Quasi-Periodic Bifurcations and Chaos

Abstract

A natural phenomenon in applications is the interaction of quasi-periodic solutions of dynamical systems in a dissipative setting. We study the interactions of two of such ODE systems based on the construction of a nonlinear oscillator with thermostatic (energy) control. This leads to the emergence of complexity, torus doubling, and chaos. We find canards; 1-, 2-, and 3-tori; chaos, and hyperchaos. Detailed analysis is possible in the case of small oscillations and small interactions. Large-scale phenomena are studied by the construction of charts of parameter space using Lyapunov exponents.

1. Introduction

Interactions of two or more nonlinear oscillators generally produce complex dynamics with periodic solutions, tori, tori doubling, and chaos. A natural feature in real-life models is that the interactions are quasi-periodic, which means that the individual components have frequencies that are incommensurable. For a discussion of diophantine frequency vectors and the measure of Cantor sets in two-frequency systems, see [1], with many references also inspired by the conservative setting leading to families of invariant tori; we mention in this setting the basic paper [2].

Our focus is different from KAM theory and, with some exceptions, also from dissipative KAM theory, as we are especially interested in the practical context where we start with dissipative systems that are subsequently perturbed and lead to complex behaviour. This can produce bifurcations of isolated tori and their qualitative changes, whereas classical KAM theory is involved with conservative, usually Hamiltonian, systems where tori arise in dense sets, as well as families with positive measure. As we shall see, we will consider for our analysis a system with damping and thermostatic control, where the combination of two of such systems adds forcing and bifurcation phenomena.

Multifrequency oscillations arise in many applications of various disciplines such as mechanical engineering, laser systems, and electronic circuits; for a useful list of such applications in many fields, see [3] (in particular their references 1-23) and [4]. See also [5,6]. In [3,4], the emphasis is on the construction of charts of Lyapunov exponents for interacting self-excited systems. Such numerically obtained charts yield enormous inspiration for further analysis, but in our approach, we combine both analytic methods (averaging) and numerical bifurcation theory. This hybrid approach will be fruitful. A number of recent papers on quasi-perioidc bifurcations is concerned with maps; such an approach produces important general insight, but the relation with ODEs has yet to be established.

Important analytic tools are mathematically sound approximation techniques; see, for instance, [7] (ch. 9) and bifurcation theory; see, for instance, [8]. It is generally known that the equilibrium of a system of differential equations that for a certain parameter value has two imaginary eigenvalues can generate a periodic solution through Poincaré–Andronov–Hopf bifurcation. The generating periodic solution is a guiding center of the torus.

In a similar way, a periodic solution that is characterised by two imaginary eigenvalues or corresponding Lyapunov exponents may generate a torus through Neimark–Sacker bifurcation.

In the case of quasi-periodic interactions, it is natural to discover and identify tori. Usually, they branch off equilibria and periodic solutions. The first question is then whether they are normally hyperbolic or not. Starting with an autonomous system of ODEs, there will be at least one zero Lyapunov exponent. Are there more zeros, and when varying parameters, do we find Hopf and other bifurcations, tori doubling, and cascades of such phenomena leading to chaos?

Apart from normal form theory and averaging, we used numerical methods implemented in Mathematica, Matlab, and Matcont, see also [9,10]. The last one uses continuation methods to follow parameter changes that cause bifurcations. It turns out that the hybrid combination of analytic and numerical tools can be very inspiring.

What Is New?

To obtain insight in dissipative quasi-periodic bifurcations, we modify in Section 2 an important thermostatic control problem (Sprott A), which is a chaotic system formulated in [11,12]. Earlier studies of Sprott A giving more insight are [13,14]; in [15,16], the thermostatic control problem of [12] is linked to tori families, as is well known for conservative systems. The context of conservative systems is classical and interesting but in general less suitable for applications in engineering and other science fields that involve dissipation.

A new step in Section 2 is to introduce a two-dimensional basic oscillator with periodic solutions that are asynchronous; when adding a one-dimensional thermostatic control and assuming small oscillations, we find canard behaviour. Admitting larger amplitudes, we can identify periodic solutions in a resonance manifold through slow–fast dynamics.

In Section 3, we consider the six-dimensional interaction of two such components. If dissipation and thermostatic control are excluded, we have a system with two interacting quasi-periodic oscillators. To study the bifurcations of this system with dissipation, interaction and control are the central parts of this paper. When considering small oscillations and small interactions, one can use averaging to find single tori with one or two zero Lyapunov exponents. Combining canard initial values, the solutions converge to a flattened torus. Torus doubling arises, leading to a cascade of torus doublings. To obtain an overview of the phenomena, we constructed charts for larger values of dissipation and interactions. For each value of two parameters, one obtains either a 2-torus, a 3-torus, chaos, or hyperchaos.

Conclusions and a discussion finish the main part of the paper.

Appendix A lists the results that carry directly over from the two-components case to systems with n components.

