Another new chaotic system: bifurcation and chaos control
JJune 27, 2020 6:4 arxiv
International Journal of Bifurcation and Chaosc (cid:13)
World Scientific Publishing Company
Another new chaotic system: bifurcation and chaos control
Arnob Ray and Dibakar Ghosh * Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, India
We propose a new simple three-dimensional continuous autonomous model with two nonlin-ear terms and observe the dynamical behavior with respect to system parameter. This systemchanges the stability of fixed point via Hopf bifurcation and then undergoes a cascade of period-doubling route to chaos. We analytically derive the first Lyapunov coefficient to investigate thenature of Hopf bifurcation and also investigate well-separated regions for different kind of attrac-tors in two-dimensional parameter space. Next, we introduce a time-scale ratio parameter andcalculate the slow manifold using geometric singular perturbation theory. Finally, the chaoticstate is annihilated by decreasing the value of time-scale ratio parameter.
Keywords : Chaos, Hopf-bifurcation, first Lyapunov coefficient, slow-fast dynamics.
1. Introduction
Finding a new chaotic system began from the last six decades to understand different natural and man madesystems. Still scientific communities are investigating new chaotic system due to its practical applicationsin engineering [Szemplinska-Stupnicka & Troger, 1991], finance [Guegan, 2009], plasma [Ray et al. , 2008,2009], time series analysis [Kodba et al. , 2005; Nazarimehr et al. , 2018a] and natural observations like,climatology [Selvam, 2007], biology [May, 1987; Ghosh et al. , 2007], geology [Turcotte, 1997] etc. Afterdiscovery of Lorenz model [Lorenz, 1963] in 1963, there are several chaotic models were investigated like,R¨ossler system [R¨ossler, 1976], Chen system [Chen et al. , 1999], Chua’s attractor [Chua et al. , 1986],Sprott system [Sprott, 1994, 2000], logistic model [May, 1976], predator-prey based model [Hastings &Powel, 1991; Ginoux et al. , 2019] and many more. At first, theoretical background behind generation ofchaotic solution in deterministic system was not developed. But, now different ways are brought to light tounderstand and realize the chaotic attractors and its dynamics, both from analytical as well as numericalapproaches [Eckmann & Ruelle, 1985; Perc , 2005; Silva et al. , 2018]. Initially, three well-known routes werereported for generating chaos in dynamical systems, i . e . , period-doubling, quasi-periodic and intermittentroutes [Ott, 2002]. Nowadays, researchers are fascinated for finding chaotic attractor in a different manner.One interesting topic is to find multistable chaotic attractors [Bayani et al. , 2019] which has importantapplication and impact to explore a broad field of research [Dudkowski et al. , 2016]. Besides a new conceptof attractor arises connected with multistability like attractors with no equilibria [Wei, 2011; Jafari etal. , 2013b; Pham et al. , 2014], line equilibria [Jafari & Sprott, 2013a; Nazarimehr et al. , 2018b], plane ofequilibrium [Jafari et al. , 2016] or coexistence of more than one chaotic attractors [Dudkowski et al. , 2016;Ray et al. , 2009] etc. Basin of attraction of chaotic attractor does not touch with small neighborhood ofthe unstable fixed point, which is quite different from the traditional chaotic attractors (called self-excitedchaotic attractors ). This kind of attractor is familiar as hidden attractor as finding of hidden attractor is ∗ [email protected] 1 a r X i v : . [ n li n . C D ] N ov une 27, 2020 6:4 arxiv A. Ray and D. Ghosh difficult task usually [Dudkowski et al. , 2016]. Researches are also eager to control of the multistability[Pisarchik & Feudel, 2014]. But finding a simple chaotic system is still a challenging task with a great impactfor basic research [Liu et al. , 2004; L¨u et