2. Ultra-chaos in the ABC flows

Let $x(t)$, $y(t)$ and $z(t)$ represent the location coordinates of a fluid particle, and $\dot {x}(t)$, $\dot {y}(t)$ and $\dot {z}(t)$ denote their temporal derivatives. Thus, in the Lagrangian sense, the motion of a fluid particle in ABC flow (1.1) is governed by

(2.1)\begin{equation} \left\{\begin{array}{@{}l@{}} \dot{x}(t)=A\sin[z(t)]+C\cos[y(t)], \\ \dot{y}(t)=B\sin[x(t)]+A\cos[z(t)], \\ \dot{z}(t)=C\sin[y(t)]+B\cos[x(t)], \end{array} \right. \end{equation}

with the initial condition

(2.2)\begin{equation} (x(0), y(0), z(0) ) = {\boldsymbol{r}}_0, \end{equation}

where ${\boldsymbol {r}}_0$ denotes a starting point of the fluid particle. Equation (2.1) describes a typical conservative (i.e. volume-preserving) dynamical system. Without loss of generality, let us consider the case of $A=1$ and different values of $B$ and $C$. It should be emphasized here that, by means of CNS, we invariably obtain a reproducible/reliable trajectory of the chaotic motion of a fluid particle of the ABC flow over a sufficiently long interval of time. To investigate the influence of small disturbance on trajectory of the fluid particle in ABC flow (1.1) starting from $\boldsymbol {r}_{0}= (x(0), y(0), z(0))$, we compare the trajectories of two close fluid particles of the ABC flow, starting from the initial positions $\boldsymbol {r}_0$ and ${\boldsymbol {r}}_{0}'={\boldsymbol {r}}_{0}+(0,0,1)\times \delta$, respectively, where $\delta = |{\boldsymbol {r}}_{0} - {\boldsymbol {r}}_{0}' |$ is a tiny constant. Note that $\delta = 0$ when ${\boldsymbol {r}}_{0} = {\boldsymbol {r}}_{0}'$, corresponding to non-disturbance.

For example, without loss of generality, let us consider the motion of a fluid particle of the ABC flow (in the Lagrangian sense) starting from the point $\boldsymbol {r}_{0} = (0,0,0)$ in the case for $A=1$ and different values of $B$ and $C$. In order to investigate its chaotic property, we compare the trajectory with that starting from a very close one $\boldsymbol {r}_{0}'=\boldsymbol {r}_{0}+(0,0,1)\times \delta$, where we choose either $\delta = 10^{-5}$ or $10^{-10}$. In each case, the chaotic simulation remains reproducible over the long interval $t\in [0,10\,000]$ by means of a parallel algorithm of the CNS using the $200$th-order Taylor expansion with the time step $\Delta t = 0.01$ and representing all data in $500$-digit multiple-precision floating-point arithmetic, whose replicability/reliability is guaranteed via another CNS result with even smaller background numerical noise, given by the $205$th-order Taylor expansion (with the same time step) and $520$-digit multiple-precision floating-point arithmetic.

When $A=1$, $B=0.7$ and $C=0.42$, a fluid particle starting from $\boldsymbol {r}_{0}=(0,0,0)$ (corresponding to $\delta = 0$) experiences chaotic motion (with the maximum Lyapunov exponent $\lambda _{max}=0.01$) in a restricted spatial domain, as shown in figure 1(a) for its phase plot $(x,z)$ (considering the linear increase of value of $y$). For $\delta =10^{-5}$ and $\delta =10^{-10}$, although the chaotic trajectories of the two fluid particles, separately starting from the points $\boldsymbol {r}'_{0}$ very close to $\boldsymbol {r}_{0}=(0,0,0)$, are rather sensitive to the starting point, their phase plots and statistical properties such as the p.d.f. are almost the same as those given by the chaotic trajectory starting from $\boldsymbol {r}_{0}=(0,0,0)$ that corresponds to $\delta =0$, as shown in figures 1(b), 1(c) and 1(d), respectively. Note that here we show the p.d.f.s (as well as other statistics) only of the $z$-coordinate values due to the similar properties of their $x$-coordinate counterparts. Therefore, for $A=1$, $B=0.7$ and $C=0.42$, the motion of the fluid particle starting from $\boldsymbol {r}_{0}=(0,0,0)$ is a normal-chaos, since its statistical properties such as the p.d.f. of $z(t)$ are stable, i.e. not sensitive, to a very small disturbance of the starting point.

Figure 1. Influence of tiny disturbances on the phase plot $x$–$z$ and the probability density function (p.d.f.) of a normal-chaotic motion of a fluid particle in ABC flow. The curves are based on CNS results in $t\in [0,10\,000]$ of a normal-chaotic fluid particle, governed by ABC flow (2.1) with (2.2) for $A=1$, $B=0.7$ and $C=0.42$ (with the maximum Lyapunov exponent $\lambda _{max}=0.01$), from the starting point $\boldsymbol {r}_{0}'=(0,0,0)+(0,0,1) \times \delta$ when $\delta =0$ (red), $\delta =10^{-5}$ (black) and $\delta =10^{-10}$ (blue), respectively. (a) Phase plot $(x,z)$ when $\delta =0$; (b) phase plot $(x,z)$ when $\delta =10^{-5}$; (c) phase plot $(x,z)$ when $\delta =10^{-10}$; (d) p.d.f.s of $z(t)$.

However, for a small change in $C$, i.e. $\Delta C = 0.01$, such that $A=1$, $B=0.7$ and $C=0.43$, the chaotic motion (with the maximum Lyapunov exponent $\lambda _{max}=0.06$) of the fluid particle of the ABC flow starting from $\boldsymbol {r}_{0} = (0,0,0)$ becomes quite different from that for $A=1$, $B=0.7$ and $C=0.42$: the fluid particle moves farther and farther away from $\boldsymbol {r}_{0}$ and besides its phase plot $(x,z)$ becomes very sensitive to the small disturbance of the starting position, as shown in figure 2(a). These are quite different from the results in the case of $A=1$, $B=0.7$ and $C=0.42$. Since the ABC flow is periodic, we normalize the values of $z(t)$ to $[-{\rm \pi},{\rm \pi} )$, i.e.