2. One-Component Oscillations with Energy Control

An autonomous one-degree-of-freedom nonlinear oscillator with damping can be described by

$$\ddot{q}+b\dot{q}+q+f\left(q\right)=0,$$

(1)

with dissipation parameter $b\ge 0$. A dot above a variable is short for differentiation with respect to time. We assume that the function $f\left(q\right)$ is analytic; near $q=0$, it has a power series expansion starting with quadratic terms. If $b=0$, we have no energy loss; in this case, the oscillator has the energy integral

$$E={\displaystyle \frac{1}{2}}(q^2+{\dot{q}}^2)+{\int }_0^{q}f\left(s\right)ds,$$

(2)

with E a constant parameter depending on the initial conditions. In the conservative case ($b=0$), Equation (1) has an infinite number of periodic solutions, with its period depending on $E=E_0$. If $f\left(q\right)$ is deleted in the equation, we have the synchronised case of the harmonic equation with all periods being equal ($2\pi$).

In chemical physics, see [17], one introduces a thermostatic control by adding an equation controlling the dissipative term $b\dot{q}$ by a new variable z, replacing $b\dot{q}$ with $bz\dot{q}$. The control $z\left(t\right)$ can become negative, producing excitation if the energy (or another suitable quantity) characterising the oscillator is smaller than a chosen threshold, and $z\left(t\right)$ can become positive, causing increased damping if the energy is larger than the threshold. Apart from using the energy of a nonlinear equation, we have with Equation (1), which defines the natural case of periodic solutions with the period dependent on the energy. To fix the ideas, we will choose

$$f\left(q\right)=c{q}^3.$$

The thermostatic control z for the energy leads to the system

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\ddot{q}+bz\dot{q}+q+c{q}^3\hfill & =0,\hfill \\ \dot{z}\hfill & =q^2+{\dot{q}}^2+{\displaystyle \frac{c}{4}}{q}^4-a,\hfill \end{array}\right.\end{array}$$

(3)

with parameters $a,c>0$. The expression $q^2+{\dot{q}}^2+{\displaystyle \frac{c}{4}}{q}^4$ is not exactly representing the energy but measures the energy.

In [16], system (3) was studied in the case of $c=0$; this system is called Sprott B. We note that for small oscillations ($q,\dot{q}$ small) the solutions of the Sprott B system will upon first approximation describe the solutions of system (3) correctly; this will be made more precise later on. Replacing the equation for z by

$$\dot{z}={\dot{q}}^2-a,$$

with $c=0$, the system is called Sprott A; see [11,12,15] and many references therein.

Choosing $c>0$ implies that we have no saddle equilibrium in the conservative oscillator ($b=0$). Choosing negative c would change the dynamics for larger values of the initial energy; in Figure 1, we illustrate this for $c=1$, where all solutions are periodic, and $c=-1$, producing saddles in the phase plane.

Stability and Lyapunov exponents

In the sequel, we will look for periodic solutions and their stability. In Section 5.4 of [18], it is shown that if we have located a periodic solution $\varphi \left(t\right)$ of an autonomous system (such as (3)), linearisation near $\varphi \left(t\right)$ produces as one solution of the linearised system $\dot{\varphi }\left(t\right)$. In the case of system (3), this means that one of the three Lyapunov exponents will be zero.

Exact analytical solutions

We note that system (3) has no equilibria (critical points of the vector field). We find an unbounded invariant manifold if $x\left(0\right)=\dot{x}\left(0\right)=0$, with dynamics given by

$$z\left(t\right)=z\left(0\right)-at.$$

The following result is well known.

Lemma 1.

Consider system (3) with $b=0$; as $c>0$, all solutions $q\left(t\right)$ are periodic but not synchronous.

A few examples of energy $E_0$ with $b=0,c=1$, and corresponding period P are $(E_0,P)=(0.01,6.4),(0.125,5.8),(0.5,5.1),(1.125,4.4),(2,4),(4.5,3.3),(8,3),(12.5,2.7).$ The cycles in the phase plane are shown in Figure 1 (upper); increasing $E_0$ shortens P.

Time reversal

We will use the concept of the time reversibility of system (3). Select $\dot{q}=v$. The system is characterised by time reversal if it is invariant for the following transformtion:

$$q\to -q,v\to v,z\to -z,t\to -\tau .$$

It is clear that system (3) shows time reversal. This concept will be used to apply dissipative KAM theory to system (3). For the general theory, see the introductions and statements in [19,20,21].

A second observation is that replacing $x,\dot{x}$ with $-x,-\dot{x}$ keeps the system invariant.

The phase flow is globally characterised by writing system (3) as a first-order ODE in 3-space $\dot{x}=F\left(x\right)$ and taking the divergence ($Div$) of the vector field as

$$DivF\left(x\right)=-bz.$$

(4)

Suppose first that $z\left(t\right)$ tends to a fixed nonzero number. If $z\left(t\right)$ is positive definite, the flow is contracting, and if $z\left(t\right)$ is negative definite, the flow is expanding. In the first case, we may find attracting solutions.

The symmetry conditions we have obtained allow in specific phase-space regions solutions with alternating $z\left(t\right)$ and in particular $z\left(t\right)$ T-periodic. If in the periodic case we have

$${\displaystyle \frac{1}{T}}{\int }_0^{T}z\left(t\right)dt=0,$$

together with the time reversal characteristic, the dissipative KAM theory will conclude to the existence of invariant tori.

Note that related but different types of control exist in neurodynamics that are called ‘gating’. An electro-physical signal fires a neuron if it exceeds a potential typical for the gates of the particular neuron. Such a control is different from thermostatic or energy control, as in realistic neuron models, a potential that is too small will leave the neuron inert. Such neuronal dynamical systems serve, however, as an inspiration in the present paper.