al. , 2002; Wang & Chen, 2012]. We have also knowledge aboutchaotic systems in which different timescales are separated which have huge impact in various fields, likechemical oscillatory model [Epstein, 1983], Hindmarsh-Rose neuronal model [Hindmarsh & Rose, 1984],prey-predator model [Kuwamura et al. , 2009] and so on. Well known features of chaotic attractor aresensitive dependence of initial conditions, boundedness, aperiodic and appearance of it is confirmed byexistence of positive Lyapunov exponent in the parameter region [Strogatz, 2014]. Researchers are alsovery much interested to study the chaotic dynamics in delay systems due to its impactful applicability[Wei et al. , 2019; Khajanchi et al. , 2018; Banerjee et al. , 2019; Ghosh et al. , 2017]. With studying aboutgeneration of chaotic system, parallel researchers have been focused about control of chaos in non-delaychaotic systems [Ott et al. , 1990; Perc & Marhl, 2006; Jun et al. , 2008] as well as delay chaotic systems[Ghosh et al. , 2008; Bhowmick et al. , 2014]. Its application includes chemical reaction oscillations, turbulentfluids etc. [Boccaletti et al. , 2000]. From the above discussion, the finding of a new chaotic system andexploring different dynamical states deserve a special attention.In this paper, we propose a simple nonlinear system which exhibits single-scroll chaotic attractor ina certain parameter region. At first, we calculate the fixed points of the system and analyze their linearstability analyses. From these analyses, we observe that the two fixed points out of three always unstable,whereas one fixed point changes its stability via Hopf bifurcation and finally goes to chaotic state throughperiod-doubling route. We also analytically derive the first Lyapunov coefficient to check the stabilityof the limit cycle. Then, we introduce slow-fast ratio parameter in the proposed model and investigatethe previously mentioned system parameter region by varying this slow-fast parameter. We analyticallycalculate the slow manifold with the help of geometric singular perturbation theory. The chaotic behaviorcompletely annihilate if the time scale ratio parameter is low.The rest of the paper is organized as follows: Section-2 devotes the introduction of a new chaoticsystem and linear stability analysis of the fixed points. We also discuss the bifurcation diagram, Lyapunovexponent and two-dimensional parameter regions by varying the system parameters. In Sec-3, we discussthe effect of slow-fast parameter and controlling of chaos. Finally, conclusions are drawn in Sec-4.
2. Model and system dynamics
We propose a three-dimensional model with two nonlinear turns in the following form,˙ x = − x + y + z, ˙ y = xy − z, ˙ z = − axz + y + b, (1)where a, b are two positive parameters and x, y and z are state variables. Here dot denotes time derivativewith respect to time t . We first study the linear stability analysis of the equilibrium points of the abovesystem. The equilibrium points are in the form ( x ∗ , f ( x ∗ ) , g ( x ∗ )), where f ( x ) and g ( x ) are given by f ( x ) = x x , g ( x ) = x x . Here x ∗ satisfies the following equation ax ∗ − ( b + 1) x ∗ − b = 0 . (2)Solving Eqn. (2), we have three values of x ∗ corresponding to three roots x ∗ , x ∗ and x ∗ , x ∗ = 2 (cid:113) b +13 a cos ( θ ) ,x ∗ = − (cid:113) b +13 a cos ( π − θ ) ,x ∗ = − (cid:113) b +13 a cos ( π + θ ) , whereune 27, 2020 6:4 arxiv Another new chaotic system: bifurcation and chaos control a - b parameter space where region A denotes the existence of three fixed points, whereas one fixed point existsin the region B. (b) Variation of maximum real part of the eigenvalues corresponding to the three fix points by varying theparameter a . Here maximum eigenvalues corresponding to two points x ∗ and x ∗ are always positive (red and magenta curves)and x ∗ is negative up to a < .