(2.3)\begin{equation} z'(t) = z(t)+2{\rm \pi} n_z, \end{equation}

where $n_z$ is an integer, and $-{\rm \pi} \leq z' < +{\rm \pi}$. Note that, as illustrated in figure 2(b), tiny disturbances in starting position can lead to huge deviations in the p.d.f.s of the normalized chaotic simulations $z'(t)$ in $t\in [0,10\,000]$. In other words, for $A=1$, $B=0.7$ and $C=0.43$ in the ABC flow (2.1), even statistical properties of the chaotic motion of the fluid particle starting from $\boldsymbol {r}_{0} = (0,0,0)$ are very sensitive to the initial position, and thus the corresponding motion of the particle is a kind of ultra-chaos. Obviously, this kind of ultra-chaos is at a higher level of disorder than that of the normal-chaos, as shown in figures 1 and 2. This example illustrates that ultra-chaos indeed exists in the ABC flow.

Figure 2. Influence of tiny disturbances on the phase plot $x$–$z$ and the p.d.f. of an ultra-chaotic motion of a fluid particle in ABC flow. The curves are based on CNS results in $t\in [0,10\,000]$ of an ultra-chaotic fluid particle, governed by ABC flow (2.1) with (2.2) for $A=1.0$, $B=0.7$ and $C=0.43$ (with the maximum Lyapunov exponent $\lambda _{max}=0.06$) from the starting point $\boldsymbol {r}_{0}'=(0,0,0)+(0,0,1) \times \delta$ when $\delta =0$ (red), $\delta =10^{-5}$ (black), and $\delta =10^{-10}$ (blue), respectively. (a) Phase plots $(x,z)$; (b) p.d.f.s of the normalized results $z'(t)$.

Let us further investigate some other statistics such as the variance $\sigma ^2$, the kurtosis $\gamma _2$ and the ACF to demonstrate the higher disorder of the ultra-chaos than the normal-chaos mentioned above, as shown in table 2 for the normal-chaos and table 3 for the ultra-chaos. Obviously, the statistics of the normal-chaos for $C=0.42$ (see table 2) are stable to small disturbances. Conversely, the statistics of ultra-chaos for $C=0.43$ (see table 3) are sensitive to tiny disturbances and thus unstable. In addition, the ACF of the ultra-chaos is also sensitive to small disturbances, compared with that of the normal-chaos, as shown in figure 3.

Table 2. Influence of tiny disturbances (i.e. $\delta$) on variance $\sigma ^2$ and kurtosis $\gamma _2$ of the statistic results of $z(t)$ of the corresponding normal-chaotic trajectory of fluid particle in ABC flow for $A=1$, $B=0.7$ and ${C=0.42}$. These results are obtained by solving the chaotic dynamic system (2.1) with (2.2) in $t\in [0,10\,000]$ by means of the CNS, from the starting point $\boldsymbol {r}_{0}'=(0,0,0)+ (0,0,1) \times \delta$ when $\delta =0$, $\delta =10^{-5}$ and $\delta =10^{-10}$, respectively.

Table 3. Influence of tiny disturbances (i.e. $\delta$) on variance $\sigma ^2$ and kurtosis $\gamma _2$ of the statistic results of $z(t)$ of the corresponding ultra-chaotic trajectory of fluid particle in ABC flow for $A=1$, $B=0.7$ and $C=0.43$. These results are obtained by solving the chaotic dynamic system (2.1) with (2.2) in $t\in [0,10\,000]$ by means of the CNS, from the starting point $\boldsymbol {r}_{0}'=(0,0,0)+ (0,0,1) \times \delta$ when $\delta =0$, $\delta =10^{-5}$ and $\delta =10^{-10}$, respectively.

Figure 3. Influence of tiny disturbances on the autocorrelation function (ACF) of $z(t)$ of normal-chaotic or ultra-chaotic motion of a fluid particle in ABC flow. The ACFs are based on CNS results in $t\in [0,10\,000]$ of a normal-chaotic or an ultra-chaotic fluid particle in ABC flow (2.1) with (2.2) for $A=1$, $B=0.7$ and either $C=0.42$ or $C=0.43$ from the starting point $\boldsymbol {r}_{0}'=(0,0,0)+(0,0,1) \times \delta$ when $\delta =0$ (red), $\delta =10^{-5}$ (black) and $\delta =10^{-10}$ (blue), respectively. (a) Variation in ACF with $\tau$ of normal-chaotic particle when $C=0.42$, and (b) variation in ACF with $\tau$ of ultra-chaotic particle when $C=0.43$, where $\tau$ denotes the lag.

Furthermore, let us consider the ensemble average of chaotic trajectories of a fluid particle starting from the point $\boldsymbol {r}_{0}' = \boldsymbol {r}_{0}+ (0,0,1) \times \delta _{i}$ with 1000 different tiny disturbances $\delta _{i}$ ($i=1,2,3,\ldots,1000$), which are given by the Gaussian random number generator with a standard deviation $\sigma _{d}=\sqrt {\langle \delta ^{2}_{i}\rangle }$ and a zero mean, i.e. $\mu _d =\langle \delta _{i}\rangle = 0$, where $\langle \,\rangle$ denotes the average operator. For $A=1$, $B=0.7$ and $C=0.42$ and $\boldsymbol {r}_{0} = (0,0,0)$, corresponding to the normal-chaotic motions of a fluid particle, the ensemble averages of the phase plots $x$–$z$, which are given, respectively, either by $\sigma _{d}=10^{-5}$ or $\sigma _{d}=10^{-10}$, are almost the same, as shown in figures 4(a) and 4(b). On the contrary, for $A=1$, $B=0.7$ and $C=0.43$, the ensemble averages of the phase plots $x$–$z$ (of the ultra-chaotic motions of a fluid particle), which are given by either $\sigma _{d}=10^{-5}$ or $\sigma _{d}=10^{-10}$, are totally different, as shown in figure 4(c). Furthermore, the p.d.f. of the ensemble-averaged trajectory of the ultra-chaotic fluid particle starting from $\boldsymbol {r}'_{0}$ is also very sensitive to the standard deviation $\sigma _{d}$ of the starting position, which is completely different from that given by the normal-chaotic fluid particle, as illustrated in figure 5. These results indicate that, unlike a normal-chaos, even ensemble-averaged quantities and their corresponding p.d.f.s for an ultra-chaos in ABC flow are unstable, i.e. rather sensitive to tiny disturbances. Indeed, the ultra-chaotic motion is at a higher level of disorder than that of a normal-chaos in ABC flow.

