cup {\bar{R}}$ for all time.

In [1] it is shown that instead of proving conditions (a) and (b) in Theorem 7, the hypotheses (H1)–(H5) can be proved which imply (a) and (b).

• (H1) The gradient field W, restricted to ${\mathcal {A}}$ in $R {\setminus }\{P\}$ points towards P, and above ${\mathcal {A}}$ in R has values in the upper half-plane defined by ${\mathcal {A}}$.

• (H2) $div\left( \frac{J W_x}{\Vert J W_x\Vert }\right) > 0$ in the interior of R, or equivalently, the gradient field $W_x$ rotates counterclockwise along gradient curves [in the direction $W_x$) in this region.

• (H3) The level curve $W(X)=h_0$, where $h_0= W(P)$ is the critical energy, is a straight line issuing from P that bisects the region ${\mathcal {A}}$. For energies $h > h_0$, the level curves $W = h$ lie above $W = h_0$, and are concave up with respect to the axis of symmetry. For $h < h_0$, the level curves $W = h$ lie below $W = h_0$, and are concave down with respect to the axis of symmetry.

• (H4) All orbits x(t) of (5.3), with energy $h > h_0$, which originate on $W = h$ and enter the subregion of $R {\setminus }\{P\}$ between and including the lines ${\mathcal {A}}$ and $W = h$ intersect the T-curves in this region transversely. Let $T= - J W_{xx} J W_x$. The integral curves of the vector field T are called T-curves.

• (H5) Consider any T-curve in the subregion of $R {\setminus } \{P\}$ between ${\mathcal {A}}$ and $W = h_0$, including ${\mathcal {A}}$ and $W = h_0$. For any vector u obtained as the unit tangent vector of an orbit of energy h rising to $W = h$, as it crosses this T-curve, $\langle T, J u\rangle (x) >0$ and $T(G_u(x)) < 0$, for all x on this T-curve between the point of intersection and P. Here $G_u(x)= \left( \frac{\langle W_x, J u\rangle [e_1(W)]^2}{2(h- W) \langle T, J u\rangle }\right) (x)$. $e_1$ denote a constant unit vector field which, when restricted to the axis of symmetry ${\mathcal {A}}$ to the left of P, points towards the critical point and $e_2= -J e_1$.

Remark 2

From hypothesis (H1)–(H3) it is easy to show that, at points to the left of M and on $M {\setminus }\{q\}$ in R, the acceleration field $-\nabla W(x)$ points to the left of M.

When the potential function W in (5.3) is an homogeneous function of degree n it was proved in [1] the following result in Corollary 4.2.

Corollary 1

Assume that the region R satisfies the previous requirement. Let $\nabla W \ne 0$ and $\lambda = det Hess(W) < 0$ in $R {\setminus } \{0\}$. If hypothesis (H1) holds and there is no gradient line of the potential W in the interior of R, then hypotheses (H2)–(H5) hold.