53 and change the sign after that which is shown in inset figure. Here one fixed point changesits stability and other two are always unstable. Here b = 0 . θ = tan − (cid:16) b (cid:114) b + 1) − ab a (cid:17) . The above three roots of the Eqn. (2) are real if0 < a < b + 1) b . In Fig. 1(a), we draw a curve a = b +1) b . Left side of the curve in a - b parameter space (denoted by A),system possesses three fixed points and only one fixed point exists at the right side of the curve (denotedby B). Now, we check the qualitative nature of equilibrium point ( x ∗ , f ( x ∗ ) , g ( x ∗ )) of the system (1) bylinear stability analysis. The Jacobian matrix J of the system at the equilibrium point ( x ∗ , f ( x ∗ ) , g ( x ∗ )) isas follows, J = − f ( x ∗ ) x ∗ − − ag ( x ∗ ) 1 − ax ∗ . So, the characteristic equation of the matrix J is λ + a λ + a λ + a = 0 , (3)where a = 1 − x ∗ + ax ∗ ,a = 1 − ax ∗ + ( a − x ∗ + ( a − x ∗ x ∗ ,a = 1 − ax ∗ − ax ∗ x ∗ . Solving Eqn. (3), the three eigenvalues are λ = ( z + z ) − a ,λ = − z + z − a + i √ z − z )2 ,λ = − z + z − a − i √ z − z )2 , (4)une 27, 2020 6:4 arxiv A. Ray and D. Ghosh where z = ( q + ( p + q ) ) , z = ( q − ( p + q ) ) , p = a − a and q = a a − a − a . If p + q >
0, Eqn.(3) gives one real root with a pair of complex conjugate eigenvalues and another case, i.e. , p + q <
0, theequation gives three real roots.Now, we fix the system parameter b at b = 0 .
25 and consider a as a bifurcation parameter. In Fig.1(b), we plot the maximum real part of the eigenvalues of J at x ∗ , x ∗ , and x ∗ with respect to the systemparameter a . From this figure, the maximum real part of the eigenvalue (Re( λ )) of J at third root, i.e., x ∗ shows negative in the range a ∈ (0, 1.53) (blue line) and after that turns to positive (clearly shown in theinset figure of Fig. 1(b)), but for remaining two roots, i.e., x ∗ , x ∗ , their associated maximum eigenvalues(Re( λ , )) are always positive (magenta and red lines). Interestingly, we see that at a = 4 .
63, there areone maximum real part of the eigenvalues corresponding to x ∗ , x ∗ , and x ∗ exists. This means for a > . B region in Fig. 1(a)). So it can be concluded that the Eqn. (1)has only one stable fixed point corresponding to x ∗ in the range a ∈ (0 , . a = 0 . b = 0 .
25, we calculate x ∗ and associated eigenvalues λ , , as follows: x ∗ : x ∗ ≈ . , ( λ , λ , λ ) ≈ (1 . , − . ± . i ); x ∗ : x ∗ ≈ − . , ( λ , λ , λ ) ≈ ( − . , . , − . x ∗ : x ∗ ≈ − . , ( λ , λ , λ ) ≈ ( − . , − . ± . i ) . Fig. 2. (a) Bifurcation diagram of the new system Eqn. (1) for a ∈ [2 . , . a ∈ [2 . , . b = 0 .
25. Initial condition istaken as ( x (0) , y (0) , z (0)) = (0 . , . , . . Using the sign of eigenvalues, the fixed points corresponding to x ∗ , x ∗ , and x ∗ of the system (1) areunstable saddle focus, unstable saddle and stable focus, respectively at a = 0 . b = 0 .
25. So we focuson the fixed point corresponding to x ∗ and study Hopf bifurcation analysis. The characteristic equation atthe point x ∗ becomes, λ + r λ + r λ + r = 0 , (5)where r = 1 + (1 − a )( √ a a ) cos ( θ ) ,r = 1 − ( a − √ a a cos ( θ ) + a − a cos θ + √ a a cos θ − ( √ a a ) cos ( θ ) ,r = 1 − cos ( θ ) + √ a a cos ( θ )1 − ( √ a a ) cos ( θ ) , and, θ = (cid:112) a (125 − a )9 a . une 27, 2020 6:4 arxiv Another new chaotic system: bifurcation and chaos control a : a = 2 .