Figure 4. Influence of tiny disturbances on the $x$–$z$ phase plot of ensemble-averaged trajectory of a normal-chaotic ($C=0.42$) or an ultra-chaotic ($C=0.43$) fluid particle in ABC flow (2.1) for $A=1$ and $B=0.7$. They are based on CNS results in $t\in [0,10\,000]$ from the starting point $\boldsymbol {r}_{0}=(0,0,0)+(0,0,1)\times \delta _i$, $1\leq i\leq 1000$, with $\sigma _{d}=\sqrt {\langle \delta ^{2}_{i}\rangle }=10^{-5}$ (black) and $\sigma _{d} = 10^{-10}$ (blue), respectively. (a) The $x$–$z$ phase-plot of the normal-chaotic fluid particle when $C=0.42$ with $\sigma _{d}=10^{-5}$; (b) the $x$–$z$ phase-plot of the normal-chaotic fluid particle with $\sigma _{d}=10^{-10}$; (c) the $x$–$z$ phase-plot of the ultra-chaotic fluid particle when $C=0.43$ with either $\sigma _{d}=10^{-5}$ or $10^{-10}$.

Figure 5. Influence of tiny disturbances on the p.d.f. of ensemble-averaged trajectory of a normal-chaotic ($C=0.42$) or an ultra-chaotic ($C=0.43$) fluid particle in ABC flow (2.1) with (2.2) for $A=1$ and $B=0.7$. The p.d.f.s of the ensemble-averaged trajectories are based on the CNS results in $t\in [0,10\,000]$ from the starting point $\boldsymbol {r}_{0}=(0,0,0)+(0,0,1) \times \delta _i$, $1\leq i\leq 1000$, with $\sigma _{d}=\sqrt {\langle \delta ^{2}_{i}\rangle }=10^{-5}$ (black) and $\sigma _{d} = 10^{-10}$ (blue), respectively. (a) The p.d.f.s of $z(t)$ of the normal-chaotic fluid particle when $C=0.42$; (b) the p.d.f.s of the normalized results $z'(t)$ of the ultra-chaotic fluid particle when $C=0.43$.

It should be emphasized that the main characteristic of ultra-chaos is that some statistics such as p.d.f. are extremely sensitive to tiny disturbances (Liao & Qin Reference Liao and Qin2022). Thus, in this paper we analyse the influence of tiny disturbances in starting position on the chaotic motions of fluid particles in ABC flow. According to the results mentioned above, other statistical properties (such as variance, kurtosis, ACF, ensemble-averaged trajectory and ensemble-averaged trajectory's p.d.f.) of ultra-chaos in the ABC flow are also very sensitive to tiny disturbances in the starting position. On the contrary, these statistics, given by CNS results of a normal-chaotic fluid particle in the ABC flow, are not sensitive to tiny disturbances. This indicates that the statistics of a normal-chaos are stable.

For $A=1$ and varying the values of $B$ and $C$, we find that non-chaos, normal-chaos and ultra-chaos all exist in the fluid particle trajectory starting from $\boldsymbol {r}_{0} = (0,0,0)$, as shown in figure 6. For a non-chaotic motion, the fluid particle trajectory is stable to tiny disturbances in the starting position. For a normal-chaotic motion, although the trajectory is rather sensitive to tiny disturbances in the starting position, i.e. unstable, the phase plot and the statistical properties are stable to the tiny disturbances. However, for an ultra-chaotic motion, even the statistical properties are unstable, say, sensitive to tiny disturbances in the starting position. As we can see in figure 6, for $A=1$ and $B=0.7$, a small change between $C=0.42$ and $C=0.43$ triggers the transition from normal-chaotic motion to ultra-chaotic motion, which is the reason why we present the cases in figures 1 and 2. Note that for the normal-chaotic motion, the fluid particle starting from $\boldsymbol {r}_{0}=(0,0,0)$ always moves in a restricted spatial domain (such that its position is in a restricted domain of the phase plot $x$–$z$ as shown in figure 1). However, for an ultra-chaotic motion, the fluid particle starting from $\boldsymbol {r}_{0}=(0,0,0)$ progressively departs from its starting point, further illustrating that ultra-chaotic motion in ABC flow has higher disorder than normal-chaotic motion, although the velocity field of the ABC flow as a whole is inherently periodic and steady-state.

Figure 6. Classification of trajectories of fluid particles in ABC flow (2.1) starting from $\boldsymbol {r}_{0} = (0,0,0)$ for different values of $B$ and $C$ when $A=1$: grey domain, non-chaos; blue domain, normal-chaos; red domain, ultra-chaos.

On the other hand, keeping $A=1$, $B=0.7$ and $C=0.43$ and using various positions of the starting point $\boldsymbol {r}_{0}=(x(0),y(0),z(0))$ in ABC flow, where $-{\rm \pi} \leq x(0),y(0),z(0)\leq +{\rm \pi}$, it is found in a similar way that both normal-chaos and ultra-chaos (for the motions of fluid particles starting from different $\boldsymbol {r}_{0}$) widely exist, and these two states of chaos coexist simultaneously in the ABC flow, as shown in figure 7. The statistical values of their maximum Lyapunov exponents $\lambda _{max}$ are given in table 4. Statistically speaking, the maximum Lyapunov exponents $\lambda _{max}$ of the ultra-chaotic motions of fluid particles in ABC flow are approximately two orders of magnitude larger than those of the normal-chaos.

Figure 7. Chaotic states of trajectories of the fluid particles starting from different points $\boldsymbol {r}_{0}=(x(0),y(0),z(0))$ in ABC flow (2.1) for $A=1$, $B=0.7$ and $C=0.43$. Here (a) $z(0)=0$; (b) $z(0)={\rm \pi} /8$; (c) $z(0)={\rm \pi} /4$; (d) $z(0)=3{\rm \pi} /8$; (e) $z(0)=7{\rm \pi} /16$; ( f) $z(0)={\rm \pi} /2$. Blue points, normal-chaos; red points, ultra-chaos.