We expect that the study of systems with energy control will also be of use in other biophysical systems.

2.1. Small Oscillations near the z Axis, Canards

We scale $q=\sqrt{\epsilon }x,\dot{q}=\sqrt{\epsilon }\dot{x},a=\epsilon a_0,b=\epsilon b_0$. System (3) becomes

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\ddot{x}+\epsilon b_{0}z\dot{x}+x\hfill & =-\epsilon c{x}^3,\hfill \\ \dot{z}\hfill & =\epsilon (x^2+{\dot{x}}^2-a_0)+O\left({\epsilon }^2\right),\hfill \end{array}\right.\end{array}$$

(5)

We apply averaging to system (5); see [22] or [7] for the theory. Here and in the sequel, we formulate and will formulate variational equations in amplitude–phase variables $r,\varphi$. Considering coupled systems of oscillators, $r,\varphi$ will be vectors with subscripts $r_1,r_2,{\varphi }_1$, etc. Selecting $x=r\left(t\right)cos(t+\varphi \left(t\right)),\dot{x}=-r\left(t\right)sin(t+\varphi \left(t\right))$ and rescaling produces

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\dot{r}\hfill & =-\epsilon sin(t+\varphi )(-c{r}^3{cos}^3(t+\varphi )+b_{0}zrsin(t+\varphi )),\hfill \\ \dot{\varphi }\hfill & =-{\displaystyle \frac{\epsilon }{r}}cos(t+\varphi )(-c{r}^3{cos}^3(t+\varphi )+b_{0}zrsin(t+\varphi )),\hfill \\ \dot{z}\hfill & =\epsilon (r^2-a_0)+O\left({\epsilon }^2\right).\hfill \end{array}\right.\end{array}$$

(6)

Averaging over t while keeping $r,\varphi$ fixed, we find

$$\dot{r}=-\epsilon {\displaystyle \frac{b_0}{2}}zr,\dot{\varphi }=\epsilon {\displaystyle \frac{3}{8}}c{r}^2,\dot{z}=r^2-a_0.$$

(7)

The solutions of system (6) are approximated to $O\left(\epsilon \right)$ on an interval of time $O(1/\epsilon )$ by the solutions of system (7).

The averaged system (7) contains a periodic approximate solution of the form

$$x\left(t\right)=\sqrt{a_0}cos(t+\epsilon {\displaystyle \frac{3}{8}}{a}_{0}t),z\left(t\right)=0.$$

(8)

The approximation of the solution of system (5) with initial conditions $x\left(0\right)=\sqrt{a_0},\dot{x}\left(0\right)=0,z\left(0\right)=0$ found by expressions (8) has the error $O\left(\epsilon \right)$ on the long timescale $1/\epsilon$. It corresponds with the exact periodic solution obtained with nearby initial conditions.

There is a theoretical advantage giving more insight by using the timelike variable $s=t+\varphi$ instead of t; s is indeed timelike, as the phase $\varphi \left(t\right)$ is varying slowly. Repeating the calculation, we have by averaging over s the averaged system

$${\displaystyle \frac{dr}{ds}}=-\epsilon {\displaystyle \frac{b_0}{2}}zr,{\displaystyle \frac{dz}{ds}}=\epsilon (r^2-a_0).$$

(9)

We have $r=\sqrt{a_0},z=0$ as a critical point of the averaged system (9). The determinant in the critical point does not vanish for $a_0>0$, so the approximation (8) corresponds with an existing periodic solution of the original system that is $2\pi$-periodic in s. The eigenvalues in the critical point are $\pm i\sqrt{a_0{b}_0}$, so the periodic solution is with the first approximation neutrally stable. A second-order approximation in $\epsilon$ does not change the picture qualitatively, as the time reversibility plays an essential role in producing an infinite set of KAM tori; see Figure 2. In these cases, $z\left(t\right)$ alternates around $z=0$. The periodic solution obtained above serves as an organising centre. The solutions $z\left(t\right)$ are symmetric with respect to $z=0$ as predicted.

Families of invariant tori were observed earlier for the dissipative Sprott A system in [13,14], while the mathematical explanation by time reversal and symmetry was given in [15].

Canards Near the z Axis

The dynamics near the z axis, as shown in Figure 2, were obtained with parameter a $O\left(\epsilon \right)$. System (3) was designed to find a periodic solution by averaging close to the z axis. The system can also be interpreted as a slow–fast system with fast variables $q,\dot{q}$ and slow variable z. According to the theory of slow–fast systems, see [23] and further references therein, a slow manifold will be given by $x=\dot{x}=0$ (the z axis). The slow manifold is not normally hyperbolic, so the attraction or repelling is not necessarily exponential, but the explicit form makes characterisation easy. If we start with a positive value of $z\left(0\right)$ and with $r^2\left(0\right)=x^2\left(0\right)+{\dot{x}}^2\left(0\right)<a_0$, the solution will move closer to the slow manifold. As long as $z\left(t\right)>0$, $r^2\left(t\right)$ will decrease because of dissipation. At time $t=t_1$, $z\left(t_1\right)=0$ and the term $bz\dot{x}$ will produce excitation for $t>t_1$. The slow manifold becomes unstable, but as the solutions are very close to the z axis, they will persist for some time in motion near the z axis; this is the canard phenomenon. The symmetry of the equations causes the jump-off point of the solutions to mirror the approach point; see Figure 2 (lower) again.