4, period-1 (magenta); a = 2 .
6, period-2 (black); a = 2 .
75, period-4(blue) and a = 2 .
9, chaotic state (red). From the phase diagrams, period-doubling route to chaos is easily observed. Here b = 0 . According to the Routh-Hurwitz stability criterion, the real parts of all the eigenvalues λ become negativeif r > , r > , and r r > r . Numerically, we calculate the Hopf bifurcation point at a ∗ ≈ . Fig. 4. Basin of attraction of chaotic attractor at (a) z = 0 . y = 0 . x = 0 . x (0) = 0 . , y (0) = 0 . , and z (0) = 0 .
1. Parameter valuesare fixed at a = 2 . b = 0 . Now, we discuss about the nature of Hopf bifurcation [Kuznetsov, 2004]. For this purpose, first we checkthe transversality condition and then calculate the value of first Lyapunov coefficient [Kuznetsov, 2004;une 27, 2020 6:4 arxiv A. Ray and D. Ghosh
Mello & Coelho, 2009; Wei et al. , 2014]. The transversality condition, i.e. , real part of dλ ( a ∗ ) da | λ =1 . i ≈ . (cid:54) = 0) is satisfied here. The first Lyapunov coefficient is l ≈ . >
0) (detail calculations are inappendix). So the fixed point (corresponding to x ∗ ) deals with subcritical Hopf bifurcation at a ∗ ≈ . x ∗ with respect tothe parameter a after crossing the value a = a ∗ . After crossing a ∗ , the fixed point loses its stability anda stable limit cycle of period-1 appears and persists up to a ≈ . a ≈ . et al. , 1983] as the chaotic attractorcollides with unstable fixed point (corresponding to x ∗ ). We take a zoom version of periodic window atthe neighborhood of a = 2 .
845 (shaded region in Fig. 2(a)) in Fig. 2(b). System changes from chaoticoscillation to periodic oscillation through limit point bifurcation of cycles (LPC). Chaotic oscillation againrevives via PD bifurcation and amplifies via interior crisis (IC). Then we plot maximum Lyapunov exponent(LE max ) in Fig. 2(c) for verifying our result in Fig. 2(a). Positive value of LE max after a certain value of a signifies the presence of chaos. For four different values of a , (i.e., a = 2 . , . , .
75 and 2 . i.e. , limit cycle of period-1 (magenta), period-2 (black), period-4 (blue),and chaos (red), respectively. The occurrence of period-doubling cascade for generating chaos is verifiedby calculating Feigenbaum universal constant [Strogatz, 2014]. The ratio of distances between successiveintervals between the bifurcate values of the system parameter, represented by a constant factor δ = lim k →∞ a k +1 − a k a k +2 − a k +1 approaches to 4.66992.... The value of Feigenbaum constant for our case are given in the following table:k Period Bifurcation parameter a k Ratio a k +1 − a k a k +2 − a k +1 z = 0 . y = 0 . x = 0 . a, b ) = (2 . , .
25) are depictedin Figs. 4(a-c). We get fully connected basin of attractions for respective three cases. At the same set ofparameter value, we also plot first return map, i.e. , x n vs. x n +1 in Fig. 4(d), where x n denotes the n -thlocal maxima of the time series. Unimodal nature of this first return map indicates the PD bifurcationroute to generate chaos.To explore the complete study by varying the two parameters a and b simultaneously, we plot a twoparameter ( a - b ) bifurcation diagram [Ghosh & Roy, 2015] for the range a ∈ [0 ,
5] and b ∈ [0 . , .
5] in Fig.5. This demonstrates the whole dynamics of the system (1). In this phase diagram, cyan, magenta, black,blue, and red regions represent steady state, limit cycle of period-1, period-2, period-4, and greater thanperiod-4 (or chaotic), respectively. White portion represents unbounded solution for higher values of a and b . Interestingly, well-separated parameter space is observed in ( a - b ) bifurcation diagram.