Table 4. Statistical values of maximum Lyapunov exponents $\lambda _{max}$ of the normal-chaotic and ultra-chaotic trajectories of fluid particles in the ABC flow, gained by solving the chaotic dynamic system ABC flow (2.1) in $t\in [0,10\,000]$ for $A=1$, $B=0.7$ and $C=0.43$ by means of CNS, using various starting points $\boldsymbol {r}_{0}=(x(0),y(0),z(0))$ of the fluid particles, where $-{\rm \pi} \leq x(0),y(0),z(0)\leq +{\rm \pi}$.

Note that, when $z(0)$ of the starting point $\boldsymbol {r}_{0}=(x(0),y(0),z(0))$ increases from $0$ to ${\rm \pi} /2$, there exists a kind of structure constituted by the starting positions $(x(0),y(0))$ of fluid particles with normal-chaotic motion (corresponding to blue points) and ultra-chaotic motion (red points), which undergoes continuous deformation, as shown in figure 7. Although normal-chaotic motion is qualitatively different from ultra-chaotic motion so that it is not difficult for us to obtain the general structure in figure 7, we still require a criterion by which to quantitatively determine the boundary of the structure. Let $f(z')$ denote the p.d.f. of a normalized result $z'\in [-{\rm \pi},+{\rm \pi} )$ given by a chaotic motion of fluid particle, and $f^*(z')$ the p.d.f. of another one with a tiny disturbance to the starting position. A criterion based on the following relative error:

(2.4)\begin{equation} \frac{\displaystyle\int\nolimits_{{-}{\rm \pi}}^{{+}{\rm \pi}}\mid f(z')-f^*(z')\mid {\rm d}z'}{\displaystyle\int\nolimits_{{-}{\rm \pi}}^{{+}{\rm \pi}}f(z')\,{\rm d}z'}= \int_{-{\rm \pi}}^{+{\rm \pi}}\mid f(z')-f^*(z')\mid {\rm d}z' \leq \gamma \end{equation}

is usually adopted to determine the boundary between normal-chaos and ultra-chaos, as shown in figure 7. According to our experience, $\gamma =5\,\%$ is often suitable to distinguish between a normal-chaotic motion and an ultra-chaotic motion. Figure 7 illustrates that the normal-chaotic and ultra-chaotic states coexist at the same time, which is reasonable in a volume-preserving ABC flow that has different types of chaotic trajectories for the motions of fluid particles (Dombre et al. Reference Dombre, Frisch, Greene, Hénon, Mehr and Soward1986), which will be discussed later in detail.

Let $\alpha (x(0),y(0),z(0))=0$ or $1$ denote either a normal-chaotic motion or an ultra-chaotic motion of a fluid particle starting from $\boldsymbol {r}_{0}=(x(0),y(0),z(0))$, respectively. Then, according to our computations, for $-{\rm \pi} \leq x(0) \leq +{\rm \pi}$, there exist the symmetries

(2.5)\begin{equation} \alpha(x(0),y(0),z(0))=\alpha({-}x(0),y(0),{\rm \pi}-z(0)), \end{equation}

where $y(0) \in [-{\rm \pi}, +{\rm \pi} ]$, $z(0)\in [{\rm \pi} /2, {\rm \pi}]$;

(2.6)\begin{equation} \alpha(x(0),y(0),z(0))=\alpha(x(0),{\rm \pi}-y(0),-z(0)), \end{equation}

where $y(0)\in [0, {\rm \pi}]$, $z(0)\in [-{\rm \pi}, 0]$; and

(2.7)\begin{equation} \alpha(x(0),y(0),z(0))=\alpha(x(0),-{\rm \pi}-y(0),-z(0)), \end{equation}

where $y(0)\in [-{\rm \pi}, 0]$, $z(0)\in [-{\rm \pi}, 0]$, respectively.

Considering the fact that the normal-chaotic and ultra-chaotic states coexist simultaneously as shown in figure 7, without loss of generality, we choose two starting points $\boldsymbol {r}_{0,n}=(0,-0.1,0)$ and $\boldsymbol {r}_{0,u}=(-0.1,0.1,0)$ of fluid particles in ABC flow to illustrate a normal-chaotic motion (left) and an ultra-chaotic motion (right) via a movie (see the supplementary movie available at https://doi.org/10.1017/jfm.2023.190) in the case of $A=1, B =0.7, C=0.43$ within $t\in [0,5000]$. As shown in the left part of the movie (corresponding to a normal-chaos), the fluid particle starting from $\boldsymbol {r}_{0,n}$ always moves in a restricted spatial domain and the corresponding trajectory resembles weak chaos. On the contrary, the fluid particle starting from $\boldsymbol {r}_{0,u}$ departs the starting point far away and even its position normalized by periodic condition appears to be in disorder, as shown in the right part of the movie (corresponding to ultra-chaos). All of these clearly illustrate that an ultra-chaotic motion in the ABC flow is completely different from a normal-chaotic motion: an ultra-chaos has indeed a much higher disorder than a normal-chaos.

Let $\beta$ denote the ratio of the number of the starting fluid particles with ultra-chaotic motion to the total number of particles in $-{\rm \pi} \leq x,y,z \leq +{\rm \pi}$. In theory

(2.8)\begin{equation} \beta = \frac{1}{(2{\rm \pi})^{3}}\int_{-{\rm \pi}}^{+{\rm \pi}} \int_{-{\rm \pi}}^{+{\rm \pi}}\int_{-{\rm \pi}}^{+{\rm \pi}} \alpha(x,y,z)\, {{\rm d} x}\,{{\rm d} y}\,{\rm d}z, \end{equation}

with either $\alpha =0$ for a normal-chaos or $\alpha =1$ for an ultra-chaos, respectively. In practice, we use the Monte-Carlo method to estimate the ratio

(2.9)\begin{equation} \beta \approx \frac{N_{ultra}}{N_{all}}, \end{equation}

where $N_{all}$ denotes the total number of the randomly selected starting fluid particles

(2.10)\begin{equation} \boldsymbol{r}_{0}\in\varOmega=\left\{(x,y,z): -{\rm \pi} \leq x,y,z \leq +{\rm \pi}\right\} \end{equation}

and $N_{ultra}$ is the number of starting fluid particles with ultra-chaotic motion. Obviously, the larger $N_{all}$, the more accurate the result of $\beta$ given by the Monte-Carlo method. For $A=1$, $B=0.7$, $0 \leq C \leq 0.43$ and $N_{all}=8000$, it is found that the ratio $\beta$ is dependent upon the value of $C$, as shown in table 5. Notably, when $C\leq 0.1$, it is found that there exists a power-law relationship between $\beta$ and $C$, say,

(2.11)\begin{equation} \beta \approx C^{0.4}, \end{equation}

as illustrated in figure 8. Thus, when the parameter $C$ decreases, the value of $N_{ultra}$, i.e. the number of the starting fluid particles with ultra-chaotic motion, decreases until $N_{ultra} = 0$ when $C = 0$. This is reasonable since the ABC flow for $C=0$ is stable and thus chaotic motion of fluid particles does not exist at all in $C=0$.