2.2. Generalisations

The results obtained thus far for the case $f\left(q\right)=c{q}^3$ and small oscillations carry over to the more general case with $f\left(q\right)$ being polynomial or analytic, as well as odd, and $f\left(0\right)=f^{\prime }\left(0\right)=0$. In particular, we have time reversibility producing invariant tori near the z axis around a periodic solution as the organising centre. Close to the z axis, we will find canard behaviour.

In this section, we drop the assumption of small oscillations.

A Family of Periodic Solutions

Consider again system (3) with small damping coefficient b and small nonlinear force; select $b=\epsilon b_0,c=\epsilon c_0$. System (3) becomes

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\ddot{q}+\epsilon b_{0}z\dot{q}+q+\epsilon c_0{q}^3\hfill & =0,\hfill \\ \dot{z}\hfill & =q^2+{\dot{q}}^2+\epsilon {\displaystyle \frac{c_0}{4}}{q}^4-a,\hfill \end{array}\right.\end{array}$$

(10)

If $\epsilon =0$, the solutions for $q\left(t\right)$ are harmonic. Transforming system (10) for $\epsilon \ge 0$ to amplitude–phase variables, it becomes a slow–fast system with slow variables $r,\varphi$ and fast variable z as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\dot{r}\hfill & =-\epsilon sin(t+\varphi )(\left(c_0{r}^3{cos}^3(t+\varphi )\right)+b_{0}zrsin(t+\varphi )),\hfill \\ \dot{\varphi }\hfill & =-{\displaystyle \frac{\epsilon }{r}}cos(t+\varphi )\left(c_0{r}^3{cos}^3(t+\varphi )\right)+b_{0}zrsin(t+\varphi )),\hfill \\ \dot{z}\hfill & =r^2-a+O\left(\epsilon \right).\hfill \end{array}\right.\end{array}$$

(11)

We can identify $r=\sqrt{a}$ as a resonance manifold of system (11); for the theory, see [23], ch. 12, or [7], ch. 7. Another approach is to use iteration of an integral equation; see [23], ch. 10.2, or [7] in ch. 3. This would be an application of the Poincaré–Lindstedt (continuation) method.

We will use the asymptotic analysis of resonance manifolds, where the parameter $\sqrt{\epsilon }$ plays an essential role. The standard theory can be found in [23] in ch. 12 or [7] in ch. 7. Consider a neighbourhood of the resonance manifold by introducing a local variable $\xi$:

$$\xi ={\displaystyle \frac{r-\sqrt{a}}{\delta \left(\epsilon \right)}},r=\sqrt{a}+\delta \left(\epsilon \right)\xi ,$$

(12)

with $\delta \left(\epsilon \right)=o\left(1\right)$ as $\epsilon \to 0$. Introducing $\xi$ in system (11), we find

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\delta \left(\epsilon \right)\dot{\xi }\hfill & =-\epsilon sin(t+\varphi )(c_0{(\sqrt{a}+\delta \xi )}^3{cos}^3(t+\varphi )+b_{0}z(\sqrt{a}+\delta \xi )sin(t+\varphi )),\hfill \\ \dot{\varphi }\hfill & =-{\displaystyle \frac{\epsilon }{(\sqrt{a}+\delta \xi )}}cos(t+\varphi )\left(c_0{(\sqrt{a}+\delta \xi )}^3{cos}^3(t+\varphi )\right)+b_{0}z(\sqrt{a}+\delta \xi )sin(t+\varphi )),\hfill \\ \dot{z}\hfill & ={(\sqrt{a}+\delta \xi )}^2-a+O\left(\epsilon \right).\hfill \end{array}\right.\end{array}$$

(13)

A significant degeneration in the sense of singular perturbation theory gives the choice $\delta \left(\epsilon \right)=\sqrt{\epsilon }$. Expanding while keeping terms $O\left(\sqrt{\epsilon }\right)$, we have

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\dot{\xi }\hfill & =-\sqrt{\epsilon }sin(t+\varphi )(c_0{(\sqrt{a}cos(t+\varphi ))}^3+b_{0}z\sqrt{a}sin(t+\varphi ))+O\left(\epsilon \right),\hfill \\ \dot{\varphi }\hfill & =O\left(\epsilon \right),\hfill \\ \dot{z}\hfill & =2\sqrt{a\epsilon }\xi +O\left(\epsilon \right).\hfill \end{array}\right.\end{array}$$

(14)

Averaging over time produces $O\left(\sqrt{\epsilon }\right)$ as

$$\dot{\xi }=-\sqrt{\epsilon }{\displaystyle \frac{b_0}{2}}\sqrt{a}z,\dot{\varphi }=0,\dot{z}=\sqrt{\epsilon }2\sqrt{a}\xi .$$

(15)

Consider $t+\varphi$ as a timelike variable and the dynamics of the variables $\xi ,z$. The Jacobian of the averaged equations of $\xi ,z$ is not singular (its value is $\epsilon b_{0}a$), so according to the implicit function theorem, we have the existence of the periodic solution continued from the harmonic solution for $\epsilon =0$.