3. Slow-fast dynamical system and controlling chaos
In this section, we introduce a slow-fast time scale ratio parameter to create slow-fast dynamics in theproposed system. We try to understand the effect of this time scale parameter. A general framework of aune 27, 2020 6:4 arxiv
Another new chaotic system: bifurcation and chaos control a - b parameter space. Here Cyan: stable steady state, magenta: period-1, black:period-2, blue: period-4, and red: greater than period-4 or chaotic state. White region is for unbounded solution. The initialcondition is x (0) = 0 . , y (0) = 0 .
2, and z (0) = 0 . slow-fast dynamical system [Guckenheimer, 2008] can be written as, (cid:15) dxdt = f ( x, y, η ); dydt = g ( x, y, η ) , where x ∈ R m and y ∈ R n are fast and slow variables, respectively, η ∈ R p is a model parameter, 0 < (cid:15) < (cid:15) =0, the trajectory of above equation converges to the solution ofdifferential algebraic equation f ( x, y, η ) = 0 and ˙ y = g ( x, y, η ), where S = { ( x, y ) ∈ R m × R n | f ( x, y, η ) = 0 } is a critical manifold.Now, in our model we introduce a time-scale ratio parameter (cid:15) (0 < (cid:15) <
1) in the Eqn. (1) as follows (cid:15) ˙ x = − x + y + z, ˙ y = xy − z, ˙ z = − axz + y + b, (6)so that x is fast variable and y, z are slow variables. We rescale the above Eqn. (6) by setting τ = t/(cid:15) as x (cid:48) = − x + y + z,y (cid:48) = (cid:15) ( xy − z ) ,z (cid:48) = (cid:15) ( − axz + y + b ) , (7)where x (cid:48) ≡ dxdτ . By setting (cid:15) = 0, the trajectory of Eqn. (6) converges to the solution of the differentialalgebraic equation 0 = − x + y + z, ˙ y = xy − z, ˙ z = − axz + y + b. (8)So, as per above definition, S = { ( x, y, z ) ∈ R : x = y + z } is called a critical manifold. From Fenicheltheory [Fenichel, 1979], we know that a subset S (cid:15) of S exists for (cid:15) ( >
0) if S (cid:15) is normally hyperbolic.Our system is normally hyperbolic because of real part of ∂ ( − x + y + z ) ∂x = − (cid:54) = 0) , ( x, y, z ) ∈ S (cid:15) . So thereexists a local invariant manifold (slow manifold) S (cid:15) and we derive this equation by using geometric singularperturbation method [Fenichel, 1979; Guckenheimer, 2008; Desroches et al. , 2012].In S , relation among variables x, y and z is x = y + z . Let in S (cid:15) , this relation becomes x = y + z + O ( (cid:15) ) . (9)une 27, 2020 6:4 arxiv A. Ray and D. Ghosh
Fig. 6. Plot of the critical manifold S and slow manifold S (cid:15) at (cid:15) = 0 .
85 are illustrated by green and red colors, respectively.Chaotic attractor (blue line) is drawn for a = 3 .
17 and b = 0 .
26. The position of the unstable fixed point (solid dot):( − . , − . , . By Taylor series expansion in (cid:15) , we get x = y + z + (cid:15)H ( y, z ) + O ( (cid:15) ) . (10)Differentiate Eqn. (9) and then multiply both sides by (cid:15) , we have, (cid:15) ˙ x = (cid:15) ˙ y + (cid:15) ˙ z + O ( (cid:15) ) . (11)Using Eqns. (6) and (9) in Eqn. (11), we get (cid:15) ˙ x = (cid:15) (cid:16) y − az + (1 − a ) yz + y − z + b (cid:17) + O ( (cid:15) ) . (12)Again from first equation of Eqn. (6), (cid:15) ˙ x = − x + y + z. Now using Eqn. (10) in above equation, we get (cid:15) ˙ x = − ( (cid:15)H ( y, z ) + O ( (cid:15) )) . (13)Comparing Eqns. (12) and (13), finally we obtain H ( y, z ) = − (cid:16) y − az + (1 − a ) yz + y − z + b (cid:17) . (14)We use the above value of H ( y, z ) in Eqn. (10) to calculate the approximated slow manifold S (cid:15) . Weplot the critical manifold S (in green color), an approximated slow manifold S (cid:15) (in red color) in Fig. 6.The chaotic attractor for the set of values a = 3 . , b = 0 .