Table 5. Values of the parameter $C$ versus $N_{ultra}/N_{all}$, where $N_{ultra}$ denotes the number of starting points corresponding to an ultra-chaotic trajectory of fluid particle in the ABC flow and $N_{all}$ denotes the total number of equidistant starting points. The results are obtained by solving the chaotic dynamical system (2.1) in $t\in [0,10\,000]$ for $A=1.0$, $B=0.7$ and $0 \leq C \leq 0.43$ by means of CNS, using various starting points $\boldsymbol {r}_{0}=(x(0),y(0),z(0))$ of the fluid particles, where $-{\rm \pi} \leq x(0),y(0),z(0)\leq +{\rm \pi}$.

Figure 8. The ratio of the number of the starting fluid particles with ultra-chaotic motion to the total number of particles versus the value of $C$ of ABC flow for $A=1$, $B=0.7$ and $C\leq 0.1$.

2.1. Difference between ultra-chaos and sensitivity of statistics to parameters

It is well known that a chaotic trajectory is unstable, i.e. sensitive to small disturbances. For normal-chaos, a trajectory is unstable but its statistics are stable to small disturbances. However, for ultra-chaos, even its statistics are unstable, i.e. sensitive to very small disturbances. The stability of different types of dynamic system is listed in table 1. Obviously, as illustrated by many examples (Liao & Qin Reference Liao and Qin2022; Yang et al. Reference Yang, Qin and Liao2023), ultra-chaos involves higher disorder than normal-chaos.

It should be emphasized that, unlike sensitivity to parameters, the concept of ultra-chaos focuses on the stability of statistics of a dynamic system, while all physical parameters are fixed, to small disturbances that can be very tiny. Certain dynamic systems exhibit high sensitivity in statistics to physical parameters (Broer, Simó & Vitolo Reference Broer, Simó and Vitolo2002; Ashwin et al. Reference Ashwin, Wieczorek, Vitolo and Cox2012; Lucarini & Bódai Reference Lucarini and Bódai2019; Śliwiak, Chandramoorthy & Wang Reference Śliwiak, Chandramoorthy and Wang2021), which, however, is essentially different from ultra-chaos, i.e. instability of statistics to small disturbances. For example, Broer et al. (Reference Broer, Simó and Vitolo2002) investigated bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing,

(2.12)$$\begin{gather} \dot{x} ={-}a x -y^{2}-z^{2} + a F \left(1+\epsilon \cos \omega t \right), \end{gather}$$

(2.13)$$\begin{gather}\dot{y} ={-}y + x y - b x z + G \left( 1+\epsilon \cos \omega t \right), \end{gather}$$

(2.14)$$\begin{gather}\dot{z} ={-}z + b xy + xz, \end{gather}$$

where $\omega = 2{\rm \pi} /T$ and $a, b, T, F, G$, $\epsilon$ are physical parameters. Without loss of generality, Broer et al. (Reference Broer, Simó and Vitolo2002) considered the cases of $a=1/4, b = 4, T=73$ with varying $F\in [0,12], G\in [0,9]$ and $\epsilon \in [0,0.5]$, and found that there exist Hopf bifurcations and some high sensitivity of statistics to parameters $F, G$ and $\epsilon$. However, we found that all statistic results given by the above-mentioned Lorenz-84 climate model with seasonal forcing are stable to small disturbances, in that they are either non-chaotic or normal-chaotic. In other words, even when high sensitivity of statistics to physical parameters exists, the corresponding dynamic system is stable and thus is not an ultra-chaos! In fact, like the famous three-dimensional Lorenz equation (with one positive Lyaponov exponent) and the four-dimensional Rössler system (with two positive Lyaponov exponents) (Liao & Qin Reference Liao and Qin2022), the above-mentioned Lorenz-84 climate model has normal-chaotic trajectories at most. This is a good example to illustrate the essential difference between ultra-chaos and high sensitivity of statistics to physical parameters: they are quite different things!

For an ultra-chaotic system, its statistics are unstable to any types of disturbances. For example, in the case of $A=1$, $B=0.7$ and $C=0.43$, the trajectory starting from $(0,0,0)$ is still ultra-chaotic even if there is no disturbance to the starting point but a small environmental disturbance, governed by

(2.15)\begin{equation} \left\{\begin{array}{@{}l@{}} \dot{x}(t)=A\sin[z(t)]+C\cos[y(t)] , \\ \dot{y}(t)=B\sin[x(t)]+A\cos[z(t)] , \\ \dot{z}(t)=C\sin[y(t)]+B\cos[x(t)] + \varepsilon(t), \end{array} \right. \end{equation}

with the initial condition

(2.16)\begin{equation} (x(0), y(0), z(0) ) = \boldsymbol{r}_0, \end{equation}

where $\varepsilon (t)$ is a normally random noise with a small standard deviation (at the order of magnitude $10^{-10}$), and $\boldsymbol {r}_0$ is the starting point, respectively. We found that, even for the fixed values of $A=1, B=0.7, C=0.43$ and the exact starting position $\boldsymbol {r}_0 = (0,0,0)$, the statistics of the corresponding trajectory are unstable, i.e. rather sensitive to the normally random noise $\varepsilon (t)$. In this case, the trajectory is ultra-chaotic, but there exists no sensitivity of statistics to parameters, because the starting position and all physical parameters have exactly the same values. This further indicates the essential differences between ultra-chaos and sensitivity of statistics to parameters.

Figure 9. Poincaré section at $z'=0$ for several normalized trajectories $(x'(t),y'(t),z'(t))$ in $t\in [0,10\,000]$ of fluid particles starting from different points $\boldsymbol {r}_{0}=(x(0),y(0),z(0))$ (as listed in table 6) obtained by means of CNS for ABC flow (2.1) in the case of $A=1$, $B=0.7$ and $C=0.43$.

