Nonlinear stochastic dynamics research on a Lorenz system with white Gaussian noise based on a quasi‐potential approach

Environmental noise can lead to complex stochastic dynamical behaviors in nonlinear systems. In this paper, a Lorenz system with the parameter region with two stable fixed points and a chaotic saddle subject to white Gaussian noise is investigated as an example. Noise‐induced phenomena, such as noise‐induced quasi‐cycle, three‐state intermittency, and chaos, are observed. In the intermittency process, the optimal path used to describe the transition mechanism is calculated and confirmed to pass through an unstable periodic orbit, a chaotic saddle, a saddle point, and a heteroclinic trajectory in an orderly sequence using generalized cell mapping with a digraph method constructively. The corresponding optimal fluctuation forces are delineated to uncover the effects of noise during the transition process. Then the process will switch frequently between the attractors and the chaotic saddle as noise intensity increased further, that is, noise induced chaos emerging. A threshold noise intensity is defined by stochastic sensitivity analysis when a confidence ellipsoid is tangent to the stable manifold of the periodic orbit, which agrees with the simulation results. It is finally reported that these results and methods can be generalized to analyze the stochastic dynamics of other nonlinear mechanical systems with similar structures.


| INTRODUCTION
The interaction between nonlinearity and randomness in dynamical systems can lead to novel behaviors, which have no equivalence in the deterministic case. The last several decades have witnessed an increasing number of investigations in the analysis of noise-induced phenomena, such as the so-called noise-induced chaos, 1,2 intermittent switching behavior, 3,4 noise-induced transition in multistable systems, [5][6][7] noise-induced excitability, 8,9 and complex behaviors in quantum phase transitions 10,11 and in various biological and physical systems. [12][13][14][15] Under random perturbations, even structurally stable systems may show such phenomena. Thus far, the main research methods related to the problem of noise-induced chaos have focused on the numerical evaluation of the largest Lyapunov exponent 2,16 , which is used to describe the chaotic properties of signals and their fractal dimensions. Gao et al. 17 studied a noiseinduced Hopf-bifurcation-type sequence and transition to chaos in the Lorenz system within the parameter regime where two stable fixed points and a strange attractor coexist. By carrying out the calculation of the largest Lyapunov exponent, Lai et al. 2 explored the phenomenon of noise-induced chaos for the systems with both a regular attractor and a non-attracting chaotic set.
Recently, some researchers investigated noise-induced chaos based on the concept of quasi-potential, which is a crucial concept for the problem of noise-induced escape. Tél and Lai 18 utilized this concept to confirm analytically that the scaling law for the largest Lyapunov exponent is satisfied. Furthermore, Kong and Liu 19 used the mean first passage time (MFPT) from the nonchaotic attractor to the transient chaos as an indicator to characterize the phenomenon of noise-induced chaos for a piecewise linear system.
In fact, this phenomenon can be essentially regarded as switching between the phenomena of escaping from the attractor to the transient chaos and relaxation along the deterministic flow recurrently. Therefore, MFPT, which describes the time consumed trapped in the basin of the attractor, plays a significant role in this process.
The large deviation theory proposed by Freidlin and Wentzell 20 has established crucial results to elaborate the long-term effects of small random perturbations. Specifically, it builds on the fact that almost unlikely events, when they occur, will do so along a specific path with overwhelming probability, referred to as the optimal path. In fact, this optimal path gives rise to the minimum of a certain action functional that characterizes the difficulty of the random process passing through the neighborhood of a given path. Then, the quasipotential for a specific point is generated by the action functional along the optimal path connecting it to the attractor. It is well known that the magnitudes of both MFPT and the stationary probability distribution are dominated exponentially by the value of the quasi-potential.
Consequently, the optimal path and quasi-potential are two representative quantities used to characterize the escaping scenarios and the costing time, respectively, for the problems of noise-induced escape or transition.
Based on the second-order approximation of the quasi-potential, the stochastic sensitivity function and the confidence ellipse method were proposed and have been successfully applied to many systems, such as the Hodgkin-Huxley model, 21 a hair bundle model, 22 the prey-predator plankton system, 23 the Morris-Lecar system, 24 and the Higgins model. 25 In particular, by constructing the confidence ellipses that reflect the main features of the spatial arrangement of random states, Bashkirtseva and Ryashko 26 studied the noise-induced transitions between coexisting periodic attractors of the Lorenz model. In this work, we continue to discuss the nonlinear stochastic dynamics of white noise on the Lorenz system using the large deviation principle and the stochastic sensitivity technique.
Due to the complicated structures of the chaotic invariant sets, solving the escaping or transition problems driven by noise is a challenging task, and consequently, limited research efforts have been devoted to the solution of these problems. Noise-induced escape from a non-hyperbolic chaotic attractor in a periodically excited nonlinear oscillator was considered by Luchinsky 27 by statistical analysis of the escaping paths. The same system was also studied by Chen et al. 28 using the generalized cell mapping with digraph (GCMD) method to uncover that the escape occurs through a hierarchical sequence of crossings between stable and unstable manifolds of saddle cycles.
Kraut and Feudel 29 demonstrated the effect of enhancement of noiseinduced escape via the existence of a chaotic saddle based on the use of an Ikeda map. Furthermore, noise-induced transitions crossing different kinds of fractal basin boundaries were investigated theoretically and numerically in discrete and continuous dynamical systems by Silchenko et al. 30 As a typical quasi-hyperbolic system, many researchers studied the stochastic effects of noise on the Lorenz system. For example, Anishchenko et al. 31 investigated the phenomenon of stochastic resonance of a Lorenz system perturbed by white noise and a harmonic force. Toral et al. 32 performed theoretical and numerical studies of noise-induced synchronization phenomena of the Lorenz system and analyzed its structural stability. Moreover, the escaping event from a chaotic attractor in the Lorenz system was studied by constructing the prehistory probability distribution. 33 This system was also considered by Zhou and Weinan 34 using an adaptive minimum action method to confirm the existence and uniqueness of the optimal path and reveal its successive series of the transition process. However, there are still difficulties in determining the positions of unstable invariant sets in a three-dimensional phase space that are important in the analysis of escaping mechanisms. To address this challenge, the GCMD method is applied to a specific Poincaré section of the Lorenz system in this paper, which can be generalized to more three-dimensional problems. This paper is organized as follows: The bifurcation behavior and the phase diagram of the Lorenz model are discussed in Section 2.
Numerical simulations are performed in Section 3 for various noise intensities to divide the effects of noise into regions of a noise-induced quasi-cycle, intermittency, and chaos. In Section 4, the large deviation theory is used to derive the Hamiltonian formalism satisfied by the optimal transition path and the Hamilton-Jacobi equation for a quasipotential. Then, noise-induced transitions are investigated in Section 5 to depict the optimal path combining the geometry minimum action method (GMAM) and the GCMD method. In Section 6, stochastic sensitivity analysis is used to discuss the impacts of noise theoretically for different strengths. The conclusions are presented in Section 7.