26 and (cid:15) = 0 .
85 is shown by blue line. Thisattractor lies on the both sides of the manifold S and S (cid:15) . We take the value (cid:15) = 0 .
85 for S (cid:15) (though thisvalue is not so small enough, but we choose this value due to the appearance of chaos).Next we study the effect of time-scale parameter (cid:15) of the system (6) in ( a - b ) parameter space for thepreviously mentioned range. For this, we first set (cid:15) = 0 . a - b ) parameter space compare to Fig. 5 for (cid:15) = 1 . (cid:15) , then the chaotic region is completely annihilated in the ( a - b ) parameterspace. Figure 7(b) shows that the region of chaotic state is completely annihilated whereas region of stablefixed point is expanded and rest parts of the region are filled by period-1 limit cycle for (cid:15) = 0 .
1. So byintroducing the slow-fast parameter (cid:15) in our original system (1), we can control chaos. Now we take two setsof parameter values of { a, b } , i.e., { . , . } and { . , . } from the Figs. 7(a, b), we have illustratedune 27, 2020 6:4 arxiv Another new chaotic system: bifurcation and chaos control a - b parameter spaces for (a) (cid:15) = 0 .
9, and (b) (cid:15) = 0 .
1. Here, Cyan:stable steady state, magenta: period-1, black: period-2, blue: period-4, and red: greater than period-4 or chaotic state. Whiteregion is for unbounded solution. Lower panel: bifurcation diagrams with respect to the time-scale ratio parameter (cid:15) for (c) { a, b } = { . , . } and (d) { a, b } = { . , . } . The initial condition is x (0) = 0 . , y (0) = 0 .
2, and z (0) = 0 .
1. From thisfigure, it is noticed that chaotic state is annihilated by decreasing the value of time-scale parameter (cid:15) and stable steady stateand periodic solutions dominate the whole parameter region. two bifurcation diagrams by varying slow-fast ratio parameter (cid:15) ∈ [0 ,
1] for understanding the variouschange of dynamics in slow-fast system in Figs. 7(c, d), respectively. For the first set of parameter values,the period-doubling route to chaos is observed, whereas for second case, the chaotic behavior in the systemis terminated and two states (stable fixed point and period-1 limit cycle) are dominated.