Table 6. Positions $\boldsymbol {r}_{0}=(x(0),y(0),0)$ of the starting particles, chosen for the Poincaré section shown in figure 9.

2.2. Possible relationship between ultra-chaos and Poincaré section

Following Dombre et al. (Reference Dombre, Frisch, Greene, Hénon, Mehr and Soward1986), we obtain the Poincaré section of the ABC flow (2.1) for $A=1$, $B=0.7$ and $C=0.43$, as shown in figure 9. Here, we applied CNS to obtain the trajectories $(x(t),y(t),z(t))$ of fluid particles in $t\in [0,10\,000]$, starting from several selected points (listed in table 6). For each trajectory, we have a point $(x',y')$ when $z'(t) = 2 n {\rm \pi}$ for an arbitrary integer $n$, corresponding to a point $(x^{*},y^{*})$ in the square domain

(2.17a,b)\begin{equation} x^{*} \in[-{\rm \pi}, +{\rm \pi}),\quad y^{*}\in[-{\rm \pi}, +{\rm \pi}), \end{equation}

by means of the periodic condition in $x'$ and $y'$ directions, say,

(2.18a,b)\begin{equation} x' = x^{*} + 2 m {\rm \pi}, \quad y' = y^{*} + 2 k {\rm \pi}, \end{equation}

where $m$ and $k$ are integers. The set of all these points $(x^{*},y^{*})$ gives the Poincaré section of the ABC flow (2.1), as shown in figure 9. For details, please refer to Dombre et al. (Reference Dombre, Frisch, Greene, Hénon, Mehr and Soward1986).

As shown in figure 9, there exist elliptic islands (or Kolmogorov–Arnold–Moser (KAM) tori) and a chaotic sea in the Poincaré section. By convention, it is widely believed that points in an elliptic island correspond to quasiperiodic orbits or weakly chaotic orbits, but points in a chaotic sea correspond to strongly chaotic orbits, respectively (Kuznetsov & Zaslavsky Reference Kuznetsov and Zaslavsky2000; Skokos Reference Skokos2001; Lukes-Gerakopoulos, Voglis & Efthymiopoulos Reference Lukes-Gerakopoulos, Voglis and Efthymiopoulos2008). Interestingly, the Poincaré section (as shown in figure 9) is rather similar to figure 7(a). So, it is reasonable for particles starting from the elliptic islands (or KAM tori) to represent a kind of normal-chaotic property, because their maximum Lyapunov exponents $8.5\times 10^{-5} \leq \lambda _{max} \leq 1.3\times 10^{-2}$ (listed in table 4) indeed correspond to a weak chaos. This numerical fact reveals the following relationship: the normal-chaotic (starting) points (at $z=0$) of the ABC flow correspond to the elliptic islands (or KAM tori) in the Poincaré section, but the ultra-chaotic ones invariably correspond to the chaotic sea. According to our computations, this kind of relationship is true for almost all fluid particles in the ABC flow. Besides, this numerical experiment also supports our conclusion that an ultra-chaos is a higher disorder than a normal-chaos. Thus, the classification of chaos into normal-chaos and ultra-chaos provides a new explanation of elliptic islands (or KAM tori) and chaotic sea in Poincaré section of a dynamic system.

Note that Poincaré section has a close relationship with KAM theory that is valid for an integrable Hamiltonian system only. However, the classification of chaos into ultra-chaos and normal-chaos is generally valid for all dynamic systems, even if they are not Hamiltonian, or not integrable. Therefore, this classification has a more general meaning, given that ultra-chaos reveals higher disorder than normal-chaos. An example of such higher disorder related to statistical sensitivity to small disturbances has been recently reported: small disturbances can lead to large-scale deviations of simulations of a turbulent flow not only in spatiotemporal trajectories but also in statistics, even leading to different types of flow (Qin & Liao Reference Qin and Liao2022).

3. Possible relationship between ultra-chaos and turbulence

The velocity $\boldsymbol {u}_{ABC}$ of the ABC flow (1.1) was first discovered by Arnold (Reference Arnold1965) as a class of steady-state solutions of the Euler equations, and, moreover, with external force per unit mass, it also satisfies the NS momentum and continuity equations

(3.1)$$\begin{gather} \frac{\partial\boldsymbol{u}}{\partial t}+(\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla})\boldsymbol{u}={-}\boldsymbol{\nabla} p +\frac{1}{Re}\Delta\boldsymbol{u}+\boldsymbol{f}, \end{gather}$$

(3.2)$$\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{u}=0, \end{gather}$$

where $t\geq 0$ denotes the time, $\boldsymbol {\nabla }$ is the Hamilton operator, $\varDelta$ is the Laplace operator, $Re$ is the Reynolds number, $p$ denotes the pressure and

(3.3)\begin{equation} \boldsymbol{f}=\frac{\boldsymbol{u}_{ABC}}{Re} \end{equation}

is the given external force per unit mass, with the periodic boundary conditions at $x =\pm {\rm \pi}$, $y =\pm {\rm \pi}$, and $z =\pm {\rm \pi}$.

Many papers (Dombre et al. Reference Dombre, Frisch, Greene, Hénon, Mehr and Soward1986; Mezić Reference Mezić2002; Podvigina, Ashwin & Hawker Reference Podvigina, Ashwin and Hawker2006) have been published in this field. For example, Podvigina et al. (Reference Podvigina, Ashwin and Hawker2006) analysed the bifurcation of the ABC flow and reported the supercritical Hopf bifurcation and route to chaos through tori doubling. Without loss of generality, let us consider here the ABC flow in the case of $A=1$, $B=0.7$ and $0 \leq C \leq 0.43$. Unlike other researchers (Dombre et al. Reference Dombre, Frisch, Greene, Hénon, Mehr and Soward1986; Mezić Reference Mezić2002; Podvigina et al. Reference Podvigina, Ashwin and Hawker2006), here we mainly focus on the ultra-chaotic motion of fluid particles. As reported by Podvigina & Pouquet (Reference Podvigina and Pouquet1994), the Reynolds number $Re=50$ corresponds to a turbulent flow if the initial velocity field $\boldsymbol {u}_{ABC}$ experiences small disturbances of the order of magnitude $10^{-3}$. This kind of turbulent flow is solved numerically in $t\in [0,500]$: the spatial domain $[-{\rm \pi}, +{\rm \pi} )^3$ is discretized by a uniform mesh with $128^3$ points for the spatial Fourier expansion, where the maximum grid spacing is less than the minimum Kolmogorov scale (Pope Reference Pope2001), and the $3/2$ rule for dealiasing (Pope Reference Pope2001) is used, with the time step $\Delta t=10^{-3}$.