| MODEL
In this paper, the Lorenz system driven by white Gaussian noise is chosen as a model for investigation Before considering the stochastic dynamics of the Lorenz model, its deterministic structure and bifurcation behavior are first reviewed briefly. At ρ = 1, a pitchfork bifurcation occurs, which destroys the stability of origin O and generates two stable equilibriums: Then, a chaotic saddle will arise after a homoclinic explosion occurs at ρ ≈ 13.92. Next, this chaotic saddle will gain stability to be transformed into a chaotic attractor at ρ ≈ 24.06. Subsequently, a subcritical Hopf bifurcation at ρ ≈ 24.74 results in the loss of stability of F1 and F2. More detailed analyses and discussions about its bifurcations can be found in Ref. 35.
The case for ρ = 19.375 is of interest in this paper. Figure 1A shows a saddle point O and two stable fixed points F1 and F2. L1 and

| THE EFFECTS OF NOISE
When noise is present, some interesting phenomena occur that are essentially different from the deterministic case. Originating from the fixed point F2, the stochastic trajectories are determined by performing Monte Carlo simulations. Figure 2A shows the relationship between coordinate x + y after the integration of the time 10 000 s and noise intensity. Figure 2B shows the time series of three representative stochastic paths for D = 0.3, 1.5, 3.
First, if the noise is sufficiently weak, then the system fluctuates within the small vicinity of the fixed point, as shown in the first subplot of Figure 2B. Due to the fact that F2 is a focus, the process will circle around it to generate a behavior called a quasi-cycle phenomenon. This can be readily seen from the power spectral density of the time series, as shown in Figure 3, where a sharp peak can be observed. This implies that a stochastic cycle is induced by the white noise, although the corresponding deterministic system has no limit cycle. If noise intensity is increased gradually, the amplitude of the oscillation will be also increase accordingly.
With the sustained increase of the noise intensity, the escape from the basin of attraction of F2 driven by noise will arise occasionally. Then, the random state will enter the fractal region and move with the motion of the transient chaos for some time.
Subsequently, it may be attracted to the basin of F1. Therefore, it can be observed that the system randomly switches between the two fixed points and the transient chaos, referred to as the phenomenon of noise-induced intermittency, as shown in the second subplot of Figure 2B. From Figure 2A, it can be seen that there is an obvious threshold strength of noise D ≈ 0.6 1 separating the regions of quasicycle and intermittency. In fact, even for the noise below D 1 , noiseinduced escape from the basin of the fixed point will also occur with probability one, according to the large deviation theory. 20 However, the MFPT is so large that it is hard to observe it within an appropriate time scale.
It is well known that the MFPT decreases exponentially with the increase of the noise intensity. Therefore, the time spent within the basin of the fixed points for the system will become increasingly less and the transitions induced by noise will be increasingly more frequent. After the noise intensity skips over another threshold value D ≈ 2.5 2 , the system shows chaotic-like motions that are presented in the third subplot of Figure 2B for D = 3. It can be seen that under noise, the motion of the system is similar to that of a chaotic attractor, which is referred to as noise-induced chaos.