4. Conclusions
To summarize, in this paper, we have proposed a new simple chaotic system with two nonlinear terms.We have elaborately discussed the nature of fixed points through linear stability analysis. We have alsoanalytically derived the first Lyapunov coefficient to show the nature of Hopf bifurcation by which fixedpoint changes its stability. Using bifurcation diagram, we have investigated the period-doubling route tochaos and also verified the results using Lyapunov exponent and first return map. We also separate theregion of different dynamical behavior in the parameter space. Finally, we introduce slow-fastness in ourmodel and try to understand the regarding changes of dynamics. Variation of dynamics has been capturedin ( a - b ) parameter space for different values of the slow-fast parameter (cid:15) . Most interesting fact is that wecan control the chaotic and higher periodic orbits by adjusting the value of (cid:15) . Acknowledgments:
The authors would like to thank Syamal K. Dana, Chittaranjan Hens, GourabKumar Sar, Sayeed Anwar, Soumen Majhi, Srilena Kundu, Sayantan Nag Chowdhury, Subrata Ghosh andSarbendu Rakshit for helpful discussions and comments.une 27, 2020 6:4 arxiv A. Ray and D. Ghosh
Appendix: Calculation of first Lyapunov coefficient
Let us consider the differential equation ˙ x = f ( x, α ) , (15)where x ∈ R and α ∈ R represent the state variables and system parameter respectively, and f : R × R → R is a smooth function. Let the Eqn. (15) has a fixed point x = x at α = α . Now, replacing x − x by X and Eqn. (15) becomes ˙ X = F ( X ) = f ( X, α ) . (16)Here, F ( X ) is also a smooth function and we can expand it in Taylor series in terms of symmetric functionsas follows, F ( X ) = AX + B ( X, X ) + C ( X, X, X ) + O ( || X || ) , (17)where A = f x (0 , α ) is the Jacobian matrix and for i=1,2,3, B i ( X, Y ) = (cid:80) j,k =1 ∂ F i ( ξ ) ∂ξ j ∂ξ k | ξ =0 X j X k ,C i ( X, Y, Z ) = (cid:80) j,k,l =1 ∂ F i ( ξ ) ∂ξ j ∂ξ k ∂ξ l | ξ =0 X j X k X l . (18)In our case, multilinear symmetric functions are B ( X, Y ) = (0 , x y + x y , − a ∗ ( x y + x y )) ,C ( X, Y, Z ) = (0 , , . (19)We assume that A has a pair of complex eigenvalues λ , = ± iw ( w >
0) on the imaginary axisat ( x , α ). Let generalized eigen space of A be T C which is the largest invariant subspace spanned byeigenvectors corresponding to λ , . Let p, q ∈ C be such eigen vectors then Aq = iw q,A T p = iw p,< p, q > = (cid:80) i =1 ¯ p i q i = 1 , (20)where A T denotes the transpose of the matrix A and < ... > is the standard scalar product over C . Forour system (1), we obtain q ≈ ( − . − . i, . , − . − . i ) ,p ≈ ( − . − . i, . − . i, . − . i ) , where w ≈ . y ∈ T C which is represented as y = wq + ¯ w ¯ q , where w = < p, q > ∈ C . Thetwo-dimensional center manifold associated with the eigenvalues λ , = ± iw can be parameterized by w and ¯ w in the form X = H ( w, ¯ w ). Here H : C → R can be expanded as follows H ( w, ¯ w ) = wq + ¯ w ¯ q + (cid:80) ≤ j + k ≤ j ! k ! h jk w j ¯ w k + O ( | w | ) , (21)where h jk ∈ C and h jk = ¯ h kj . Differentiating H ( w, ¯ w ) with respect to t and substituting this into Eqn.(15) and using Eqn. (16), we get the following differential equation H w ˙ w + H ¯ w ˙¯ w = F ( H ( w, ¯ w )) . (22)Using Eqns. (16), (21) and (22), we determine the complex coefficients h ij and from the Eqn. (22), the w evolves on the center manifold, ˙ w = iw w + G w | w | + O ( | w | ) , (23)where G ∈ C . Substituting Eqn. (23) into Eqn. (22) and using Eqns.˜(17) and (18), we get the followingcoefficients, h = − A − B ( q, ¯ q ) ,h = (2 iw I − A ) − B ( q, q ) , (24)une 27, 2020 6:4 arxiv REFERENCES where I is the 3 × G is obtained from the condition of existence of solutionof the equation for h . So, G = < p, C ( q, q, ¯ q ) + B (¯ q, h ) + 2 B ( q, h ) >, where we use < p, q > = 1. So the first Lyapunov coefficient is defined as l = 12 Re ( G ) , which determines the nonlinear stability of a nondegenerate codimension one Hopf bifurcation. Now, forcalculating l , we first find h ≈ (0 . , . , − . ,h ≈ ( − . . i, − . − . i, − . − . i ) . using Eqns. (18), (24) and (24). Finally,we get l ≈ . > References
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