sol=NDSolve[{x''[t]==-x[t] - 
                  0.1 (4 x[t]^3 y[t]^2 - 2 x[t] y[t]^4),y''[t]==-y[t] + 
                  0.1 (2 x[t]^4 y[t] - 4 x[t]^2 y[t]^3),x'[0]==1,y'[0]==0,x[0] == 0,y[0] == 1},{x, y},{t, 70}]

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 70}, \ 
      PlotRange -> All]\)

sol=NDSolve[{x''[t]==-x[t] - 
                  0.1 (4 x[t]^3 y[t]^2 - 2 x[t] y[t]^4),y''[t]==-y[t] + 
                  0.1 (2 x[t]^4 y[t] - 4 x[t]^2 y[t]^3),x'[0]==1,y'[0]==0,x[0] == 0,y[0] == 1},{x, y},{t, 10}]

x[0] /. Out[33]

x[0]

x[0] /. sol

x'[0]^2  - y'[0]^2  /. sol

{1/2} (x'[0]^2  - y'[0]^2)  + {1/2} 
          (x[0]^2 - y[0]^2) + 
        0.1 ( - x[0]^2 y[0]^4  +  x[0]^4 y[0]^2)  /. sol

{1/2} (x'[1]^2  - y'[1]^2)  + {1/2} 
          (x[1]^2 - y[1]^2) + 
        0.1 ( - x[1]^2 y[1]^4  +  x[1]^4 y[1]^2)  /. sol

{1/2} (x'[10]^2  - y'[10]^2)  + {1/2} 
          (x[10]^2 - y[10]^2) + 
        0.1 ( - x[10]^2 y[10]^4  +  x[10]^4 y[10]^2)  /. sol

t = 6

{1/2} (x'[t]^2  - y'[t]^2)  + {1/2} 
          (x[t]^2 - y[t]^2) + 
        0.1 ( - x[t]^2 y[t]^4  +  x[t]^4 y[t]^2)  /. sol

H = {1/2} (x'[t]^2  - y'[t]^2)  + {1/2} 
            (x[t]^2 - y[t]^2) + 
          0.1 ( - x[t]^2 y[t]^4  +  x[t]^4 y[t]^2)  /. sol

t = 60

H =.

H

t =.

sol=NDSolve[{x''[t]==-x[t] - 
                  0.1 (4 x[t]^3 y[t]^2 - 2 x[t] y[t]^4),y''[t]==-y[t] + 
                  0.1 (2 x[t]^4 y[t] - 4 x[t]^2 y[t]^3),x'[0]==0.5,y'[0]==0,x[0] == 0,y[0] == 1},{x, y},{t, 70}]

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 10}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 70}, \ 
      PlotRange -> All]\)

t = 0

H = {1/2} (x'[t]^2  - y'[t]^2)  + {1/2} 
            (x[t]^2 - y[t]^2) + 
          0.1 ( - x[t]^2 y[t]^4  +  x[t]^4 y[t]^2)  /. sol

{{-0.375`} }   This is energy

t = 10

H = {1/2} (x'[t]^2  - y'[t]^2)  + {1/2} 
            (x[t]^2 - y[t]^2) + 
          0.1 ( - x[t]^2 y[t]^4  +  x[t]^4 y[t]^2)  /. sol

{{-0.374997583714709081`}}

t = 70

H = {1/2} (x'[t]^2  - y'[t]^2)  + {1/2} 
            (x[t]^2 - y[t]^2) + 
          0.1 ( - x[t]^2 y[t]^4  +  x[t]^4 y[t]^2)  /. sol

-0.374980136245943773`

V = {1/2} (x^2  -  y^2) +  
        k ( - x^2 y^4  +  x^4 y^2)

Plot3D[
      (1/2) ((x^2) - (y^2)) + 0.1 ((x^4) (y^2) - (x^2) (y^4)), {x, 
        -2, 2}, {y, -2, 2}]

sol=NDSolve[{x''[t]==-x[t] - 
                  0.1 (4 x[t]^3 y[t]^2 - 2 x[t] y[t]^4),y''[t]==-y[t] + 
                  0.1 (2 x[t]^4 y[t] - 4 x[t]^2 y[t]^3),x'[0]==0.5,y'[0]==0,x[0] == 0,y[0] == 1},{x, y},{t, 200}]

t =.