| QUASI-POTENTIAL APPROACH
The three-state intermittency behavior can be explained as follows: the system starting from an attractor is pushed into the transient chaos driven by noise and then relax to the original or another attractor. Thus, it is essentially the process of noise-induced transition. Therefore, its mechanism will become clear if we are able to uncover the transition process from F2 and F1.
For this purpose, assume that the asymptotic solution of the stationary probability density has the following WKB form: where x C ( ) is a prefactor not investigated in this paper and x W ( ) is the quasi-potential that characterizes the activation energy of fluctuations to the vicinity of the point x in the state space.
Specifically, the quasi-potential is defined as the infimum of the action functional through all the absolutely continuous trajectories connecting the fixed point x and x, that is, Here with b being the vector field of the system. The functional S φ [ ] T estimates the difficulty of the random path in passing through a small neighborhood of the trajectory φ, in the sense that The substitution of Equation (2) into the stationary Fokker-Planck equation and then the collection of the terms of the lowest order in D yield the Hamilton-Jacobi equation According to the results of classical mechanics, the value of the quasi-potential can be integrated along the trajectory of an auxiliary Hamiltonian system The trajectories providing solutions of Equation (7) span a three-dimensional unstable Lagrangian manifold in a six-dimensional extended phase space. Lying in this manifold, the Hamiltonian flow given by Equation (7) dominates the optimal paths of system (1) connecting the initial and final states.
On the other hand, the MFPT for escaping from an attraction domain exponentially depends on the difference in the quasi-potential between the fixed point and the boundary, that is, where U ∂ denotes its basin boundary. Consequently, all the issues considered are focused on the computation of the optimal transition path and quasi-potential.