Solving the averaged system, we find ${\xi }^2+{\displaystyle \frac{b_0}{4}}{z}^2=$ to be constant, corresponding with neutral stability for critical point $\xi =z=0$ in the resonance manifold. However, a first-order approximation in a resonance manifold will always yield conservative dynamics even if the original system is dissipative; see, for the general theory again, [23] in ch. 12 or [7] in ch. 7. So, we have to construct a second-order approximation to establish stability.

Expansion to $O\left(\epsilon \right)$ leads to the system

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\dot{\xi }\hfill & =-\sqrt{\epsilon }sin((t+\varphi )[-a^{3/2}{cos}^3(t+\varphi )+b_0\sqrt{a}zsin(t+\varphi )]\hfill \\ & -\epsilon sin(t+\varphi )[-3a\xi {cos}^3(t+\varphi )+b_{0}z\xi sin(t+\varphi )]+{\epsilon }^{3/2},\hfill \\ \dot{\varphi }\hfill & =-\epsilon cos(t+\varphi )[-a{cos}^3(t+\varphi )+b_{0}zsin(t+\varphi )]+{\epsilon }^{3/2},\hfill \\ \dot{z}\hfill & =\sqrt{\epsilon }2\sqrt{a}\xi +\epsilon {\xi }^2.\hfill \end{array}\right.\end{array}$$

(16)

However, the second-order averaging (see [23] or [7]) of system (13) does not change the neutral stability.

Numerical explorations for $a=1,2,4,9$ and nearby initial conditions show instability of $\xi =z=0$ and convergence to other solutions. See Figure 3 for the cases of orbits starting near $a=1,z=0$ and $a=2,z=0$.

3. A Coupled System of Two Controlled Oscillators

We extend system (3) to two coupled systems with a simple direct coupling. The coupling is not inspired by pendulum couplings but by the process of transmitting impulses to neigbouring components, as in neural systems. Consider the coupled system

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\ddot{q_1}+b{z}_1\dot{q_1}+q_1+c{q}_1^3\hfill & =\beta q_2,\hfill \\ \dot{z_1}\hfill & =q_1^2+{\dot{q_1}}^2+{\displaystyle \frac{c}{4}}{q}_1^4-a_1,\hfill \\ \ddot{q_2}+b{z}_2\dot{q_2}+q_2+c{q}_2^3\hfill & =\beta q_1,\hfill \\ \dot{z_2}\hfill & =q_2^2+{\dot{q_2}}^2+{\displaystyle \frac{c}{4}}{q}_2^4-a_2.\hfill \end{array}\right.\end{array}$$

(17)

The interaction constant $\beta \ge 0$ will determine the interaction force. In system (17), the first component activates the second, and the second one activates the first. We assume that $a_1,a_2>0$.

The result of Lemma 1 extends in the following form.

Lemma 2.

If $b=\beta =0$, the corresponding solutions $q_1\left(t\right),q_2\left(t\right)$ of system (17) are quasi-periodic, and they are periodic if the initial conditions are equal.

As we shall see, system (17) will produce for $\beta >0$ interactions of quasi-periodic oscillations with complicated dynamics. In [3,4], such two-frequency interactions were studied involving torus bifurcations and chaotic dynamics characterised by Lyapunov exponents. The nature of the coupling in [3,4] is different from our system (17) with more complicated dissipation. A common feature is that in both model systems, a form of self-excitation takes place.

System (17) can be written as a first-order system $\dot{x}=F\left(x\right)$ in 6-space with divergence:

$$DivF\left(x\right)=-b(z_1+z_2).$$

(18)

So, in a region where $z_1+z_2>0$, the flow contracts if, in the region $z_1+z_2<0$, the flow expands.

Lemma 3.

If $c=\beta =0$ in system (17), we have a family of harmonic synchronised periodic solutions in the manifold $z_1=z_2=0$ of the form

$$q_1\left(t\right)=\sqrt{a_1}cost,{\dot{q}}_1\left(t\right)=-\sqrt{a_1}sint,q_2\left(t\right)=\sqrt{a_2}cost,{\dot{q}}_2\left(t\right)=-\sqrt{a_2}sint.$$

(19)

If $\beta \ne 0$, we expect instability of the periodic solution because of resonance. It is natural to study the cases $c,\beta \ll 1$.

3.1. Small Interactions and Small Oscillations

Consider the case of small interactions $\beta =\epsilon {\beta }_0$, small deflections, and consequently small nonlinearity $c{q}^3$; select $b=\epsilon b_0$.

As in Section 2.1, we can scale for the deflections $q_{1,2}=\sqrt{\epsilon }{x}_{1,2}$ for the velocities ${\dot{q}}_{1,2}=\sqrt{\epsilon }{\dot{x}}_{1,2},a_{1,2}\to \epsilon a_{1,2}$. Introducing amplitude–phase coordinates as before and by averaging as in Section 2.1, we find the system for $O\left(\epsilon \right)$ approximations given by