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 120}, \ 
      PlotRange -> All]\)

t = 120

H = {1/2} (x'[t]^2  - y'[t]^2)  + {1/2} 
            (x[t]^2 - y[t]^2) + 
          0.1 ( - x[t]^2 y[t]^4  +  x[t]^4 y[t]^2)  /. sol

x'[120] /. sol

y'[120] /. sol

x[120] /. sol

y[120] /. sol

sol=NDSolve[{x''[t]==-x[t] - 
                  0.1 (4 x[t]^3 y[t]^2 - 2 x[t] y[t]^4),y''[t]==-y[t] + 
                  0.1 (2 x[t]^4 y[t] - 4 x[t]^2 y[t]^3),x'[120]==0.0233369307654103286`  ,y'[120]== -0.954547115960078329` ,x[120] ==  -0.457371414236554763` ,y[120] ==  0.220972056664788096`},{x, y},{t, 120, 200}]

t =.

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 120, 200}, \ 
      PlotRange -> All]\)

t = 200

H = {1/2} (x'[t]^2  - y'[t]^2)  + {1/2} 
            (x[t]^2 - y[t]^2) + 
          0.1 ( - x[t]^2 y[t]^4  +  x[t]^4 y[t]^2)  /. sol

t =.

sol=NDSolve[{x''[t]==-x[t] - 
                  0.1 (4 x[t]^3 y[t]^2 - 2 x[t] y[t]^4),y''[t]==-y[t] + 
                  0.1 (2 x[t]^4 y[t] - 4 x[t]^2 y[t]^3),x'[120]==0.0233369307654103286`  ,y'[120]== -0.954547115960078329` ,x[120] ==  -0.457371414236554763` ,y[120] ==  0.220972056664788096`},{x, y},{t, 120, 300}]

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 120, 220}, \ 
      PlotRange -> All]\)

sol=NDSolve[{x''[t]==-x[t] - 
                  0.1 (4 x[t]^3 y[t]^2 - 2 x[t] y[t]^4),y''[t]==-y[t] + 
                  0.1 (2 x[t]^4 y[t] - 4 x[t]^2 y[t]^3),x'[0]==0,y'[0]==0.5,x[0] == 1,y[0] == 0},{x, y},{t, 200}]

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 120}, \ 
      PlotRange -> All]\)

t = 0

H = {1/2} (x'[t]^2  - y'[t]^2)  + {1/2} 
            (x[t]^2 - y[t]^2) + 
          0.1 ( - x[t]^2 y[t]^4  +  x[t]^4 y[t]^2)  /. sol

t = 100

H = {1/2} (x'[t]^2  - y'[t]^2)  + {1/2} 
            (x[t]^2 - y[t]^2) + 
          0.1 ( - x[t]^2 y[t]^4  +  x[t]^4 y[t]^2)  /. sol

x'[120] /. sol

y'[120] /. sol

x[120] /. sol

y[120] /. sol

sol=NDSolve[{x''[t]==-x[t] - 
                  0.1 (4 x[t]^3 y[t]^2 - 2 x[t] y[t]^4),y''[t]==-y[t] + 
                  0.1 (2 x[t]^4 y[t] - 4 x[t]^2 y[t]^3),x'[120]==-0.954547115960078329` ,y'[120]== 0.0233369307654103286` ,x[120] ==  0.220972056664788096`,y[120] ==  -0.457371414236554763`},{x, y},{t, 120, 220}]

t =.

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 120, 220}, \ 
      PlotRange -> All]\)

Plot3D[
      (1/2) ((x^2) - (y^2)) + 0.1 ((x^4) (y^2) - (x^2) (y^4)), {x, 
        -2, 2}, {y, -2, 2}]

Plot3D[
      (1/2) ((x^2) - (y^2)) + 0.1 ((x^4) (y^2) - (x^2) (y^4)), {x, 
        -4, 4}, {y, -4, 4}]

Plot3D[
      (1/2) ((x^2) - (y^2)) + 0.1 ((x^4) (y^2) - (x^2) (y^4)), {x, 
        -10, 10}, {y, -10, 10}]