| NOISE-INDUCED TRANSITION
As mentioned previously, it's possible that the escape event of a fixed point from the attraction domain will occur after the trajectory moving around the fixed point for an exponentially long time, of which a representative simulation trajectory is shown in Figure 4A.
It can be seen that the fluctuations push the system spiraling outward first, entering the neighborhood of the saddle O, and finally approaching another fixed point along the unstable manifold of O.
Similar to the two-dimensional case, the saddle point still acts as the exit location in the escaping scenario.
To confirm our discovery, the GMAM, 38 an iterative path-based method requiring an initial guess for the path, has been used to search for the optimal path. Its main idea relies on the re-parametrization of the action functional by the arc length instead of time. 34 F1 and F2 are chosen as the endpoints and discretizing the path into 10 5 points is sufficient to obtain the optimal path. The results shown in Figure 4B confirm our previous conjectures about the transition mechanisms of passing through the saddle and following its unstable manifold subsequently. On the other hand, analogous to the two-dimensional system, the transition path approaches the saddle tangent to its stable manifold. However, there is an obvious difference: it has a two-dimensional stable manifold. According to our results, it is tangent to the eigenvector corresponding to the principal eigenvalue, the one closer to zero.
Besides, it is clear from Figure 4B that the optimal trajectory has passed through the unstable periodic orbit C2 before arriving at the saddle point. On the other hand, we also aimed to determine how significant a role the fractal boundary plays in the escaping scenario.
However, it is rather difficult to determine the exact position of the unstable invariant sets in a three-dimensional phase space. For this purpose, one can select a specific Poincaré section and construct a Poincaré map to reduce it to a two-dimensional problem, as shown in Figure 5. According to the deterministic structure of the system, a rectangular region between F1 and F2 and is chosen as the section and the GCMD method is used to describe its global structure in this section.
The main idea of the GCMD method is to discretize the phase space into a cell state space consisting of finite cells so that the mapping ordering between original points is transformed into the one between cells. Actually, these cells constitute a homoge- The attractors F1 and F2 can be calculated analytically so that there is no need to identify these attractors in the Poincaré section.
Then, the global structures evaluated by the GCMD method are shown in Figure 6 in the absence of noise. Note that the pink regions denote an unstable chaotic saddle (UCS) with its stable manifold forming the fractal boundary, which is indicated by the yellow area. Similarly, one can define these periodic orbits as accessible periodic orbits whose two-dimensional stable manifold separates the fractal boundary and the regular basin of the attractor.
Keeping the global dynamical behavior in mind, we continue to consider the transition problem driven by noise. As shown in Figure 7, the intersection points between the optimal transition path by  Figure 9A. It can be seen that the three directions of noise have almost the same magnitude, meaning that they all play a nonnegligible role in pushing the system to transit. However, things would be different if the three characteristic directions of the attractor F2 are considered. As one can easily see from Figure 9B, the magnitude of one of the momenta is much less than the two other. In fact, the fixed point has three eigenvalues, λ i = −0.18 ± 8.58 1,2 and λ = −13.3 3 . It is obvious that the third direction has a much higher compression rate than the other two directions so that it is too difficult to overcome this attractive force to escape. Consequently, the transition path nearly lies in a specific plane before arriving at the saddle point.
From Figure 9B, it can also be found that p + and p − representing the momenta along the directions corresponding to λ 1 and λ 2 alternate to achieve their maxima. Specifically, when p + achieves its maximal value, p − nearly approaches zero and vice versa. Actually, this stems from the fact that the deterministic vector field spirals into the equilibrium, while the transition path spirals out with the same rotation direction. In other words, the effects of fluctuations preserve the rotation component of the vector field but just modify the convergent component into divergence. Therefore, when the system state moves to the highest or lowest position, p + will drive it down or up while p − has little influence. Similarly, p − will force the process to move more right or left, with p + being nearly zero if it is at the same height as the attractor. can be approximated by its quadratic term of the Taylor expansion: According to the Hamilton-Jacobi equation of Equation (6), the matrix V , referred to as the stochastic sensitivity matrix, 21,24,43 is the unique solution of the following equation: This matrix describes the spatial arrangement and size of the confidence ellipsoid for a three-dimensional case where the function K P ( ) is the inverse function to the fiducial This implies that the random states will be located at this domain with a high probability. Equation (10) where λ i are the eigenvalues of V with corresponding eigenvectors v i , For the sake of convenience of description, the confidence ellipsoid is depicted in the Poincaré section mentioned previously such that it is replaced by an ellipse. As shown in Figure 10 In addition, the relationships between the three eigenvalues of the stochastic sensitivity matrix versus the parameter ρ are plotted in Figure 11. It can be observed that the first two eigenvalues increase markedly as ρ approaches the subthreshold Hopf bifurcation value ρ ≈ 24.74 where the fixed point loses its stability. This extraordinary phenomenon agrees with our intuition that the fixed point will be highly sensitive near the bifurcation. However, λ 3 remains nearly constant since the third direction of the fixed point still remains stable even after the bifurcation occurs. Therefore, the eigenvalues of the stochastic sensitivity matrix can be regarded as measures of stability of the fixed point.

| CONCLUSIONS
To gain insights into the effects of noise for three-dimensional systems, we focused on a Lorenz system driven by white Gaussian noise with the parameter region with two stable fixed points and a transient chaos. For various noise intensities, the phenomena of noise-induced quasi-cycle, intermittency, and chaos can be observed.
Their separations rely on the magnitude of MFPT, or equivalently, the concentration of the stationary probability distribution on the attractor, which are both dominated substantially by the size of the quasi-potential.
If noise is extremely weak, it is impractical to observe the escaping event due to MFPT being very large. Thus, the random states will fluctuate around the attractor with a small amplitude to form the quasi-cycle regime. Next, slightly bigger noise can lead to a transition occasionally between the two stable fixed points through a chaotic saddle, resulting in a three-state intermittency.
The calculation of the optimal transition path provides a detailed description of the transition mechanism and uncovers the role played by those unstable invariant sets. Besides, further increase of noise intensity causes the process to switch frequently between the attractors and the chaotic saddle, which is considered as noise-induced chaos.
Specifically, the generalized cell mapping with the digraph method is applied to the Poincaré section of the phase space to delineate its global structure. Combined with the optimal path, it is found that the transition trajectory would pass through the unstable periodic orbit, nonattracting chaotic set, and the saddle point in an orderly sequence, and follows the heteroclinic trajectory converging to another fixed point. During this process, it has been analyzed as to how the stochastic force drives the system to transit in every stage.
The unstable periodic orbit plays the role of the accessible periodic orbit whose two-dimensional stable manifold separates the regular and fractal basins of attraction. Besides, the existence of the chaotic saddle is found to reduce the escaping energy remarkably.
In addition, the stochastic sensitivity analysis method is applied to provide a confidence ellipsoid that is used to elaborate the stationary distribution according to the second-order approximation of the quasi-potential. When this ellipsoid is tangent to the stable manifold of the periodic orbit, a threshold noise intensity is defined to divide noise-induced chaos and intermittency, which agrees well with the numerical simulations. It should be finally remarked that these procedures and methods can be generalized to analyze other three-dimensional systems with similar structures.