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\dot{r}}_1\hfill & =-\epsilon (r_1{\displaystyle \frac{b_0}{2}}{z}_1+{\displaystyle \frac{{\beta }_0}{2}}{r}_{2}sin({\varphi }_1-{\varphi }_2)),\dot{{\varphi }_1}=\epsilon ({\displaystyle \frac{3}{8}}c{r}_1^2-{\displaystyle \frac{{\beta }_0{r}_2}{2r_1}}cos({\varphi }_1-{\varphi }_2)),\hfill \\ {\dot{r}}_2\hfill & =-\epsilon (r_2{\displaystyle \frac{b_0}{2}}{z}_2+{\displaystyle \frac{{\beta }_0}{2}}{r}_{1}sin({\varphi }_2-{\varphi }_1)),\dot{{\varphi }_2}=\epsilon ({\displaystyle \frac{3}{8}}c{r}_2^2-{\displaystyle \frac{{\beta }_0{r}_1}{2r_2}}cos({\varphi }_2-{\varphi }_1)),\hfill \\ {\dot{z}}_1\hfill & =\epsilon (r_1^2-a_1)+O\left({\epsilon }^2\right),{\dot{z}}_2=\epsilon (r_2^2-a_2)+O\left({\epsilon }^2\right).\hfill \end{array}\right.\end{array}$$

(20)

Structurally stable critical points of the approximating vector field correspond with periodic solutions close to the critical points; see [7].

We have the critical values $r_1=\sqrt{a_1},r_2=\sqrt{a_2}$. Select for the combination angle $\chi ={\varphi }_1-{\varphi }_2$. The equation for $\chi$ becomes

$${\displaystyle \frac{d\chi }{dt}}={\displaystyle \frac{\epsilon }{2}}\left[{\displaystyle \frac{3}{4}}c(r_1^2-r_2^2)-{\beta }_0({\displaystyle \frac{r_2}{r_1}}-{\displaystyle \frac{r_1}{r_2}})cos\chi \right].$$

(21)

The case $a_1=a_2=a$

This is clearly a degenerate case, as from (21), we have in the critical points $r_1=r_2$, and so $d\chi /dt=0$. The only requirement is $z_1=-z_2$, but with $\chi$, it is not determined. The time series $q_1\left(t\right),q_2\left(t\right)$ corresponding to the critical points is shown in Figure 4. As expected from the analysis of one component in Section 2, see also Figure 2, the canard behaviour becomes less prominent when decreasing $z_1\left(0\right),z_2\left(0\right)$. This is illustrated in Figure 4 and Figure 5, where $a_1=a_2=0.0025$. Choosing $z_1\left(0\right)=2,z_2\left(0\right)=-2$, we have canard behaviour. Decreasing the initial z values, we find the irregular pattern shown in Figure 6 for $q_1\left(t\right)$ (and the same for $q_2\left(t\right)$). This irregularity originates from the exponential closeness of the orbits to the slow manifold at the z axis, which is not normally hyperbolic.

Starting close to the critical point, the dynamics show instability. See Figure 5 for an example; the numerics based on system (17) suggest the presence of a torus.

• The case $a_1\ne a_2$

This case also yields nontrivial equilibria. Requiring $\dot{\chi }=0$ gives the following condition:

$$cos\chi =-{\displaystyle \frac{3c\sqrt{a_1{a}_2}}{4{\beta }_0}}$$

(22)

Note that Equation (22) implies that $|3c\sqrt{a_1{a}_2}/\left(4{\beta }_0\right)|\le 1$. Looking for roots of the amplitude equations yields

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}sin\chi =\hfill & -{\displaystyle \frac{\sqrt{a_1}{b}_0{z}_1}{\sqrt{a_2}{\beta }_0}}\hfill \\ sin\chi =\hfill & {\displaystyle \frac{\sqrt{a_2}{b}_0{z}_2}{\sqrt{a_1}{\beta }_0}}.\hfill \end{array}\right.\end{array}$$

(23)

This gives the two nontrivial equilibria solutions in the case where $a_1\ne a_2$.

$$\begin{array}{cc}\hfill r_1=& \sqrt{a_1},r_2=\sqrt{a_2},\hfill \end{array}$$

$$\begin{array}{cc}{z}_2=\hfill & -{\displaystyle \frac{a_1}{a_2}}{z}_1,\hfill \end{array}$$

(24)

$$\begin{array}{cc}{z}_1=\hfill & \pm \sqrt{{\displaystyle \frac{16a_2{\beta }_0^2-9c^2{a}_1{a}_2^2}{16a_1{b}_0^2}}},\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill \chi =& \pi +arctan\left({\displaystyle \frac{4b_0{z}_1}{3c{a}_2}}\right),\mathrm{if}{\beta }_{0}c>0,\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill \chi =& arctan\left({\displaystyle \frac{4b_0{z}_1}{3c{a}_2}}\right),\mathrm{if}{\beta }_{0}c\le 0\hfill \end{array}$$

(27)

The Jacobian matrix of system (20) at the nontrivial equilibriun in the case where $a_1\ne a_2$ and small oscillations is defined as follows:

$$DF\left(x\right)=\left(\begin{array}{ccccc}-{\displaystyle \frac{b{z}_1}{2}}& {\displaystyle \frac{3}{8}}c{a}_2\sqrt{a_1}& {\displaystyle \frac{\sqrt{a_1}b{z}_1}{2\sqrt{a_2}}}& -{\displaystyle \frac{\sqrt{a_1}b}{2}}& 0\\ {\displaystyle \frac{3c(a_1-a_2)}{8\sqrt{a_1}}}& {\displaystyle \frac{b{z}_1(a_1-a_2)}{2a_2}}& {\displaystyle \frac{3c(a_1-a_2)}{8\sqrt{a_2}}}& 0& 0\\ -{\displaystyle \frac{\sqrt{a_1}b{z}_1}{2\sqrt{a_2}}}& -{\displaystyle \frac{3}{8}}c{a}_1\sqrt{a_2}& {\displaystyle \frac{a_{1}b{z}_1}{2a_2}}& 0& -{\displaystyle \frac{\sqrt{a_2}b}{2}}\\ 2\sqrt{a_1}& 0& 0& 0& 0\\ 0& 0& 2\sqrt{a_2}& 0& 0\end{array}\right)$$