First, let us use the unstable ABC flow for $A=1.0$, $B=0.7$ and $C=0.43$ as the initial condition of the NS equations (3.1) and (3.2) (with the external force per unit mass $\boldsymbol {f}=\boldsymbol {u}_{ABC}/Re$ as mentioned above), under small disturbances of initial velocity at the order of magnitude $10^{-3}$. It is found that, the flow is initially rather similar to the ABC flow at times that are too short for the tiny velocity disturbances to transfer to the macrolevel; in this situation, approximately 49 % starting fluid particles are ultra-chaotic, according to table 5. The transition from laminar flow to turbulence occurs approximately at $t \approx 50.0 = T_{tran}$, as shown in figure 10, where $T_{tran}$ denotes the time of the transition occurrence. Given the velocity field $\boldsymbol {u}$ of (3.1) and (3.2), we can investigate the chaotic property of trajectory (i.e. whether it is ultra-chaotic or normal-chaotic) of a fluid particle starting from $\boldsymbol {r}_{0}$ in a similar way as mentioned in § 2. When $t=50$, we randomly choose 10 000 starting fluid particles in $-{\rm \pi} \leq x,y,z < +{\rm \pi}$ and find that all trajectories of the fluid particles starting from them are ultra-chaotic. This numerical experiment strongly suggests that a necessary condition of turbulence is that almost all fluid particles should move along ultra-chaotic trajectories.

Figure 10. (a) Total kinetic energy; (b–f) modulus $|\boldsymbol {\omega }|$ of instantaneous vorticity field at $t=30$, $t=50$, $t=60$, $t=100$, or $t=300$, respectively, governed by the NS equations (3.1) and (3.2) with $R_{e}=50$ using the ABC flow (1.1) (for $A=1$, $B=0.7$ and $C=0.43$) under a small disturbance at the order of magnitude $10^{-3}$ as the initial solution.

Similarly, let us consider the stable ABC flow in the case of $A=1$, $B=0.7$ and $C=0$. Using the ABC flow $\boldsymbol {u}_{ABC}$ subject to small disturbances at the order of magnitude $10^{-3}$ as the initial condition, we numerically solve the NS and continuity equations (3.1) and (3.2) in the time interval $t\in [0,2000]$ and investigate the chaotic property of trajectories of the 10 000 randomly chosen fluid particles. It is found that, when $C=0$, transition from the laminar flow to turbulence never occurs, and, moreover, there exists no ultra-chaotic motion for all of these fluid particles throughout $t\in [0,2000]$. This further confirms our suggestions that ultra-chaotic trajectories of fluid particles should have some relationships with turbulence.

In addition, let us further consider the ABC flows for $A=1$, $B=0.7$ and different values of $C$, and for each case we use the Monte-Carlo method to randomly choose 10 000 starting points in $-{\rm \pi} \leq x,y,z < +{\rm \pi}$. It is found that the transition time $T_{tran}$ when the flow alters from laminar flow to turbulence increases as $C$ decreases from $0.1$ to $0.0001$, as shown in table 7. We found that, when $0 < C\leq 0.1$, there exists a linear relationship

(3.4)\begin{equation} T_{tran}\approx{-}10\log_{10}(C)+40, \end{equation}

as illustrated in figure 11, indicating that $T_{tran} \to +\infty$ as $C\to 0$. Hence, indeed the transition from laminar flow to turbulence should never occur when $C=0$, which agrees with our numerical simulation in the case of $C=0$ mentioned above. In all cases of $A=1$, $B=0.7$ and $0 < C \leq 0.43$ under consideration, it is found that all trajectories of the fluid particles starting from the randomly chosen 10 000 fluid particles are ultra-chaotic after the flow becomes fully turbulent. Besides, the smaller the number of ultra-chaotic particles at the beginning (corresponding to an unstable ABC flow with a smaller value of $C$), the longer the transition time of $T_{tran}$. In other words, more time is needed for all fluid particles to become ultra-chaotic at $t =T_{tran}$. The foregoing again suggests that a necessary condition of the occurrence of transition from laminar flow to turbulence is that nearly all fluid particles should move along ultra-chaotic trajectories.

Table 7. Transition occurrence time $T_{tran}$ for different values of $C$, gained by numerically solving the NS equations (3.1) and (3.2) using the ABC flow (1.1) (for $A=1$, $B=0.7$ and $0\leq C\leq 0.1$) as the initial condition plus a small disturbance of velocity field at the order of magnitude $10^{-3}$.

Figure 11. Relationship between $C$ and $T_{tran}$ obtained for ABC flow with $A=1$, $B=0.7$ and $0< C\leq 0.1$ subject to the ABC flow (1.1) plus a small disturbance at order of magnitude $10^{-3}$ as the initial condition used when solving the NS and continuity equations (3.1) and (3.2), where $T_{tran}$ denotes the time of transition occurrence.

Here, regarding the ultra-chaotic motion as a new property of fluid particle, we simply report some results of numerical experiments governed by (3.1) and (3.2) subject to periodic boundary conditions. Frankly speaking, it is not yet understood why almost all fluid particles should move along ultra-chaotic trajectories when the flow becomes fully turbulent: What happens to the trajectory property of fluid particles when the transition from laminar flow to turbulence occurs, and so on? We firmly believe that the concept of ultra-chaos could enable us to gain new insight into viscous flows. We intend to pursue this in future work.

It should be emphasized that the transition from laminar flow to turbulence is a key problem in fluid mechanics. Hopefully, ultra-chaos as a new concept could provide us a completely new viewpoint to investigate and understand the mechanism of the transition to turbulence.