(28)

The trace of the Jacobian (28) is

$$Tr\left(DF\right)=b{z}_1{\displaystyle \frac{a_1-a_2}{a_2}}.$$

(29)

The trace vanishes to produce degenerate dynamics in the two cases as

$$a_1=a_2,16{\beta }_0^2=9c^2{a}_1{a}_2.$$

(30)

In the second case, we have $z_1=z_2=0$; see Figure 5 again.

An analysis of the eigenvalues of the nontrivial equilibria for the case where $a_1\ne a_2$ reveals that the equilibria are of an unstable focus type. For the parameter values $a_1=0.25,a_2=0.15,b=1,\beta =1,c=1,ϵ=1$, we identify two nontrivial equilibria with $z_1=\pm 0.766384$ and corresponding eigenvalues:

$$\begin{array}{cc}\hfill {\Lambda }_1=& \left\{-0.0495\pm 0.5553i,0.1838\pm 0.3056i,0.2423\right\},\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill {\Lambda }_2=& \left\{0.0495\pm 0.5553i,-0.1838\pm 0.3056i,0.2423\right\}.\hfill \end{array}$$

(32)

Further continuation of the equilibria with respect to the system’s parameters does not reveal any significant bifurcations.

3.1.1. Quasi-Periodic Motion and Associated Bifurcations

Interestingly, the averaged system (20) contains in the case where $a_1\ne a_2$ a stable periodic orbit, which is unrelated to the equilibria mentioned so far. This implies the presence in the original system of quasi-periodic motion in the form of a stable two-dimensional torus that can be depicted numerically starting in the neighborhood of the averaged stable cycle; see Figure 7 and Figure 8 for the Lyapunov exponents.

Continuation of the limit cycle in the averaged system using $a_2$ as a control parameter reveals a supercritical period doubling at the critical value $a_2=0.2285909$ from which a stable cycle with double the period ($T=27.9445$) emerges. The corresponding stable double torus in the original system has been numerically located near the double cycle as expected. See Figure 9, Figure 10 and Figure 11.

3.1.2. Chaos Through the Cascade of Period Doubling in Tori

Further continuation produces cascades of periodic doublings and corresponding tori, and so also chaos see Figure 12 and Figure 13.

3.1.3. Coexisting Period 2 Orbit

Tracking periodic orbits and analyzing their stability through continuation methods is crucial for detecting bifurcations and understanding the resulting dynamics, as well as the routes to chaos in the system under investigation. Using the numerical methods outlined in [24], an additional period 2 orbit was identified in the averaged system, which coexists with the period 1 cycle shown in Figure 7. This cycle has four complex multipliers on the unit circle and one real multiplier equal to 1. Because the cycle is only Lyapunov-stable, there is no guarantee that the corresponding two-dimensional torus exists in the original system. See Figure 14.

Searching for the corresponding dynamics in the original system, starting near the Lyapunov-stable cycle of the averaged system, the following invariant set was identified. See Figure 15 and Figure 16.

By searching for period 1 and period 2 periodic orbits in the averaged system with initial values $r_1=0.3$, $\chi =0$, and $r_2$ within the interval $(0,6]$ and $z_1,z_2$ ranging from −3 to 3 in increments of $1/50$, eight additional periodic orbits were identified that coexist with the two cycles previously found. Four stable and four unstable orbits were identified. See illustrations in Figure 17.

3.1.4. Interactions Involving Canards

As seen in Figure 18, the evolution of the two-particle system tends to canard behaviour. The dynamical evolution starts to stabilise after 15,000 time steps.

The canard character of the flow is shown in Figure 19, where the projected dynamics suggests a flattened torus; the variable $z_1\left(t\right)$ takes negative and positive values, while $z_2\left(t\right)$ takes mostly positive ones producing both attraction and repelling.

The Lyapunov exponents of the flow shown in Figure 19 have four exponents clustered near zero. In Figure 20 on the right side, we present an enlarged picture of the small exponents. We find both positive and negative spiky behaviour, suggesting a small fractal structure of the flattened torus of Figure 19. Additional information is shown in Figure 21, where we show the alternating divergence of the flow corresponding with alternating attraction and repelling.

3.2. Interactions of Larger Quasi-Periodic Solutions

Consider the dynamics when leaving the region of small oscillations. In the case of friction parameter b and interaction parameter $\beta$ still being small, we consider with $a_1\ne a_2$ interactions of quasi-periodic solutions. Selecting $O\left(1\right)$ initial conditions for the coordinates and parameters $a_1,a_2$, we obtain chaotic solutions; see Figure 22, where we have chosen $b=\beta =0.1$. The Lyapunov exponents for the case $a_1=1,a_2=0.5$ are depicted in Figure 23. As usual $c=1$; varying c changes the dynamics but keeps chaos. The exponents are ${\lambda }_1=0.86164049916299\times {10}^{-3};{\lambda }_2=0.28784389624742\times {10}^{-2};{\lambda }_3=0.33455860114490\times {10}^{-3};{\lambda }_3=-0.14209055102249\times {10}^{-3};$ ${\lambda }_5=-0.24875434928298\times {10}^{-2}$, ${\lambda }_6=-0.10024345648336\times {10}^{-1}.$ The dynamics turn out to be hyperchaotic with two positive exponents.

Larger Interaction Parameter $\beta$

Exploring systematically the case of $O\left(1\right)$ values of $\beta$, we find tori, chaos, and hyperchaos. The hyperchaos found in the system of Figure 23 with two associated L exponents was investigated with other phenomena for the values of $b,\beta$ between 0 and 1; see Figure 24. We find small islands (yellow) in a sea of chaotic cases (green). The chaotic cases are associated with one L exponent. In a neighbourhood of $b=\beta =0$, we find two-dimensional tori (blue) as found earlier for the case of small interactions and small oscillations, but interestingly, we also have small dissipation sets of chaotic solutions. The picture of the phenomena of many of the cases in Figure 24 is very remarkable and not easy to predict analytically.

Consider again system (17). Assume that $0<\beta <1$, and $b=\epsilon b_0,c=\epsilon c_0$. To study the emergence of tori for forced two-frequency oscillations, consider the system

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\ddot{q_1}+\epsilon b_0{z}_1\dot{q_1}+q_1+\epsilon c_0{q}_1^3\hfill & =\beta q_2,\hfill \\ \dot{z_1}\hfill & =q_1^2+{\dot{q_1}}^2+\epsilon {\displaystyle \frac{c_0}{4}}{q}_1^4-a_1,\hfill \\ \ddot{q_2}+\epsilon b_0{z}_2\dot{q_2}+q_2+\epsilon c_0{q}_2^3\hfill & =\beta q_1,\hfill \\ \dot{z_2}\hfill & =q_2^2+{\dot{q_2}}^2+\epsilon {\displaystyle \frac{c_0}{4}}{q}_2^4-a_2.\hfill \end{array}\right.\end{array}$$

(33)

For $\epsilon =0$, we have the quasi-periodic solutions

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{q}_1\hfill & =r_{1}cos(\sqrt{1+\beta }t+{\varphi }_1)+r_{2}cos(\sqrt{1-\beta }t+{\varphi }_2),\hfill \\ q_2\hfill & =-r_{1}cos(\sqrt{1+\beta }t+{\varphi }_1)+r_{2}cos(\sqrt{1-\beta }t+{\varphi }_2),\hfill \end{array}\right.\end{array}$$

(34)

with $r_1,r_2,{\varphi }_1,{\varphi }_2$ constants determined by the initial conditions. The interactions of these quasi-periodic solutions are shown for small values of b in Figure 25.

If $\epsilon =0$, the expressions of system (34) with constant amplitudes and phases are general quasi-periodic solutions of system (33). They describe for $0<\beta <1$ tori in phase space. We are interested in the bifurcations arising for $0<\epsilon \ll 1$. In Figure 24, we characterise these bifurcations by the Lyapunov exponents. We find periodic orbits (orange), 2- and 3-tori (resp. yellow and light blue), chaos (red), and hyperchaos (dark red spots).

The chart uses colours to represent the different dynamics based on the Lyapunov exponents of the system with the following cases:

• 0: All Lyapunov exponents are negative (no zeros). This indicates that the system settles to an equilibrium state.
• 1: One Lyapunov exponent is positive. This signifies the presence of chaos in the system.
• 2: Two Lyapunov exponents are positive. This corresponds to hyperchaos, where the system exhibits even more complex behaviour.
• −1: One Lyapunov exponent is zero, and all others are negative. This reflects a periodic orbit.
• −2: Two Lyapunov exponents are zero, and all others are negative. This corresponds to a T² torus.
• −3 to −6: Increasing numbers of zero Lyapunov exponents, with all remaining exponents being negative. These represent higher-dimensional invariant tori with quasi-periodic behaviour.

As in most applications, the dissipation has to be fairly small; we zoom in for $0<b<0.2$. We obtained and show the Lyapunov exponents in Figure 25.

4. Conclusions and Discussion

• Our paper essentially shows an efficient hybrid approach. Analysis, in particular averaging–normalisation and numerical bifurcation techniques, go hand in hand. Figure 15 is an example. The periodic solutions have been given by averaging and are indicated; the associated dynamics are added by numerics.
• The use of nonlinear oscillators enables us to study isolated quasi-periodic interactions. For two components (Section 3), the dynamics are essentially different from dissipative KAM theory.
• The interaction of two components in the sense of Section 3 is surprisingly rich, producing periodic solutions, 2- and 3-tori, chaotic, and hyperchaotic behaviour.
• Considering weak interactions and small oscillations, one already finds this rich collection of bifurcations. The chart in Figure 25 shows this for small b and $\beta$.
• The thermostatic control used in the interacting systems originates from chemical physics. Applications of such a control to neural dynamics or economic models might be useful.
• In the analysis of dynamics, one can meet fractal manifolds that may show strange attraction. The use of Lyapunov exponents for fractal manifolds presents basic problems; improvements in the use of these exponents to compute Kaplan–Yorke dimensions will be discussed in a forthcoming paper [25].