4. Concluding remarks and discussions

Due to the butterfly effect (Lorenz Reference Lorenz1963), numerical noise (due to truncation error and round-off error) enlarges exponentially so that a computer-generated simulation of chaotic systems quickly becomes a mixture of a ‘true’ physical solution $s$ and a ‘false’ numerical noise $\varepsilon$, which are mostly at the same order of magnitude. In practice, statistics of chaotic systems are usually calculated using such a mixture, i.e. $s+\varepsilon$, because there is no way to separate out the ‘true’ physical solution $s$. In fact, the statistics is based on the hypothesis

(4.1)\begin{equation} \langle s+\varepsilon \rangle = \langle s \rangle, \end{equation}

where $\langle \rangle$ is a statistical operator. Here, the numerical noise $\varepsilon$ is in fact equivalent to a kind of small disturbances. Unfortunately, there exists no theoretical proof that the above hypothesis is always true in general.

By means of CNS (Liao Reference Liao2009, Reference Liao2013, Reference Liao2014, Reference Liao2017), one can gain reliable/convergent numerical simulations of chaotic systems over a long enough interval of time (Liao & Wang Reference Liao and Wang2014; Li & Liao Reference Li and Liao2017; Lin et al. Reference Lin, Wang and Liao2017; Hu & Liao Reference Hu and Liao2020; Qin & Liao Reference Qin and Liao2020; Liao & Qin Reference Liao and Qin2022; Qin & Liao Reference Qin and Liao2022; Yang et al. Reference Yang, Qin and Liao2023), during which the ‘false’ numerical noise $\varepsilon$ is much smaller than the ‘true’ physical solution $s$, say, $|\varepsilon | \ll |s|$, thus the influence of the numerical noise $\varepsilon$ is negligible compared with the ‘true’ physical solution $s$. Hence, CNS can provide us, for the first time, with a nearly ‘clean’ computer-generated simulation of chaos in an interval of time long enough for statistics, which can be used as a benchmark solution since it is very close to the ‘true’ physical solution $s$. Therefore, CNS provides us with a useful tool by which to study accurately the influence of numerical noise on statistics of different types of chaotic systems (Hu & Liao Reference Hu and Liao2020; Qin & Liao Reference Qin and Liao2020; Liao & Qin Reference Liao and Qin2022; Qin & Liao Reference Qin and Liao2022; Yang et al. Reference Yang, Qin and Liao2023).

Using CNS, it has been observed that the hypothesis $\langle s+\varepsilon \rangle = \langle s \rangle$ holds for many chaotic systems, whose statistics are stable to small disturbances. However, it has been found that statistics of some chaotic systems are indeed rather sensitive to small disturbances including artificial numerical noise (Hu & Liao Reference Hu and Liao2020; Qin & Liao Reference Qin and Liao2020, Reference Qin and Liao2022; Yang et al. Reference Yang, Qin and Liao2023), say, $\langle s+\varepsilon \rangle \neq \langle s \rangle$. Liao & Qin (Reference Liao and Qin2022) termed the former ‘normal-chaos’ and the latter ‘ultra-chaos’, respectively. The stability of trajectory and statistics of different types of dynamic systems is listed in table 1, which clearly indicates that an ultra-chaos is a higher disorder than a normal-chaos.

It is well known that the steady-state ABC flow, which satisfies the steady NS equations with a proper external force, has chaotic properties from a Lagrangian viewpoint. Naturally, it is worth investigating whether ultra-chaos exists in ABC flow, whether relationships hold between ultra-chaos and turbulence, and so on. In this paper, we illustrate that trajectories of many fluid particles in the steady-state ABC flow (1.1) are ultra-chaotic, in that their statistical properties are rather sensitive to tiny disturbances. Obviously, this kind of ultra-chaotic motion of fluid particles represents a higher disorder than normal-chaotic ones. We found that these two kinds of totally different chaos coexist widely and simultaneously in the steady-state ABC flow.

Besides, we discuss the difference between ultra-chaos and the high sensitivity of statistics to parameters, and also the relationships between ultra-chaos and Poincaré section, ultra-chaos and ergodicity, and so on. Unlike the high sensitivity of statistics to parameters, ultra-chaos focuses on the stability of statistics to very small disturbances. We have illustrated that certain model equations (such as the Lorenz-84 climate model with seasonal forcing) exhibit high sensitivity of statistics to parameters, even though all the statistics remain stable so that all the simulations involve either non-chaos or normal-chaos.

Furthermore, we discuss the possible relationship between the Poincaré section and normal/ultra-chaos. It is found that the normal-chaotic (starting) points (at $z=0$) of the ABC flow correspond to elliptic islands (or KAM tori), but the ultra-chaotic ones correspond to the chaotic sea of the Poincaré section. It should be emphasized that, considering the fact that the Poincaré section has a close relationship with the KAM theory that is usually suitable for an integrable Hamiltonian system only, the new classification of chaos into ultra-chaos and normal-chaos is more general because of its validity for all dynamic systems, even if they are not Hamiltonian, and/or not integrable. Therefore, this new classification should have more general meaning.

In addition, we discuss possible relationships between ultra-chaos and non-ergodicity. According to Birkhoff (Reference Birkhoff1931) and von Neumann (Reference von Neumann1932), time averages can be set equal to phase averages, provided the system is ergodic (i.e. metrically transitive) (Moore Reference Moore2015). However, it is very difficult to seriously prove that a system is metrically transitive. It is an open question whether or not a normal-chaos corresponds to ergodicity and an ultra-chaos corresponds to non-ergodicity, respectively. Hopefully the classification of chaos into normal-chaos and ultra-chaos could provide us a simple and practical way to investigate ergodic property of various types of dynamic systems.

In order to study the possible relationships between ultra-chaos and turbulence, we numerically solve the NS and continuity equations (3.1) and (3.2) when $Re = 50$ using the ABC flow (for $A=1, B =0.7$ and various values of $C$) plus a small disturbance as the initial condition. It is found that the trajectories of nearly all fluid particles become ultra-chaotic after the transition from laminar to turbulent flow occurs. Our numerical facts strongly suggest that turbulence is likely to be related to ultra-chaotic trajectories of fluid particles, although the detailed mechanisms are presently unknown. Considering that the chaotic property of the ABC flow is essential for the development of turbulence (Dombre et al. Reference Dombre, Frisch, Greene, Hénon, Mehr and Soward1986; Galloway & Frisch Reference Galloway and Frisch1987; Podvigina & Pouquet Reference Podvigina and Pouquet1994), we anticipate that the concept of ultra-chaos (Liao & Qin Reference Liao and Qin2022) could lead to a better understanding of fluid chaos, Poincaré section, ergodicity/non-ergodicity, turbulence and their inter-relationships.